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This article is cited in 1 scientific paper (total in 1 paper) Entropy of a unitary operator on K. A. Afonin, D. V. Treschev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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     § 1. IntroductionLet $\mathcal X$ be a nonempty set and $\mathcal B$ be a $\sigma$-algebra of subsets $X\subset\mathcal X$. Consider the measure space $(\mathcal X,\mathcal B,\mu)$, where $\mu$ is a probability measure: $\mu(\mathcal X)=1$. Consider the Hilbert space $\mathcal H=L^2(\mathcal X,\mu)$ with standard inner product and standard norm:
$$
\begin{equation*}
\langle f,g\rangle=\int_\mathcal X f\overline g \, d\mu, \qquad \|f\|=\sqrt{\langle f,f\rangle}.
\end{equation*}
\notag
$$
Let $\mathcal{L}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Let $\|W\|$ denote the operator norm of $W\in\mathcal{L}(\mathcal{H})$: For each function $g\in L^\infty(\mathcal X,\mu)$ let $\widehat g$ denote the operator of multiplication by $g$, that is,
$$
\begin{equation}
\widehat g \colon \mathcal H\to\mathcal H, \qquad \mathcal H\ni f \mapsto \widehat g f=g \cdot f.
\end{equation}
\tag{1.1}
$$
In particular, if $X\in\mathcal B$, then $\widehat{\bf 1}_X$ is the operator of multiplication by the indicator ${\bf 1}_X$ of $X$. We say that a system of sets $\chi=\{Y_1,\dots,Y_J\}$ is a (finite, measurable) partition (of $\mathcal X$) if
$$
\begin{equation*}
\begin{gathered} \, Y_j\in\mathcal B, \qquad\mu\biggl(\mathcal X\setminus \bigcup_{1\leqslant j\leqslant J} Y_j\biggr)=0, \qquad \mu(Y_j\cap Y_k)=0 \\ \text{for all }\ j,k\in\{1,\dots,J\}, \qquad k\ne j. \end{gathered}
\end{equation*}
\notag
$$
We call $\kappa=\{X_1,\dots,X_K\}$ a subpartition of the partition $\chi=\{Y_1,\dots,Y_J\}$ if for each $k\in\{1,\dots,K\}$ there exists $j\in\{1,\dots,J\}$ such that $\mu(X_k\setminus Y_j)=0$. For two arbitrary subpartitions $\chi=\{Y_1,\dots,Y_J\}$ and $\kappa=\{X_1,\dots,X_K\}$ set
$$
\begin{equation*}
\chi\vee\kappa=\{Y_j\cap X_k\}_{j=1,\dots,J,\, k=1,\dots,K}.
\end{equation*}
\notag
$$
Clearly, $\chi\vee\kappa$ is also a partition. Let $W$ be a bounded operator on $\mathcal H$. For each partition $\chi=\{Y_1,\dots,Y_J\}$ set
$$
\begin{equation}
\mathcal M_\chi(W)=\sum_{j=1}^J \mu(Y_j) \| W \widehat{\bf 1}_{Y_j} \|^2.
\end{equation}
\tag{1.2}
$$
The following definition of the $\mu$-norm1 was introduced in [28]: Recall that an operator $U$ is called an isometry (an isometric operator) if it preserves the inner product:
$$
\begin{equation*}
\langle f,g\rangle=\langle Uf,Ug\rangle, \qquad f,g\in\mathcal H.
\end{equation*}
\notag
$$
An operator $U$ is unitary if it is isometric and invertible. If $W\in\mathcal L(\mathcal H)$, $Y\in\mathcal B$ and $U$ is an isometric operator, then
$$
\begin{equation*}
\|W \widehat{\bf 1}_Y\| \leqslant \|W\|, \qquad \|UW\|=\|W\|, \quad \| \widehat{\bf 1}_Y\|=1 \quad(\text{provided that }\ \mu(Y)>0).
\end{equation*}
\notag
$$
As consequences, we obtain some obvious properties of the $\mu$-norm:
$$
\begin{equation}
\|\operatorname{id}\|_\mu=1, \qquad \|W\|_\mu\leqslant\|W\|,
\end{equation}
\tag{1.4}
$$
$$
\begin{equation}
\|\lambda W\|_\mu=|\lambda|\,\|W\|_\mu \quad\text{for each }\ \lambda\in\mathbb{C},
\end{equation}
\tag{1.6}
$$
$$
\begin{equation}
\|W\|_\mu=\|UW\|_\mu \quad \text{for each isometry }\ U.
\end{equation}
\tag{1.7}
$$
The $\mu$-norm has been introduced to extend the concept of (Kolmogorov-Sinai) metric entropy to the case of quantum systems. In connection with the definition of the $\mu$-norm note that in the case of finite sets $\mathcal X$ with measure $\mu$ that is uniformly distributed over the points, several authors (see [7] and [16]–[18]) have addressed the question of how small the quantity $\|W\widehat{\bf 1}_Y\|/\|W\|$ can be for various subsets $Y$ of $ \mathcal X$. It has usually been assumed that $W$ acts from $L^2(\mathcal X,\mu)$ to a finite-dimensional space of significantly lower dimension. Let $F\colon\mathcal X\to\mathcal X$ be an endomorphism (measure-preserving transformation) of the probability space $(\mathcal X,\mathcal B,\mu)$. This means that for each $X\in\mathcal B$ the set $F^{-1}(X)$ (full preimage of $X$) also belongs to $\mathcal B$ and $\mu(X)=\mu(F^{-1}(X))$. An invertible endomorphism is called an automorphism. Let $\operatorname{End}(\mathcal X)$ denote the semigroup of endomorphisms of $(\mathcal X,\mathcal B,\mu)$. There are two standard constructions connected with an endomorphism $F$. 1) Each endomorphism $F$ gives rise to an isometric (even unitary when $f$ is an automorphism) operator $U_F$ on $\mathcal H$ (the Koopman operator):
$$
\begin{equation*}
L^2(\mathcal X,\mu)\ni f \mapsto U_F f=f\circ F, \qquad U_F=\operatorname{Koop}(F).
\end{equation*}
\notag
$$
2) To each $F\in\operatorname{End}(\mathcal X,\mu)$ we can assign a nonnegative (but maybe equal to $+\infty$) number $h_{\mu}(F)$ called the metric entropy (or simply entropy); see [19], § 4.3. One can ask the following. How can one define (in some ‘natural way’) a nonnegative function $\mathfrak h$ on the isometry semigroup $\operatorname{Iso}(\mathcal H)$ with values on the extended real axis so that the diagram be commutative?Recall the construction of the entropy of an endomorphism $F$. Let $J_N$ be the set of multi-indices $j=(j_0,\dots,j_N)$ with components $j_n$ from the set $\{0,\dots,K\}$. For each partition $\chi=\{X_0,\dots,X_K\}$ put
$$
\begin{equation*}
\mathbf X_j=F^{-N}(X_{j_N})\cap \dots \cap F^{-1}(X_{j_1}) \cap X_{j_0},
\end{equation*}
\notag
$$
where $j=(j_0,\dots,j_N)\in J_N$. Let
$$
\begin{equation*}
h_{\mu}(F,\chi,N+1) =- \sum_{j\in J_N} \mu(\mathbf X_j) \log \mu(\mathbf X_j).
\end{equation*}
\notag
$$
Regarded as a function of the last argument, $h_\mu$ is subadditive: $h_\mu(F,\chi,n+m) \leqslant h_\mu(F,\chi,n) + h_\mu(F,\chi,m)$. Hence there exists a limit
$$
\begin{equation*}
h_\mu(F,\chi)=\lim_{n\to\infty} \frac1n h_\mu(F,\chi,n).
\end{equation*}
\notag
$$
Now, the entropy is defined by The idea is to define the entropy of a unitary operator $U$ similarly to the entropy of an endomorphism, with the following modifications. In place of the sets $\mathbf X_j$ we consider the operators
$$
\begin{equation*}
\mathfrak X_j = \widehat{\bf 1}_{X_{j_N}} U \widehat{\bf 1}_{X_{j_{N-1}}} U \dotsb U \widehat{\bf 1}_{X_{j_1}} U \widehat{\bf 1}_{X_{j_0}}
\end{equation*}
\notag
$$
and set
$$
\begin{equation*}
\mathfrak h(U,\chi,N+1) =- \sum_{j\in J_N} \|\mathfrak X_j\|_\mu^2 \log \|\mathfrak X_j\|_\mu^2.
\end{equation*}
\notag
$$
Next, as in the definition of the Kolmogorov-Sinai entropy, we set
$$
\begin{equation*}
\mathfrak h(U,\chi)=\lim_{n\to\infty} \frac1n \mathfrak h(U,\chi,n)\quad\text{and} \quad \mathfrak h(U)=\sup_\chi \mathfrak h(U,\chi).
\end{equation*}
\notag
$$
It was proved in [28] that for each automorphism $F$ we have Before we present the main results of this paper and the papers [28] and [29] devoted to the $\mu$-norm and its applications, we make some observations concerning the literature. Several attempts to transfer the concept of metric entropy to the quantum case have been made (see [8], [22], [23], [2], [3], [21] and [13]). Different authors used different approaches in different contexts. Some of these approaches were compared in [1]. In [2], [26], [24], [4] and [20] the authors considered the finite-dimensional case ($\#\mathcal X<\infty$). Our construction uses some ideas underlying the metric entropy of a doubly stochastic operator (see [9] and [10]). § 2. Previous resultsIn this section we present the main results of [28]. 1. $\| \widehat{\bf 1}_X\|_\mu^2=\mu(X)$ for each $X\in\mathcal B$. 2. If $\chi'$ is a subpartition of a partition $\chi$, then $\mathcal M_{\chi'}(W) \leqslant \mathcal M_\chi(W)$. Hence the quantities $\mathcal M_\chi(W)$ approximate the infimum in (1.3) for fine partitions. 3. Given two arbitrary bounded operators $W_1$ and $W_2$, we have the triangle inequality
$$
\begin{equation*}
\|W_1 + W_2\|_\mu \leqslant \|W_1\|_\mu + \|W_2\|_\mu.
\end{equation*}
\notag
$$
In combination with (1.6) it means that $\|\,{\cdot}\,\|_\mu$ is a seminorm on $\mathcal L(\mathcal H)$. 4. If $F$ is an automorphism of the space $(\mathcal X,\mathcal B,\mu)$ and $U_F=\operatorname{Koop}(F)$, then
$$
\begin{equation}
U_F \widehat{\bf 1}_X= \widehat{\bf 1}_{F^{-1}(X)} U_F \quad\text{for each }\ X\in\mathcal B.
\end{equation}
\tag{2.1}
$$
For all $W\in\mathcal L(\mathcal H)$ Hence $\|U_F^{-1} W U_F\|_\mu=\|W\|_\mu$. 5. $\|\,{\cdot}\,\|_\mu$ is a continuous function on $\mathcal L(\mathcal H)$ with respect to the operator norm $\|\,{\cdot}\,\|$. 6. If $\mu$ is atomless, then $\|W+W_0\|_\mu=\|W\|_\mu$ for a bounded operator $W$ and a compact operator $W_0$. In particular, the $\mu$-norm vanishes identically on the space of compact operators. 7. If $g\in L^\infty(\mathcal X,\mu)$, then $\|\widehat g\|_\mu=\|g\|$. 8. Let $\mathcal X$ be a finite set: $\mathcal X=\{1,\dots,J\}$, and assume that each point has measure $1/J$. Then $\mathcal H$ is isomorphic to the space $\mathbb{C}^J$ with the inner product $\langle f,g\rangle_J=J^{-1}\sum_{j=1}^J f(j)\overline{g(j)}$. Consider an arbitrary operator $W$ on $\mathcal H$:
$$
\begin{equation*}
f\mapsto Wf, \qquad (Wf)(k)=\sum_{j=1}^J W(k,j) f(j).
\end{equation*}
\notag
$$
Then 9. For each partition $\{X_1,\dots,X_K\}$ of the space $\mathcal X$ the function $\|\,{\cdot}\,\|_\mu^2$ is right additive:
$$
\begin{equation}
\|W\|_\mu^2=\sum_{k=1}^K \|W \widehat{\bf 1}_{X_k}\|_\mu^2,
\end{equation}
\tag{2.3}
$$
and left subadditive:
$$
\begin{equation*}
\|W\|_\mu^2\leqslant \sum_{k=1}^K \| \widehat{\bf 1}_{X_k} W\|_\mu^2.
\end{equation*}
\notag
$$
10. Let $\mathcal X$ be a compact metric space and $\mu$ be a Borel measure. Let $B_r(x)\subset\mathcal X$ denote the open ball with centre $x$ and radius $r$. Then for each $x\in\mathcal X$ the finite limit
$$
\begin{equation*}
\vartheta(x)=\lim_{\varepsilon\searrow 0} \|W \widehat{\bf 1}_{B_\varepsilon(x)}\|^2
\end{equation*}
\notag
$$
exists, where the function $\vartheta$ is measurable (and even Borel) and $\displaystyle\|W\|_\mu^2 \leqslant \smash[t]{\int_\mathcal X \vartheta \,d\mu}$. There is an example showing that this inequality is strict in general. However, if the following two conditions are satisfied.
§ 3. Main resultsConsider the probability space $(\mathbb{T}^n,\mathcal B(\mathbb{T}^n),\mu)$, where $\mathbb{T}^n{=}\,\mathbb{R}^n/(2\pi\mathbb{Z}^n)$ is the $n$-torus, $n\in\mathbb{N}$, $\mathcal{B}(\mathbb{T}^n)$ is the Borel $\sigma$-algebra of subsets of $\mathbb{T}^n$ and $\mu$ is the normalized Lebesgue measure on $(\mathbb{T}^n,\mathcal B(\mathbb{T}^n))$ ($d\mu=(2\pi)^{-n}dx$). For $n=1$ the results below were mostly established in [28] and [29]. $\bullet$ In § 4 we introduce the Banach space $\mathcal{AC}(\mathbb{T}^n)$ of continuous complex functions on $\mathbb{T}^n$ whose Fourier series are absolutely convergent. With each functional ${\lambda\in(\mathcal{AC}(\mathbb{T}^n))^*}$ we associate the convolution operator $\operatorname{Conv}_\lambda$ acting on $L^2(\mathbb{T}^n)$ by
$$
\begin{equation*}
L^2(\mathbb{T}^n)\ni f\mapsto \operatorname{Conv}_\lambda f=g, \qquad g=\sum_{k\in\mathbb{Z}^n}g_ke^{i(k,\,\cdot\,)}, \quad g_k=f_k\lambda_k,
\end{equation*}
\notag
$$
where $\lambda_k=\lambda(e^{-i(k,\,\cdot\,)})$, $k\in\mathbb{Z}^n$. The operator $\operatorname{Conv}_\lambda$ is bounded, and we have $\|\operatorname{Conv}_\lambda\|=\|\lambda\|_{\mathcal{AC}^*}=\sup_{k\in\mathbb{Z}^n}|\lambda_k|$. We show (Proposition 4.1) that
$$
\begin{equation*}
\|\operatorname{Conv}_\lambda\|^2_{\mu}=\rho(\lambda), \qquad \rho(\lambda)=\limsup_{\mathbf{I}}\frac{1}{\#\mathbf{I}}\sum_{k\in \mathbf{I}}|\lambda_k|^2,
\end{equation*}
\notag
$$
where $\mathbf{I}$ under the upper limit sign ranges over the directed set $(\mathcal P^n,\leqslant)$ (see § 7.2) of integer parallelepipeds $\mathcal P^n$. $\bullet$ In § 5 we introduce the special class $\mathcal{DT}(\mathbb{T}^n)$ of all bounded operators $W$ on $L^2(\mathbb{T}^n)$ such that
$$
\begin{equation*}
\|W\|_{\mathcal{DT}}:=\sum_{k\in\mathbb{Z}^n}\sup_{j\in\mathbb{Z}^n}|W_{k+j,j}|<\infty,
\end{equation*}
\notag
$$
where $W_{j,k}=\langle We^{i(k,\,\cdot\,)},e^{i(j,\,\cdot\,)}\rangle$, $j,k\in\mathbb{Z}^n$. Elements $W\in\mathcal{DT}(\mathbb{T}^n)$ are called operators of diagonal type. We show (Theorem 5.1) that $(\mathcal{DT}(\mathbb{T}^n),\|\,{\cdot}\,\|_{\mathcal{DT}})$ is a unital Banach algebra with star norm $\|\,{\cdot}\,\|_{\mathcal{DT}}$. $\bullet$ For each operator $W\in\mathcal{DT}(\mathbb{T}^n)$ the following inequalities hold (property (1.4) and Lemma 5.2):
$$
\begin{equation*}
\|W\|_\mu \leqslant \|W\| \leqslant \|W\|_\mathcal{DT}.
\end{equation*}
\notag
$$
$\bullet$ With an operator $W\in\mathcal{DT}(\mathbb{T}^n)$ and a point $a\in\mathbb{T}^n$ we associate the functional $L_a\in(\mathcal{AC}(\mathbb{T}^n))^*$ such that
$$
\begin{equation*}
L_a=\sum_{l\in\mathbb{Z}^n}w_l(a)e_l^*, \qquad w_l(a)=\sum_{k\in\mathbb{Z}^n}W_{l,l-k}e^{i(k,a)},
\end{equation*}
\notag
$$
where $e^*_l$ is defined by (4.3) and the series is weak-$*$ convergent. We prove the following:
$\bullet$ Let $W$ be a bounded operator on $L^2(\mathbb{T}^n)$. In § 6 we introduce the averaged trace $\mathbf{T}(W)$ of the operator $WW^*$ as follows:
$$
\begin{equation*}
\mathbf{T}(W)=\limsup_{\mathbf{I}}\frac{1}{\#\mathbf{I}} \sum_{l\in\mathbf{I},\,j\in\mathbb{Z}^n}|W_{l,j}|^2.
\end{equation*}
\notag
$$
We show that if $W\in\mathcal{DT}(\mathbb{T}^n)$, then
$\bullet$ In § 7 we introduce the set $\mathcal R(\mathbb{T}^n)$ of operators $W\in\mathcal{DT}(\mathbb{T}^n)$ such that for each pair $m,k\in\mathbb{Z}^n$ the following limit exists:
$$
\begin{equation*}
\omega_{m,k}=\lim_{\mathbf{I}}\frac{1}{\# \mathbf{I}} \sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n} W_{l+m,j} \overline W_{l,j+k}
\end{equation*}
\notag
$$
(the limit is taken over the directed set $(\mathcal P^n,\leqslant)$). Elements $W\in\mathcal R(\mathbb{T}^n)$ are called regular operators. We prove (Lemma 7.10) that $\mathcal R(\mathbb{T}^n)$ is a closed cone in $(\mathcal{DT}(\mathbb{T}^n),\|\,{\cdot}\,\|_{\mathcal{DT}})$. $\bullet$ We show that $\|W\|_\mu^2=\mathbf{T}(W)$ for each $W\in\mathcal R(\mathbb{T}^n)$ (Lemma 7.11). Let $(\mathcal X,\mathcal B,\mu)$ be a probability space and $W$ be an operator in one of the following classes:
In this paper we consider the first two classes, the third class of operators was considered in [28] and [29]. $\bullet$ In § 8 we construct a transition measure $\mu_W(\,\cdot\,{,}\,\cdot\,)$ such that for any pair of ‘sufficiently regular functions’ $g_1,g_2\colon \mathcal X\to\mathbb{C}$ we have
$$
\begin{equation*}
\|\widehat g_2 W\widehat g_1\|_{\mu}^2=\int_{\mathcal X}\!\int_{\mathcal X}|g_1(x_1)|^2\,|g_2(x_2)|^2\, \mu_W (x_1,dx_2)\,\mu(dx_1).
\end{equation*}
\notag
$$
$\bullet$ In § 10 we prove that if $W=U$ and $U$ is a unitary operator, then the following hold:
$\bullet$ In § 11, with each $U\in\mathcal N$, where the class $\mathcal N$ consists of the Koopman operators and regular unitary operators, we associate the quantity $\mathfrak{h}(U)\in\overline{\mathbb{R}}_+$, which is called the entropy of $U$. We show that
§ 4. Convolution operator on $L^2( {\mathbb{T}}^n)$4.1. Preliminary constructions4.1.1. A torus: the metric and the Lebesgue measure on a torusLet $\mathbb{T}^n=\mathbb{R}^n/(2\pi\mathbb{Z}^n)$ be the $n$-dimensional torus, $n\in\mathbb{N}$. For each $\varphi=(\varphi_1,\dots,\varphi_n)\in\mathbb{R}^n$ set $\|\varphi\|_\infty=\max\{|\varphi_j|\mid1\leqslant j\leqslant n\}$. Consider the surjective map $\pi\colon \mathbb{R}^n\to \mathbb{T}^n$ that takes a vector $\varphi=(\varphi_1,\dots,\varphi_n)\in\mathbb{R}^n$ to its coset $\pi(\varphi)\in\mathbb{T}^n$. We introduce the metric $\mathrm{dist}$ on $\mathbb{T}^n$ by setting
$$
\begin{equation}
\mathrm{dist}(x,y)=\|x-y\|, \qquad \|x\|=\inf_{k\in\mathbb{Z}^n}\|\varphi-2\pi k\|_{\infty}, \quad x=\pi(\varphi).
\end{equation}
\tag{4.1}
$$
It is well known that $(\mathbb{T}^n, +)$ is a compact Abelian topological group (the topology is generated by the invariant metric $\mathrm{dist}$, and the operation $+$ is the ordinary addition in the quotient space $\mathbb{R}^n/(2\pi\mathbb{Z}^n)$). For all $a=\pi(\psi)\in\mathbb{T}^n$, $r>0$ and $k=(k_1,\dots,k_n)\in\mathbb{Z}^n$ set
$$
\begin{equation}
\begin{gathered} \, B_{r}(a)=\{x\in\mathbb{T}^n\mid\mathrm{dist}(x,a)<r\}, \\ e^{i(k,a)}=\exp\biggl\{i\sum_{j=1}^{n}k_j\psi_j\biggr\}\quad\text{and}\quad\|k\|_1=\sum_{j=1}^{n}|k_j|. \end{gathered}
\end{equation}
\tag{4.2}
$$
Let $\mathcal{B}(\mathbb{T}^n)$ be the Borel $\sigma$-algebra of subsets of $\mathbb{T}^n$ and $\mu$ be the normalized Lebesgue measure on the measurable space $ (\mathbb{T}^n,\mathcal B(\mathbb{T}^n))$ ($d\mu\!=\!(2\pi)^{-n}\,dx$). Throughout what follows, for the probability space $(\mathbb{T}^n,\mathcal B(\mathbb{T}^n),\mu)$ we use the notation and concepts introduced in § 1. If $f\colon\mathbb{T}^n\to\mathbb{C}$ is a $\mu$-integrable function on $\mathbb{T}^n$, then we let $f_k$, $k\in\mathbb{Z}^n$, denote the Fourier coefficients of $f$, so that
$$
\begin{equation*}
f_k=\int_{\mathbb{T}^n}f(x)e^{-i(k,x)}\,\mu(dx), \qquad f\in L^1(\mathbb{T}^n), \quad k\in\mathbb{Z}^n.
\end{equation*}
\notag
$$
4.1.2. Three spaces and their duals1) Let $\mathcal{AC}(\mathbb{T}^n)$ denote the set of continuous complex function with absolutely convergent Fourier series on $\mathbb{T}^n$, so that $\mathcal{AC}(\mathbb{T}^n)$ consists of functions $f$ of the form
$$
\begin{equation*}
f(x)=\sum_{k\in\mathbb{Z}^n}f_ke^{i(k,x)}, \qquad \|f\|_{\mathcal{AC}}:=\sum_{k\in\mathbb{Z}^n}|f_k|<\infty.
\end{equation*}
\notag
$$
It is known that $\mathcal{AC}(\mathbb{T}^n)$ is a commutative Banach algebra with respect to the usual pointwise operations and the norm $\|\,{\cdot}\,\|_{\mathcal{AC}}$. Moreover, introducing convolution $*$ as multiplication in $l^1(\mathbb{Z}^n)$ we see that the map $ f\mapsto \{f_k\}_k$ is a product-preserving isometry, that is, an algebra isomorphism between $\mathcal{AC}(\mathbb{T}^n)$ and $l^1(\mathbb{Z}^n)$. 2) Let $i_1$ and $i_2$ be the natural embeddings of $\mathcal{AC}(\mathbb{T}^n)$ in $C(\mathbb{T}^n)$ and $C(\mathbb{T}^n)$ in $L^2(\mathbb{T}^n)$, respectively. It follows from the inequalities
$$
\begin{equation*}
\|i_1(\cdot)\|_C\leqslant\|\,{\cdot}\,\|_{\mathcal{AC}}\quad\text{and}\quad \|i_2(\cdot)\|\leqslant \|\,{\cdot}\,\|_C
\end{equation*}
\notag
$$
that the operators $i_1$ and $i_2$ are continuous. 3) Let $\mathcal{V}(\mathbb{T}^n)$ be one of the three Banach spaces $\mathcal{AC}(\mathbb{T}^n)$, $C(\mathbb{T}^n)$ and $L^2(\mathbb{T}^n)$. Note that the linear span of the set $\{e^{i(k,\,\cdot\,)}\mid k\in\mathbb{Z}^n\}$ is dense in $\mathcal{V}(\mathbb{T}^n)$, and if $\mathcal{V}(\mathbb{T}^n)=\mathcal{AC}(\mathbb{T}^n)$ or $\mathcal{V}(\mathbb{T}^n)=L^2(\mathbb{T}^n)$, then
$$
\begin{equation*}
\sum_{k\in\mathbb{Z}^n}f_ke^{i(k,\,\cdot\,)}=f \quad\text{for }\ f\in \mathcal{V}(\mathbb{T}^n),
\end{equation*}
\notag
$$
that is, the series on the left-hand side converges to $f$ in the norm $\|\,{\cdot}\,\|_\mathcal V$. In other words, where
$$
\begin{equation*}
S_N(f)=\sum_{\|k\|_\infty\leqslant N}f_ke^{i(k,\,\cdot\,)}, \qquad N\in\mathbb{N}.
\end{equation*}
\notag
$$
For each $k\in\mathbb{Z}^n$ and each functional $\lambda\in(\mathcal V(\mathbb{T}^n))^*$ set $\lambda_k:=\lambda(e^{-i(k,\,\cdot\,)})$. 4) Let $i_1^*$ and $i_2^*$ be the operators adjoint to $i_1$ and $i_2$, respectively. Since $\mathcal{AC}(\mathbb{T}^n)$ is dense in $C(\mathbb{T}^n)$ and $C(\mathbb{T}^n)$ is dense in $L^2(\mathbb{T}^n)$, $i_1^*$ and $i_2^*$ are continuous injective operators such that $i_1^*f=f|_{\mathcal{AC}}\in (\mathcal{AC}(\mathbb{T}^n))^*$ for $f\in (C(\mathbb{T}^n))^*$ and $i_2^*g=g|_{C}\in (C(\mathbb{T}^n))^*$ for $g\in (L^2(\mathbb{T}^n))^*$. Thus, we obtain the continuous embeddings
$$
\begin{equation*}
(L^2(\mathbb{T}^n))^*\xrightarrow[]{i_2^*}(C(\mathbb{T}^n))^* \xrightarrow[]{i_1^*}(\mathcal{AC}(\mathbb{T}^n))^*
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
(L^2(\mathbb{T}^n))^*\xrightarrow[]{i^*}(\mathcal{AC}(\mathbb{T}^n))^*, \quad\text{where }\ i=i_2i_1, \quad i^*=i^*_1i^*_2.
\end{equation*}
\notag
$$
5) We introduce an operation of convolution on $(\mathcal{AC}(\mathbb{T}^n))^*$, and we claim that, with respect to convolution multiplication, $(\mathcal{AC}(\mathbb{T}^n))^*$ is a commutative Banach algebra which is isometrically isomorphic to the algebra $l^{\infty}(\mathbb{Z}^n)$ with componentwise multiplication. To prove this we need some preparations. (a) Since the Banach spaces $\mathcal{AC}(\mathbb{T}^n)$ and $l^1(\mathbb{Z}^n)$ are isometrically isomorphic, the map
$$
\begin{equation*}
\Psi\colon(\mathcal{AC}(\mathbb{T}^n))^*\to l^\infty(\mathbb{Z}^n), \qquad \Psi\lambda:=\{\lambda(e^{-i(k,\,\cdot\,)})\}_{k}=\{\lambda_{k}\}_{k},
\end{equation*}
\notag
$$
is an isometric isomorphism. For $\lambda\in(\mathcal{AC}(\mathbb{T}^n))^*$ the series $\sum_{k\in\mathbb{Z}^n}\lambda_ke_k^*$ converges to $\lambda$ weak-$*$, where
$$
\begin{equation}
e_k^*(f):=f_{-k}=\int_{\mathbb{T}^n}f(x)e^{i(k,x)}\,\mu(dx), \qquad f\in\mathcal{AC}(\mathbb{T}^n).
\end{equation}
\tag{4.3}
$$
When we write $\lambda=\sum_{k\in\mathbb{Z}^n}\lambda_ke_k^*$ in what follows, we mean that the series on the right converges to $\lambda$ weak-$*$. (b) Consider the family of continuous automorphisms $F_x$, $x\in\mathbb{T}^n$, of the probability space $(\mathbb{T}^n,\mathcal B(\mathbb{T}^n),\mu)$ defined by
$$
\begin{equation}
F_x\colon \mathbb{T}^n\to \mathbb{T}^n, \qquad F_x(y)=x+y, \quad x,y\in\mathbb{T}^n.
\end{equation}
\tag{4.4}
$$
Note that $f\circ F_x\in\mathcal{AC}(\mathbb{T}^n)$ for $f\in\mathcal{AC}(\mathbb{T}^n)$ and $x\in\mathbb{T}^n$, because $(f\circ F_x)_k=e^{i(k,x)}f_k$ for each $k\in\mathbb{Z}^n$. Let $\lambda\in(\mathcal{AC}(\mathbb{T}^n))^*$. For each function $f\in\mathcal{AC}(\mathbb{T}^n)$ let $\overline\lambda f$ be the function defined by $\overline\lambda f(x)=\lambda(f\circ F_x)$ for each $x\in\mathbb{T}^n$. Then
$$
\begin{equation*}
\overline\lambda f(x) =\lambda\biggl(\sum_{k\in\mathbb{Z}^n}e^{i(k,x)}f_k e^{i(k,\,\cdot\,)}\biggr) =\sum_{k\in\mathbb{Z}^n}e^{i(k,x)}f_k\lambda_{-k}, \qquad x\in\mathbb{T}^n.
\end{equation*}
\notag
$$
Hence $\overline\lambda f\in\mathcal{AC}(\mathbb{T}^n)$, so that $\overline\lambda\colon f\mapsto \overline\lambda f$ is a continuous operator on $\mathcal{AC}(\mathbb{T}^n)$, and we have $\|\overline\lambda\|_{\mathcal{AC}\to\mathcal{AC}} =\|\Psi\lambda\|_\infty=\|\lambda\|_{\mathcal{AC}^*}$. We define the operation of convolution following Definition $19.1$ in [15], Ch. V, § 19. The convolution of two functionals $\lambda^1$ and $\lambda^2$ in the space $(\mathcal{AC}(\mathbb{T}^n))^*$ is the composite map $\lambda^1\circ \overline {\lambda^2}$, which we denote by $\lambda^1*\lambda^2$. It follows directly from this definition that
$$
\begin{equation*}
\lambda^1*\lambda^2\in (\mathcal{AC}(\mathbb{T}^n))^*\quad\text{and} \quad \Psi(\lambda^1*\lambda^2)=\{\Psi\lambda^1\}\cdot\{\Psi\lambda^2\}:=\{\lambda^1_{k}\lambda^2_{k}\}.
\end{equation*}
\notag
$$
Thus, $(\mathcal{AC}(\mathbb{T}^n))^*$ is a commutative Banach algebra with respect to convolution $*$. 4.1.3. The convolution operatorFirst we introduce two isometric isomorphisms which identify $L^2(\mathbb{T}^n)$ with the spaces $l^2(\mathbb{Z}^n)$ and $(L^2(\mathbb{T}^n))^*$, namely,
Let $\lambda=\sum\lambda_ke_k^*\in(\mathcal{AC}(\mathbb{T}^n))^*$ and $f=\sum f_ke^{i(k,\,\cdot\,)}\in L^2(\mathbb{T}^n)$, and let $M_{\Psi\lambda}$ be the diagonal operator
$$
\begin{equation*}
M_{\Psi\lambda}\colon l^2(\mathbb{Z}^n)\to l^2(\mathbb{Z}^n), \qquad M_{\Psi\lambda}\{\alpha_k\}=\{\lambda_k\alpha_k\}, \quad \{\alpha_k\}\in l^2(\mathbb{Z}^n).
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
\Psi(\lambda*i^*Lf)=M_{\Psi\lambda}(\mathcal F f)=\{\lambda_kf_k\}_{k}\in l^2(\mathbb{Z}^n).
\end{equation*}
\notag
$$
Set $\operatorname{Conv}_\lambda:=\mathcal F^{-1}\circ M_{\Psi\lambda}\circ \mathcal F$ and $\lambda*f:=\operatorname{Conv}_\lambda f$. We see immediately that $i^*L(\lambda*f)=\lambda*i^*Lf$, which means that
$$
\begin{equation}
(L(\lambda*f))\big|_{\mathcal{AC}}=\lambda* (Lf)|_{\mathcal{AC}}.
\end{equation}
\tag{4.5}
$$
Thus, $\lambda*f$ is a function in $L^2(\mathbb{T}^n)$ such that we have (4.5), and $\mathrm{Conv}_\lambda$ is a bounded operator on $L^2(\mathbb{T}^n)$ such that
$$
\begin{equation}
L^2(\mathbb{T}^n)\ni f\mapsto \operatorname{Conv}_\lambda f=g, \qquad g=\sum_{k\in\mathbb{Z}^n}g_ke^{i(k,\,\cdot\,)}, \quad g_k=f_k\lambda_k.
\end{equation}
\tag{4.6}
$$
Moreover, as $\mathcal F$ is an isometry, we have
$$
\begin{equation*}
\|\mathrm{Conv}_\lambda\|=\|M_{\Psi\lambda}\|_{l^2\to l^2}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation}
\|\mathrm{Conv}_{\lambda}\|=c_{\lambda}, \qquad c_{\lambda}=\sup_{k\in\mathbb{Z}^n}|\lambda_k|.
\end{equation}
\tag{4.7}
$$
We call $\mathrm{Conv}_{\lambda}$ the convolution operator, and we call $\lambda*f=\operatorname{Conv}_\lambda f$ the convolution of the functional $\lambda\in (\mathcal{AC}(\mathbb{T}^n))^*$ with the function $f\in L^2(\mathbb{T}^n)$. Examples. 1. If $g,f\in L^2(\mathbb{T}^n)$, then 2. If $\lambda_1,\lambda_2\in(\mathcal{AC}(\mathbb{T}^n))^*$, then
$$
\begin{equation*}
\operatorname{Conv}_{\lambda_1}\operatorname{Conv}_{\lambda_2} =\operatorname{Conv}_{\lambda_1*\lambda_2} =\operatorname{Conv}_{\lambda_2*\lambda_1} =\operatorname{Conv}_{\lambda_2}\operatorname{Conv}_{\lambda_1}\!.
\end{equation*}
\notag
$$
3. If $\delta_a$ is the Dirac function at $a\in\mathbb{T}^n$, then for each $f\in L^2(\mathbb{T}^n)$
$$
\begin{equation*}
\delta_a*f:=\delta_a|_{\mathcal{AC}}*f=f\circ F_{-a},
\end{equation*}
\notag
$$
where $F_{-a}$ is the automorphism defined by (4.4). Thus, $\operatorname{Conv}_{\delta_a}$ coincides with the Koopman operator $U_{F_{-a}}$; in particular, $\operatorname{Conv}_{\delta_0} = \operatorname{id}$. 4. If $\lambda=\sum\lambda_ke_k^*\in(\mathcal{AC}(\mathbb{T}^n))^*$, then
$$
\begin{equation*}
\lambda*(e^{i(m,\,\cdot\,)})=\lambda_m e^{i(m,\,\cdot\,)} \quad\text{for each }\ m\in\mathbb{Z}^n.
\end{equation*}
\notag
$$
5. For all $\lambda\in(\mathcal{AC}(\mathbb{T}^n))^*$, $f\in L^2(\mathbb{T}^n)$ and $m\in\mathbb{Z}^n$ we have
$$
\begin{equation*}
\lambda * (e^{i(m,\,\cdot\,)} f)=e^{i(m,\,\cdot\,)} \bigl( (\lambda\circ\widehat{e^{-i(m,\,\cdot\,)}}) * f\bigr),
\end{equation*}
\notag
$$
where the operator $\widehat{e^{-i(m,\,\cdot\,)}}$ is defined by (1.1) for $g=e^{-i(m,\,\cdot\,)}$. 4.1.4. Estimates for Fourier coefficients Lemma 4.1. Let $ f=\sum_{k \in \mathbb{Z}^n}f_ke^{i(k,\,\cdot\,)} \in L^2(\mathbb{T}^n)$ and $Y=B_\varepsilon (a)$, where $a\in\mathbb{T}^n$ and $\varepsilon>0$. If $g=\widehat{\bf 1}_Y f=\sum_{k \in \mathbb{Z}^n}g_ke^{i(k,\,\cdot\,)}$, then for all $m,l\in\mathbb{Z}^n$ Proof. Set $h=g - e^{i(m,\cdot-a)}g=\widehat{\bf 1}_Y(1-e^{i(m,\cdot-a)})\cdot f$. Then
$$
\begin{equation*}
\begin{aligned} \, \|h\|^2 &=\frac{1}{(2\pi)^n}\int_{Y}|1 - e^{i(m,x-a)}|^2\,|f(x)|^2\,dx \\ &\leqslant\| \widehat{\bf 1}_Y(1-e^{i(m,\cdot-a)})\|_{\infty}^2\,\|f\|^2 \leqslant\|m\|_1^2\varepsilon^2\|f\|^2, \end{aligned}
\end{equation*}
\notag
$$
which is equivalent to (4.8).
Note that
$$
\begin{equation*}
\| \widehat{\bf 1}_Y(1 - e^{-i(l,\cdot-a)}) \|^2 =\frac{1}{(2\pi)^n}\int_{Y} |1 - e^{-i(l,x-a)} |^2\,dx\leqslant\frac{1}{(2\pi)^n}\|l\|_1^2\varepsilon^2(2\varepsilon)^n,
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation*}
\begin{aligned} \, |g_m - e^{i(l,a)}g_{m+l} | &= \biggl|\frac{1}{(2\pi)^n}\int_{Y} \bigl(f(x)e^{-i(m,x)} - f(x)e^{-i(m+l,x)}e^{i(l,a)}\bigr)\,dx\biggr| \\ &\leqslant\frac{1}{(2\pi)^n}\int_{Y} \bigl| (1 - e^{-i(l,x-a)})f(x) \bigr|\,dx \\ &\leqslant \| \widehat{\bf 1}_Y (1 - e^{-i(l,x-a)}) \|\,\|f\|\leqslant\frac{\varepsilon^{n/2+1}}{\pi^{n/2}}\|l\|_1\,\|f\|. \end{aligned}
\end{equation*}
\notag
$$
Hence we have (4.9).
To prove (4.10) consider the inner product
$$
\begin{equation*}
\langle h,g\rangle=\langle g, g\rangle- e^{-i(m,a)}\langle g, e^{-i(m,\,\cdot\,)}g\rangle=\|g\|^2 - \sum_{k\in\mathbb{Z}^n}e^{-i(m,a)}g_k\overline g_{k+m}.
\end{equation*}
\notag
$$
Applying the Cauchy-Schwarz-Bunyakovskii inequality to (4.8) we obtain
$$
\begin{equation*}
|\langle h,g\rangle|\leqslant\|h\|\,{\cdot}\,\|g\|\leqslant\|m\|_1\varepsilon\|f\|^2,
\end{equation*}
\notag
$$
so that we have (4.10).
The proof is complete. 4.1.5. Integer parallelepipeds and the upper limit $\rho(\lambda)$Consider the following objects:
For each functional $ \lambda=\sum\lambda_ke_k\in(\mathcal{AC}(\mathbb{T}^n))^*$ set
$$
\begin{equation}
\rho(\lambda)= \lim_{N\to\infty}\sup_{\mathbf{I}\in\mathcal{P}_N^n}\rho_{\mathbf{I}}(\lambda), \qquad \rho_{\mathbf{I}}(\lambda)= \frac{1}{\#\mathbf{I}}\sum_{k\in \mathbf{I}}|\lambda_k|^2.
\end{equation}
\tag{4.11}
$$
It follows from the inclusion $\mathcal{P}_{N+1}^n\subset\mathcal{P}_N^n$ that the sequence
$$
\begin{equation*}
\rho_N=\sup_{\mathbf{I}\in\mathcal{P}_N^n}\rho_{\mathbf{I}}(\lambda), \qquad N\in\mathbb{N},
\end{equation*}
\notag
$$
is nonincreasing and bounded below. Hence the limit in (4.11) exists and is a finite number. 4.2. Calculating $\|\mathrm{Conv}_\lambda \widehat{\bf 1}_{B_\varepsilon(a)}\|$Lemma 4.2. If $\lambda=\sum\lambda_ke_k^*\in(\mathcal{AC}(\mathbb{T}^n))^*$, then for each point $a\in\mathbb{T}^n$ and each ${\varepsilon>0}$ where $B_\varepsilon=B_\varepsilon(0)$. Proof. From equality (2.1) for $X=B_\varepsilon(a)$ and $F=F_a$ (see (4.4)) we obtain
$$
\begin{equation*}
U_{F_a}\widehat{\bf 1}_{B_{\varepsilon}(a)}=\widehat{\bf 1}_{F_{-a}B_{\varepsilon}(a)}U_{F_a}=\widehat{\bf 1}_{B_{\varepsilon}(a)-a}U_{F_a}=\widehat{\bf 1}_{B_{\varepsilon}}U_{F_a}
\end{equation*}
\notag
$$
(the last equality holds because the metric $\mathrm{dist}$ is invariant). Then, taking examples 2 and 3 in § 4.1.3 into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \|\operatorname{Conv}_\lambda \widehat{\bf 1}_{B_{\varepsilon}(a)}\| &=\|\operatorname{Conv}_\lambda U_{F_{-a}}\widehat{\bf 1}_{B_{\varepsilon}}U_{F_a}\| =\|\operatorname{Conv}_{\lambda*\delta_a}\widehat{\bf 1}_{B_{\varepsilon}}U_{F_a}\| \\ &=\|U_{F_{-a}}\operatorname{Conv}_\lambda\widehat{\bf 1}_{B_{\varepsilon}}U_{F_a}\| =\|\operatorname{Conv}_\lambda\widehat{\bf 1}_{B_{\varepsilon}}\|. \end{aligned}
\end{equation*}
\notag
$$
The proof is complete. Let $K^*$ be a compact set in the strong topology2 of the space $(\mathcal{AC}(\mathbb{T}^n))^*$. Lemma 4.3. $\mathrm{(a)}$ The family of functions $\{\rho_{\mathbf{I}}\}_{\mathbf{I}\in\mathcal{P}^n}$ is equicontinuous on $K^*$. $\mathrm{(b)}$ The functions $\rho_N=\sup_{\mathbf{I}\in\mathcal P^n_N}\rho_{\mathbf{I}}$, $N\in\mathbb{N}$, and $\rho$ are continuous on $K^*$, and the uniform convergence holds.Lemma 4.4. For each $\gamma > 0$ there exists $\varepsilon_0 > 0$ such that for each $\varepsilon\in(0,\varepsilon_0]$ and all $\lambda\in K^*$, $a\in\mathbb{T}^n$ and $f \in L^2(\mathbb{T}^n)$ where $Y=B_\varepsilon(a)$. Proof of Lemma 4.3. The compact set $K^*$ is bounded, that is, there exists $r>0$ such that $\sup_{k\in\mathbb{Z}^n}|\lambda_k|=\|\lambda\|_{\mathcal{AC}^*}\leqslant r$ for each $\lambda\in K^*$. Then for each pair of functionals $\lambda_1,\lambda_2\in K^*$ and each parallelepiped $\mathbf{I}\in\mathcal{P}^n$ which yields $\mathrm{(a)}$.
It follows from $\mathrm{(a)}$ that the sequence of functions $\{\rho_N\}$ is equicontinuous on $K^*$, so that $\rho$ is continuous on $K^*$. Hence the sequence $\{\rho_N,\,N\in\mathbb{N}\}$ of continuous functions on $K^*$ decreases pointwise to the continuous function $\rho$. Then this sequence converges uniformly to $\rho$ on $K^*$ by Dini’s theorem (for instance, see [6], Theorem 1.7.10). The proof is complete. Proof of Lemma 4.4. It follows from Lemma 4.2 that we can limit ourselves to $a=0$. Let $\sigma >0$ be arbitrary. By assertion $\mathrm{(b)}$ in Lemma 4.3 there exists a positive integer $M'=M'(\sigma)$ such that
$$
\begin{equation}
\rho_{\mathbf{I}}( \lambda) \leqslant \rho( \lambda) + \sigma \quad\text{for all }\ \lambda\in K^*, \quad \mathbf{I}\in\mathcal{P}^{n}_{M'}.
\end{equation}
\tag{4.14}
$$
Let $\varepsilon>0$, $f\in L^2(\mathbb{T}^n)$ and $\mathbf{I}=\mathbf{I}_B$, where $B$ is a positive integer such that $2B+1\geqslant M'$. Set
$$
\begin{equation*}
Y=B_\varepsilon, \qquad g=\widehat{\bf 1}_Yf\quad\text{and} \quad G= \frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I}}e^{i(l,\,\cdot\,)}g=\frac{1}{\#\mathbf{I}} \sum_{k\in\mathbb{Z}^n,\,l\in \mathbf{I}}g_{k-l}e^{i(k,\,\cdot\,)},
\end{equation*}
\notag
$$
where the $g_k$, $k\in\mathbb{Z}^n$, are the Fourier coefficients of $g$. Using the triangle inequality and (4.8) for $a=0$, we obtain
$$
\begin{equation*}
\|g-G\| \leqslant \frac{1}{\#\mathbf{I}}\sum_{l\in \mathbf{I}}\|g-e^{i(l,\,\cdot\,)}g\| \leqslant \frac{1}{\#\mathbf{I}}\sum_{l\in \mathbf{I}}\|l\|_1\varepsilon \|f\|.
\end{equation*}
\notag
$$
If $ l=(l_1,\dots,l_n)\in\mathbf{I}$, then $\|l\|_1=\sum_{j=1}^n|l_j|\leqslant nB$. Hence
$$
\begin{equation}
\|g-G\|\leqslant nB\varepsilon\|f\|, \qquad\| \lambda * g - \lambda * G\|\leqslant r nB\varepsilon\|f\|,
\end{equation}
\tag{4.15}
$$
where $r=\sup_{\lambda\in K^*}\| \lambda\|_{\mathcal{AC}^*}$. Consider the square of the norm of $ \lambda*G$:
$$
\begin{equation}
\begin{aligned} \, \nonumber \| \lambda * G\|^2 &=\frac{1}{(\#\mathbf{I})^2}\sum_{k\in\mathbb{Z}^n}| \lambda_k|^2\sum_{l\in\mathbf{I},\,s\in \mathbf{I}}g_{k-l}\overline g_{k-s} \\ \nonumber &=\frac{1}{(\#\mathbf{I})^2}\sum_{m\in2\mathbf{I}}\sum_{\substack{l\in\mathbf{I},\\l\in (\mathbf{I}+m)}}\sum_{k\in\mathbb{Z}^n}|\lambda_{k+l}|^2 g_{k}\overline g_{k+m} \\ &=\sum_{m\in \mathbf{J}}\sum_{k\in\mathbb{Z}^n}\mathbf{b}(m)\rho_{\mathbf{I}_{k,m}}(\lambda) g_k\overline g_{k+m}, \end{aligned}
\end{equation}
\tag{4.16}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \mathbf{J}=J\times\dots\times J, \qquad J=[-2B,2B]\cap\mathbb{Z}, \qquad\mathbf{I}_{k,m}= I_{k_1,m_1}\times\dots\times I_{k_n,m_n}, \\ I_{k_j,m_j}=\begin{cases} [k_j+m_j-B, k_j+B]\cap\mathbb{Z} &\text{for }\ m_j\geqslant 0, \\ [k_j-B, k_j+m_j+B]\cap\mathbb{Z} &\text{for }\ m_j < 0, \end{cases} \qquad \mathbf{b}(m)=\frac{\# \mathbf{I}_{k,m}}{(\#\mathbf{I})^2}. \end{gathered}
\end{equation*}
\notag
$$
Note that
$$
\begin{equation*}
\begin{gathered} \, \# \mathbf{I}_{k,m}=\prod_{j=1}^{n}(2B+1-|m_j|), \qquad \rho_{\mathbf{I}_{k,m}}( \lambda)\leqslant r^2, \\ \mathbf{b}(m)=b(m_1)\dotsb b(m_n), \quad\text{where }\ b(q)=\frac{2B+1-|q|}{(2B+1)^2}, \quad q\in J, \end{gathered}
\end{equation*}
\notag
$$
and furthermore,
$$
\begin{equation}
\sum_{q\in J} b(q)=1, \quad \sum_{m\in\mathbf{J}}\mathbf{b}(m)=1, \qquad b(q)>0 \quad\text{and} \quad\mathbf{b}(m)>0.
\end{equation}
\tag{4.17}
$$
Set $\mathbf{S}=S\times\dots\times S$, where $S=[-(2B+1-M'),2B+1-M']\cap\mathbb{Z}$. Then $S\subset J$, $\mathbf{S}\subset\mathbf{J}$ and
$$
\begin{equation*}
b(q)<\frac{M'}{(2B+1)^2} \quad\text{for }\ q\in J\setminus S.
\end{equation*}
\notag
$$
In addition, by (4.14) we have ${\rho_{\mathbf{I}_{k,m}}( \lambda)\leqslant \rho( \lambda)+\sigma}$ for all $a\in\mathbb{T}^n$, $k\in\mathbb{Z}^n$ and $m\in\mathbf{S}$. It follows from (4.16) that $ \| \lambda * G\|^2\leqslant\Delta_1+\Delta_2$, where
$$
\begin{equation*}
\Delta_1=\sum_{m\in \mathbf{S}} \sum_{k\in\mathbb{Z}^n}\mathbf{b}(m)\rho_{\mathbf{I}_{k,m}}( \lambda) |g_k|\,|g_{k+m}|
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Delta_2=\sum_{m\in \mathbf{J}\setminus\mathbf{S}} \sum_{k\in\mathbb{Z}^n}\mathbf{b}(m)\rho_{\mathbf{I}_{k,m}}( \lambda)| g_k|\,|g_{k+m}|.
\end{equation*}
\notag
$$
Using (4.17) we obtain
$$
\begin{equation*}
\Delta_1=\sum_{m\in \mathbf{S}}\sum_{k\in\mathbb{Z}^n}\mathbf{b}(m)\rho_{\mathbf{I}_{k,m}}( \lambda) |g_k|\,|g_{k+m}|\leqslant(\rho( \lambda)+\sigma)\|g\|^2.
\end{equation*}
\notag
$$
Hence We represent the set $\mathbf{J}\setminus\mathbf{S}$ as a union:
$$
\begin{equation*}
\mathbf{J}\setminus\mathbf{S}=\bigcup_{r=1}^{n}H_r, \qquad H_r=\{m=(m_1,\dots,m_n)\in\mathbf{J}\mid m_r\in J\setminus S\}, \quad r=1,\dots,n.
\end{equation*}
\notag
$$
Next we estimate a sum:
$$
\begin{equation}
\begin{aligned} \, \nonumber \sum_{m\in \mathbf{J}\setminus\mathbf{S}}\mathbf{b}(m) &\leqslant\sum_{r=1}^{n} \sum_{m\in H_r}\mathbf{b}(m)=\sum_{r=1}^{n}\sum_{m_1\in J}\dotsb\sum_{m_r\in J\setminus S}\dotsb\sum_{m_n\in J}b(m_1)\dotsb b(m_n) \\ \nonumber &=\sum_{r=1}^{n}\sum_{m_1\in J}b(m_1)\dotsb\sum_{m_r\in J\setminus S}b(m_r)\dotsb\sum_{m_n\in J}b(m_n) \\ &\leqslant \sum_{r=1}^{n}2(M'-1)\frac{M'}{(2B+1)^2}=\frac{2M'(M'-1)n}{(2B+1)^2}. \end{aligned}
\end{equation}
\tag{4.19}
$$
This yields a bound for $\Delta_2$:
$$
\begin{equation}
\begin{aligned} \, \nonumber \Delta_2 &=\sum_{m\in \mathbf{J}\setminus\mathbf{S}}\sum_{k\in\mathbb{Z}^n}\mathbf{b}(m)\rho_{\mathbf{I}_{k,m}}( \lambda) |g_k|\,|g_{k+m}|\leqslant r^2\|g\|^2\sum_{m\in \mathbf{J}\setminus\mathbf{S}}\mathbf{b}(m) \\ &\leqslant \frac{2M'(M'-1)r^2n}{(2B+1)^2}\|f\|^2\leqslant \frac{2M'r^2n}{2B+1}\|f\|^2 \end{aligned}
\end{equation}
\tag{4.20}
$$
(the last inequality follows because $2B+1\geqslant M'$). Thus, using (4.18) and (4.20) we obtain
Finally, it follows from the relation
$$
\begin{equation*}
\| \lambda*g\|^2\leqslant \| \lambda*g- \lambda*G\|^2+2\| \lambda*G\|\,{\cdot}\,\| \lambda*g- \lambda*G\|+\| \lambda*G\|^2
\end{equation*}
\notag
$$
and estimates (4.15) and (4.21) that
$$
\begin{equation*}
\| \lambda*( \widehat{\bf 1}_Yf)\|\leqslant \Bigl(\rho( \lambda)+\sigma+\alpha_1+\alpha_2^2+2\alpha_2\sqrt{\rho( \lambda)}+2\alpha_2\sqrt{\sigma+\alpha_1}\Bigr)\|f\|^2,
\end{equation*}
\notag
$$
where $\alpha_1=\alpha_1(M',B)=2M'r^2 n/(2B+1)$ and $\alpha_2=\alpha_2(\varepsilon, B)=rnB\varepsilon$. Note that $\sigma>0$ and $\varepsilon>0$ can be arbitrary, $M'=M'(\sigma)$ depends only on $\sigma$, and $B\in\mathbb{N}$ is an arbitrary integer such that $2B+1\geqslant M'$. Moreover, $\alpha_1(M',B)\to0$ as $B\to\infty$ and $\alpha_2(\varepsilon, B)\to0$ as $\varepsilon\to0$ for each $B$. This observation completes the proof of Lemma 4.4. Lemma 4.5. If $\lambda=\sum\lambda_ke_k^*\in(\mathcal{AC}(\mathbb{T}^n))^*$, then for each positive number $\gamma$ and each function $f\in L^2(\mathbb{T}^n)$ there exists $m\in\mathbb{Z}^n$ such that Proof. We assume that $\rho(\lambda)\ne0$ (otherwise the statement is obvious). Let $\sigma_1>0$ and $f\in L^2(\mathbb{T}^n)$. For each $N\in\mathbb{N}$ set $S_N=\sum_{k\in\mathbf{I}}f_ke^{i(k,\,\cdot\,)}$, where $\mathbf{I}=\mathbf{I}_N$. Since $\|S_N-f\|\to0$ as $N\to\infty$, there exists $N\in\mathbb{N}$ such that $\|f-S_N\|\leqslant\sigma_1\|f\|$. Therefore, For each $m\in\mathbb{Z}^n$ consider the continuous operator $T_m=\mathrm{Conv}_{\lambda}\widehat{e^{i(m,\,\cdot\,)}}$. We have $\|\widehat{e^{i(m,\,\cdot\,)}}\|=\|e^{i(m,\,\cdot\,)}\|_\infty=1$, so
$$
\begin{equation*}
\|T_m(f)-T_m(S_N)\|=\|T_m(f-S_N)\|\leqslant\|\mathrm{Conv}_{\lambda}\|\,{\cdot}\,\|f-S_N\|\leqslant c_{\lambda}\sigma_1\|f\|.
\end{equation*}
\notag
$$
In addition,
$$
\begin{equation}
T_m(S_N)=\sum_{k\in(\mathbf{I}+m)}\lambda_k f_{k-m}e^{i(k,\,\cdot\,)}
\end{equation}
\tag{4.24}
$$
and Hence for each $m\in\mathbb{Z}^n$ we have
Consider some $\sigma_2$ and $\sigma_3$ such that $0<\sigma_2< \rho(\lambda)$ and $0<\sigma_3 <1$. Then by (4.11) there exist $M\in\mathbb{N}$ and $\mathbf{J}\in\mathcal{P}^n_M$ such that
$$
\begin{equation}
\rho_{\mathbf{J}}(\lambda) > \rho(\lambda) - \sigma_2\quad\text{and} \quad \frac{N}{M}<\sigma_3.
\end{equation}
\tag{4.26}
$$
From (4.24) we obtain
$$
\begin{equation}
\sum_{m\in\mathbf{J}}\|T_m(S_N)\|^2=\sum_{m\in \mathbf{J}}\sum_{l\in \mathbf{I}}|\lambda_{m+l}f_l|^2=\|S_N\|^2\sum_{m\in \mathbf{J}}|\lambda_m|^2 + \Delta,
\end{equation}
\tag{4.27}
$$
where
$$
\begin{equation}
\begin{aligned} \, \nonumber \Delta &= \sum_{m\in \mathbf{J}}\sum_{l\in \mathbf{I}}|\lambda_{m+l}f_l|^2 - \sum_{m\in \mathbf{J}}\sum_{l\in \mathbf{I}}|\lambda_{m}f_l|^2 \\ \nonumber &=\sum_{l\in \mathbf{I}}|f_l|^2\biggl(\sum_{m\in (\mathbf{J}+l)}|\lambda_m|^2-\sum_{m\in \mathbf{J}}|\lambda_m|^2\biggr) \\ &=\sum_{l\in \mathbf{I}}|f_l|^2\biggl(\sum_{m\in (\mathbf{J}+l)\setminus\mathbf{J}}|\lambda_m|^2 -\sum_{m\in \mathbf{J}\setminus(\mathbf{J}+l)}|\lambda_m|^2\biggr)=\Delta_1-\Delta_2, \\ \nonumber \Delta_1&=\sum_{l\in \mathbf{I}}|f_l|^2\biggl(\sum_{m\in (\mathbf{J}+l)\setminus\mathbf{J}}|\lambda_m|^2\biggr), \qquad \Delta_2=\sum_{l\in \mathbf{I}}|f_l|^2\biggl(\sum_{m\in \mathbf{J}\setminus(\mathbf{J}+l)}|\lambda_m|^2\biggr). \end{aligned}
\end{equation}
\tag{4.28}
$$
We find an estimate for $\Delta_1$. Note that for each $l=(l_1,\dots,l_n)\in\mathbf{I}$ we have
$$
\begin{equation*}
\begin{gathered} \, (\mathbf{J}+l)\setminus\mathbf{J}=\bigcup_{r=1}^{n}H_r, \\ H_r=\{m=(m_1,\dots,m_n)\in(\mathbf{J}+l)\mid m_r\notin J_r\}, \qquad r=1,\dots,n, \end{gathered}
\end{equation*}
\notag
$$
where $J_1,\dots, J_n$ are the integer segments such that $\mathbf{J}=J_1\times\dots\times J_n$. Then
$$
\begin{equation*}
\begin{aligned} \, \sum_{m\in (\mathbf{J}+l)\setminus\mathbf{J}}|\lambda_m|^2 &\leqslant \sum_{r=1}^{n}\sum_{m_1\in (J_1+l_1)}\dotsb\sum_{m_r\in (J_r+l_r)\setminus J_r}\dotsb\sum_{m_n\in (J_n+l_n)}c^2_{\lambda} \\ &=c^2_{\lambda}\sum_{r=1}^{n}|l_r|\prod_{\substack{j=1\\ j\ne r}}^{n}\#J_j=c^2_{\lambda}\sum_{r=1}^{n}\frac{|l_r|}{\# J_r}\#\mathbf{J}\leqslant nc^2_{\lambda}\frac{N}{M}\#\mathbf{J}, \end{aligned}
\end{equation*}
\notag
$$
where the last inequality holds because $|l_r|\leqslant N$ and $\# J_r\geqslant M$ for all $r=1,\dots,n$. Consequently,
$$
\begin{equation*}
\Delta_1\leqslant nc^2_{\lambda}\frac{N}{M}\#\mathbf{J}\|S_N\|^2.
\end{equation*}
\notag
$$
We also have a similar estimate for $\Delta_2$. Then, taking (4.28) and (4.26) into account we obtain
$$
\begin{equation*}
|\Delta|\leqslant \Delta_1+\Delta_2\leqslant 2nc^2_{\lambda}\frac{N}{M}\#\mathbf{J}\|S_N\|^2\leqslant 2nc^2_{\lambda}\sigma_3\#\mathbf{J}\|f\|^2.
\end{equation*}
\notag
$$
It follows from (4.27) that there exists $m_0\in \mathbf{J}$ such that
$$
\begin{equation*}
\|T_{m_0}(S_N)\|^2 \geqslant \|S_N\|^2\rho_{\mathbf{J}}(\lambda)+ \frac{\Delta}{\#\mathbf{J}}\geqslant \|S_N\|^2\rho_{\mathbf{J}}(\lambda) - 2nc^2_{\lambda}\sigma_3\|f\|^2,
\end{equation*}
\notag
$$
so that using (4.23), (4.25) and (4.26) we obtain
$$
\begin{equation*}
\begin{aligned} \, \|\lambda *(e^{i(m_0,\,\cdot\,)}f)\|^2 &\geqslant \|f\|^2\bigl((1-\sigma_1)^2(\rho(\lambda) - \sigma_2) - 2nc^2_{\lambda}\sigma_3\bigr) \\ &=\|f\|^2\bigl(\rho(\lambda)-(\rho(\lambda)\sigma_1(2-\sigma_1)+\sigma_2(1-\sigma_1)^2+2nc^2_{\lambda}\sigma_3)\bigr). \end{aligned}
\end{equation*}
\notag
$$
It remains to observe that $\sigma_1>0$, $0<\sigma_2<\rho(\lambda)$ and $0<\sigma_3<1$ are arbitrary. Lemma 4.5 is proved. From Lemmas 4.4 and 4.5 we obtain the following. Corollary 4.1. If $\lambda=\sum\lambda_ke^*_k\in(\mathcal{AC}(\mathbb{T}^n))^*$, then for each point $a\in \mathbb{T}^n$ 4.3. The $\mu$-norm of the convolution operator Proof. Set $W:=\mathrm{Conv}_\lambda $. By Corollary 4.1, for each point $a\in\mathbb{T}^n$
$$
\begin{equation*}
\vartheta(a)=\lim_{\varepsilon \searrow 0} \|W \widehat{\bf 1}_{B_\varepsilon(a)}\|^2=\rho(\lambda),
\end{equation*}
\notag
$$
so $\vartheta$ is a continuous function on $\mathbb{T}^n$. Moreover, it follows from Lemmas 4.4 and 4.5 that condition C2 in § 2, item 10, holds. Hence from the statement there we obtain
The proof is complete. § 5. Diagonal-type operators5.1. The definition and propertiesLet $W$ be an operator in the space $\mathcal L(L^2)$ of bounded operators on $L^2(\mathbb{T}^n)$. If $f\in L^2(\mathbb{T}^n)$, then
$$
\begin{equation*}
Wf=F, \quad F_j=\sum_{k\in\mathbb{Z}^n}W_{j,k}f_k \quad\text{for } \quad f=\sum_{k\in\mathbb{Z}^n}f_ke^{i(k,\,\cdot\,)}\quad\text{and}\quad F=\sum_{j\in\mathbb{Z}^n}F_j e^{i(j,\,\cdot\,)},
\end{equation*}
\notag
$$
where $W_{j,k}=\langle We^{i(k,\,\cdot\,)},e^{i(j,\,\cdot\,)}\rangle$. Set $c_k= \sup_{j\in\mathbb{Z}^n}|W_{k+j, j}|$, $k\in\mathbb{Z}^n$. We call the sequence $\{c_k\}_{k\in\mathbb{Z}^n}$ the majorizing sequence of $W$. Definition 5.1. We call $W$ an operator of diagonal type if $\sum_{k\in\mathbb{Z}^n}c_k=\mathbf{c}<\infty$. Let $\mathcal{DT}(\mathbb{T}^n)$ denote the set of operators of diagonal type and set Note that if $W\in\mathcal{DT}(\mathbb{T}^n)$, then $|W_{j,k}|\leqslant c_{j-k}$ for all $j,k\in\mathbb{Z}^n$.Let $W\in\mathcal{DT}(\mathbb{T}^n)$. For each $k\in\mathbb{Z}^n$ let $\Lambda_k$ be the continuous functionals on $\mathcal{AC}(\mathbb{T}^n)$ defined by where the $e_j^*$, $j\in\mathbb{Z}^n$, are the functionals defined in (4.3) and the series is weak-$*$ convergent on $(\mathcal{AC}(\mathbb{T}^n))^*$. Since $\|\mathrm{Conv}_{\Lambda_k}\|=c_k$ (see (4.7)), the series $\sum\widehat{e^{i(k,\,\cdot\,)}}\operatorname{Conv}_{\Lambda_k}$ is absolutely convergent in $\mathcal L(L^2)$, so that there exists a bounded operator $\widetilde W$ such that $\widetilde W=\sum_{k\in\mathbb{Z}^n}\widehat{e^{i(k,\,\cdot\,)}} \operatorname{Conv}_{\Lambda_k}$ and for all $j,m\in\mathbb{Z}^n$
$$
\begin{equation*}
\begin{aligned} \, \widetilde{W}_{j,m} &=\sum_{k\in\mathbb{Z}^n}\bigl\langle e^{i(k,\,\cdot\,)}(\Lambda_k*e^{i(m,\,\cdot\,)}), e^{i(j,\,\cdot\,)}\bigr\rangle \\ &=\sum_{k\in\mathbb{Z}^n}\langle W_{k+m,m}e^{i(k+m,\,\cdot\,)}, e^{i(j,\,\cdot\,)}\rangle=W_{j,m} \end{aligned}
\end{equation*}
\notag
$$
(the second equality holds in view of (5.1) and Example 4 in § 4.1.3). Therefore,
$$
\begin{equation}
W=\sum_{k\in\mathbb{Z}^n}\widehat{e^{i(k,\,\cdot\,)}}\operatorname{Conv}_{\Lambda_k}\!.
\end{equation}
\tag{5.2}
$$
Examples. 1. Let $\lambda=\sum\lambda_ke^*_k\in(\mathcal{AC}(\mathbb{T}^n))^*$. If $W=\mathrm{Conv}_{\lambda}$ (see (4.6)), then for all ${j,k\in\mathbb{Z}^n}$ 2. Let $g\in\mathcal{AC}(\mathbb{T}^n)$. Then $W=\widehat g$ (see (1.1)) is an operator of diagonal type, and we have $\|W\|_{\mathcal{DT}}=\|g\|_{\mathcal{AC}}$. In fact, since $W_{j,k}=g_{j-k}$, we obtain $c_k=|g_k|$, so that 3. It follows directly from the definition that $\mathcal{DT}(\mathbb{T}^n)$ is a linear space with norm $\|\,{\cdot}\,\|_{\mathcal{DT}}$, that is, if $W,G\in\mathcal{DT}(\mathbb{T}^n)$ and $\lambda\in\mathbb{C}$, then
$$
\begin{equation*}
\begin{gathered} \, \lambda W\in\mathcal{DT}, \qquad \|\lambda W\|_{\mathcal{DT}}=|\lambda|\,\| W\|_{\mathcal{DT}}, \\ (W+G)\in\mathcal{DT}, \qquad \|W+G\|_{\mathcal{DT}}\leqslant \|W\|_{\mathcal{DT}}+\|G\|_{\mathcal{DT}}. \end{gathered}
\end{equation*}
\notag
$$
Moreover, if $\|W\|_{\mathcal{DT}}=0$, then $W=0$. 4. If $W\in \mathcal{DT}(\mathbb{T}^n)$, then also $W^*\in \mathcal{DT}(\mathbb{T}^n)$ and we have $\|W^*\|_{\mathcal{DT}}=\|W\|_{\mathcal{DT}}$. In fact, it follows from the equalities $W^*_{j,k}=\overline W_{k,j}$ (for all $j,k\in\mathbb{Z}^n$) that ${c^{*}_k=c_{-k}}$, ${k\in\mathbb{Z}^n}$, where $\{c^{*}_k\}$ and $\{c_k\}$ are the majorizing sequences of $W^{*}$ and $W$, respectively. Proof. Let $\{c'_{k}\}$, $\{c''_k\}$ and $\{c_k\}$ be the majorizing sequences of the operators $W'$, $W''$ and $W'W''$, respectively. Then for all $j,l\in\mathbb{Z}^n$, Therefore, where $\mathbf{c}'=\|W'\|_{\mathcal{DT}}$ and $ \mathbf{c}''=\|W''\|_{\mathcal{DT}}$.
The proof is complete. Proof. Let $\{c_k\}$ be the majorizing sequence of $W$. Then for each function ${f\!\in\! L^2(\mathbb{T}^n)}$ we have
$$
\begin{equation*}
\begin{aligned} \, \|Wf\|^2 &=\sum_{j\in\mathbb{Z}^n}\biggl| \sum_{k\in\mathbb{Z}^n}W_{j,k}f_k\biggr|^2= \sum_{j,k,l\in\mathbb{Z}^n}W_{j,k}\overline W_{j,l}f_k\overline f_l \\ &\leqslant \sum_{j,k,l\in\mathbb{Z}^n}|W_{j,k} W_{j,l}f_k f_l| \leqslant\sum_{k,l\in\mathbb{Z}^n}\sum_{j\in\mathbb{Z}^n}c_{j-k}c_{j-l}|f_k f_l| \\ &= \sum_{k,l\in\mathbb{Z}^n}A_{k-l}|f_k f_l|=\sum_{s,l\in\mathbb{Z}^n}A_{s}|f_{s+l} f_l|, \end{aligned}
\end{equation*}
\notag
$$
where $A_s=\sum_{j\in\mathbb{Z}^n}c_{j}c_{j+s}$. Using the relations $\sum_{s\in\mathbb{Z}^n} A_s= \sum_{s,j\in\mathbb{Z}^n}c_{j}c_{j+s}=\mathbf{c}^2$ and the Cauchy-Schwarz-Bunyakovskii inequality we obtain
$$
\begin{equation*}
\|Wf\|^2\leqslant \sum_{s\in\mathbb{Z}^n}A_s\sum_{l\in\mathbb{Z}^n}|f_{s+l}|\,|f_l|\leqslant \mathbf{c}^2\|f\|^2=\|W\|_{\mathcal{DT}}^2\,\|f\|^2
\end{equation*}
\notag
$$
for any $f\in L^2(\mathbb{T}^n)$.
The proof is complete. With each bounded operator $W$ we associate a sequence $\mathbf{W}$ of elements of $l^{\infty}=l^\infty(\mathbb{Z}^n)$:
$$
\begin{equation}
\mathbf{W}\colon\mathbb{Z}^n\to l^{\infty}, \qquad \mathbf{W}(k)=\{x(k,j)\}_{j\in\mathbb{Z}^n}, \quad x(k,j)=W_{k+j,j}, \quad k,j\in\mathbb{Z}^n.
\end{equation}
\tag{5.3}
$$
Clearly, $W\in\mathcal{DT}(\mathbb{T}^n)$ if and only if $\mathbf{W}\in l^1(\mathbb{Z}^n, l^{\infty})$ (see Definition 5.1). Moreover, if $W$ has the diagonal type, then
$$
\begin{equation*}
\|\mathbf{W}\|_1=\sum_{k\in \mathbb{Z}^n}\|\mathbf{W}(k)\|_{\infty}=\sum_{k\in \mathbb{Z}^n}\sup_{j\in \mathbb{Z}^n}|x(k,j)|=\sum_{k\in \mathbb{Z}^n}c_k=\|W\|_{\mathcal{DT}},
\end{equation*}
\notag
$$
where $\{c_k\}$ is the majorizing sequence of $W$. Thus
$$
\begin{equation*}
\mathcal{J}\colon\mathcal{DT}(\mathbb{T}^n)\to l^1(\mathbb{Z}^n, l^{\infty}), \qquad \mathcal{J}W=\mathbf{W},
\end{equation*}
\notag
$$
is an isometric operator. Lemma 5.3. The normed space $(\mathcal{DT}(\mathbb{T}^n), \|\,{\cdot}\,\|_{\mathcal{DT}})$ is a Banach space. Proof. Let $\{W_m\}_{m\in\mathbb{N}}$ be a Cauchy sequence in $(\mathcal{DT}(\mathbb{T}^n), \|\,{\cdot}\,\|_{\mathcal{DT}})$. Then, as the operator $\mathcal J$ is an isometry, the sequence $\{\mathcal{J}W_m\}_{m\in\mathbb{N}}$ is Cauchy in the space $l^1(\mathbb{Z}^n, l^{\infty})$. Since $l^{\infty}$ is a complete space, the space $l^1(\mathbb{Z}^n, l^{\infty})$ is Banach (see [1], Theorem III.6.6), so there exists $\mathbf{S}\in l^1(\mathbb{Z}^n, l^{\infty})$ such that
$$
\begin{equation*}
\begin{aligned} \, 0&=\lim_{m\to\infty}\|\mathcal{J}W_m-\mathbf{S}\|_1 =\lim_{m\to\infty}\sum_{k\in\mathbb{Z}^n}\|\mathcal{J}W_m(k)-\mathbf{S}(k)\|_{\infty} \\ &=\lim_{m\to\infty}\sum_{k\in \mathbb{Z}^n}\sup_{j\in \mathbb{Z}^n}|x_m(k,j)-s(k,j)|, \end{aligned}
\end{equation*}
\notag
$$
where $x_m(k,j)=(W_m)_{k+j,j}$ and $ \mathbf{S}(k)=\{s(k,j)\}_{j\in\mathbb{Z}^n}$. Then
It follows from Lemma 5.2 that the operators $W_m$, $m\in\mathbb{N}$, form a Cauchy sequence with respect to the operator norm $\|\,{\cdot}\,\|$. Therefore, they converge to a (bounded linear) operator $W$ in this norm. Then $(W_m)_{j,k}\xrightarrow[]{}W_{j,k}$ as $m\to\infty$ for all $j,k\in\mathbb{Z}^n$, and taking (5.4) into account we obtain $\mathbf{W}=\mathbf{S}$, where $\mathbf{W}$ is defined by (5.3). Hence $W\in\mathcal{DT}(\mathbb{T}^n)$, $\mathcal{J}W=\mathbf{W}$, and moreover,
$$
\begin{equation*}
\lim_{m\to\infty}\|W_m-W\|_{\mathcal{DT}}=\lim_{m\to\infty}\|\mathcal{J}W_m-\mathcal{J}W\|_1=0.
\end{equation*}
\notag
$$
The proof is complete. The examples and lemmas above can be combined into the following statement. Theorem 5.1. The space $\mathcal{DT}(\mathbb{T}^n)$ is a unital star Banach algebra with star norm3 $\|\,{\cdot}\,\|_{\mathcal{DT}}$. Here is an example showing that $\mathcal{DT}(\mathbb{T}^n)$ is not a $C^*$ -algebras (as regards $C^*$ -algebras, see [14], Ch. IV, § 7, for instance). Let $n=1$. Consider the operator $W=\widehat g$ from example 2 in this subsection, where It is obvious that $W^*(f)=\overline gf$ and $W^{*}Wf=\overline g gf=|g|^2f$. Since $|g(x)|^2=-e^{-2ix}+3-e^{2ix}$, it follows that
$$
\begin{equation*}
\|W^{*}W\|_{\mathcal{DT}}=5\ne9=\|W\|^2_{\mathcal{DT}}.
\end{equation*}
\notag
$$
5.2. Calculating the $\mu$-normLet $W\in\mathcal{DT}(\mathbb{T}^n)$. For arbitrary $a\in\mathbb{T}^n$ and $k\in\mathbb{Z}^n$ consider the continuous functional $\Lambda_{a,k}=e^{i(k,a)}(\Lambda_k\circ \widehat{e^{i(k,\,\cdot\,)}})$ on $\mathcal{AC}(\mathbb{T}^n)$, where $\Lambda_k$ is defined by (5.1). Since $\|\Lambda_{a,k}\|_{\mathcal{AC}^*}= c_k$ and $\sum_{k\in\mathbb{Z}^n}c_k<\infty$, there exists a functional $L_a\in (\mathcal{AC}(\mathbb{T}^n))^*$ such that
$$
\begin{equation}
L_a=\sum_{k\in\mathbb{Z}^n}\Lambda_{a,k}\quad\text{and} \quad \operatorname{Conv}_{L_a}=\sum_{k\in\mathbb{Z}^n}\operatorname{Conv}_{\Lambda_{a,k}}, \quad a\in\mathbb{T}^n
\end{equation}
\tag{5.5}
$$
(the series converge with respect to the norms $\|\,{\cdot}\,\|_{\mathcal{AC}^*}$ and $\|\,{\cdot}\,\|$). Note that for all $a\in\mathbb{T}^n$ and $l\in\mathbb{Z}^n$ we have
$$
\begin{equation*}
\begin{gathered} \, \|\operatorname{Conv}_{L_a}\|=\|L_a\|_{\mathcal{AC}^*}\leqslant \sum_{k\in\mathbb{Z}^n}\|\Lambda_k\|_{\mathcal{AC}^*}=\sum_{k\in\mathbb{Z}^n}c_k=\mathbf{c}, \\ L_a(e^{-i(l,\,\cdot\,)})=\sum_{k\in\mathbb{Z}^n}e^{i(k,a)}\Lambda_k (e^{i(k-l,\,\cdot\,)})=\sum_{k\in\mathbb{Z}^n}W_{l,l-k}e^{i(k,a)}, \\ \sum_{k\in\mathbb{Z}^n}|W_{l,l-k}|\leqslant\sum_{k\in\mathbb{Z}^n}c_k=\mathbf{c}, \quad\text{where }\ \mathbf{c}=\|W\|_{\mathcal{DT}}. \end{gathered}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
L_a=\sum_{l\in\mathbb{Z}^n}w_l(a)e_l^*, \qquad w_l(a)=\sum_{k\in\mathbb{Z}^n}W_{l,l-k}e^{i(k,a)}, \quad w_l\in\mathcal{AC}(\mathbb{T}^n);
\end{equation}
\tag{5.6}
$$
moreover,
$$
\begin{equation}
|w_l(a)|\leqslant\|w_l\|_{\mathcal{AC}}\leqslant\mathbf{c} \quad\text{for all }\ l\in\mathbb{Z}^n, \quad a\in\mathbb{T}^n.
\end{equation}
\tag{5.7}
$$
Lemma 5.4. Let $W\in\mathcal{DT}(\mathbb{T}^n)$. Then the map $a\mapsto L_a$ from the metric space4 $(\mathbb{T}^n,\mathrm{dist})$ to the normed space $((\mathcal{AC}(\mathbb{T}^n))^*,\|\,{\cdot}\,\|_{\mathcal{AC}^*})$ is continuous. Proof. First we prove that the family of functions $\{w_l\}_{l\in\mathbb{Z}^n}$ is equicontinuous. In fact, it follows from (5.6) that for any pair of points $a,b\in\mathbb{T}^n$ and every $l\in\mathbb{Z}^n$ we have
$$
\begin{equation*}
\begin{aligned} \, |w_l(a)-w_l(b)| &=\biggl|\sum_{k\in\mathbb{Z}^n}W_{l,l-k}(e^{i(k,a)}-e^{i(k,b)})\biggr| \\ &\leqslant \sum_{k\in\mathbb{Z}^n}c_k|e^{i(k,a)}-e^{i(k,b)}|=\Sigma_1+\Sigma_2, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\Sigma_1=\sum_{k\in\mathbf{I}_K}c_k|e^{i(k,a)}-e^{i(k,b)}|, \quad \Sigma_2=\sum_{k\notin\mathbf{I}_K}c_k|e^{i(k,a)}-e^{i(k,b)}|, \qquad K\in\mathbb{N}.
\end{equation*}
\notag
$$
Now we find estimates for $\Sigma_1$ and $\Sigma_2$:
$$
\begin{equation*}
\begin{gathered} \, \Sigma_1\leqslant \mathrm{dist}(a,b)\sum_{k\in\mathbf{I}_K}\|k\|_1c_k\leqslant Kn\mathbf{c}\cdot\mathrm{dist}(a,b), \\ \Sigma_2 \leqslant 2\sigma_0(K), \qquad \sigma_0(K)=\sum_{k\notin\mathbf{I}_K}c_k. \end{gathered}
\end{equation*}
\notag
$$
Hence, if $\mathrm{dist}(a,b)<\gamma$ for $\gamma>0$, then Note that $\sigma_0(K)\to0$ as $K\to\infty$, because $\sum_{k\in\mathbb{Z}^n}c_k=\mathbf{c}<\infty$. Therefore, for each $\varepsilon>0$ there exist $K\in\mathbb{N}$ and $\gamma>0$ such that $2\sigma_0(K)<\varepsilon/{2}$ and $Kn\mathbf{c}^2\gamma<{\varepsilon}/{2}$. Hence if $\mathrm{dist}(a,b)<\gamma$, then Thus the functions $w_l$, $l\in\mathbb{Z}^n$, are equicontinuous.
It follows from (5.6) that for any pair of points $a,b\in\mathbb{T}^n$ we have $\|L_a-L_b\|_{\mathcal{AC}^*}=\sup_{l\in\mathbb{Z}^n}|w_l(a)-w_l(b)|$, and so, as the family of functions $\{w_l\}$ is equicontinuous, the map $a\mapsto L_a$ is continuous. The proof is complete. Consider the functions $a\mapsto\rho(L_a)$ and $a\mapsto\rho_{\mathbf{I}}(L_a)$ given by (4.11):
$$
\begin{equation}
\rho(L_a)=\lim_{N\to\infty}\sup_{\mathbf{I}\in\mathcal{P}_N^n}\rho_{\mathbf{I}}(L_a), \qquad \rho_{\mathbf{I}}(L_a)= \frac{1}{\#\mathbf{I}}\sum_{l\in \mathbf{I}}|w_l(a)|^2, \quad \mathbf{I}\in\mathcal{P}^n.
\end{equation}
\tag{5.8}
$$
It follows from Lemma 5.4 that the set $\{L_a\mid a\in\mathbb{T}^n\}$ is compact in the strong topology of $(\mathcal{AC}(\mathbb{T}^n))^*$. Hence we obtain the following result from Lemma 4.3. Lemma 5.5. Let $W\in\mathcal{DT}(\mathbb{T}^n)$. Then the sequence $\{\sup_{\mathbf{I}\in\mathcal{P}^{n}_{N}}\rho_{\mathbf{I}}(L_a)\}_{N\in\mathbb{N}}$ of continuous functions on $\mathbb{T}^n$ converges uniformly to a function $\rho(L_a)\in C(\mathbb{T}^n)$. Lemma 5.6. Let $W\in\mathcal{DT}(\mathbb{T}^n)$. Then for each $\sigma>0$ there exists $\varepsilon_0=\varepsilon_0(\sigma)>0$ such that for all $\varepsilon\in(0,\varepsilon_0]$, $ a\in\mathbb{T}^n$ and $f\in L^2(\mathbb{T}^n)$ where $Y=B_\varepsilon(a)\subset\mathbb{T}^n$. Proof. Let $\sigma>0$, $\varepsilon>0$, $Y=B_\varepsilon(a)$, $a\in\mathbb{T}^n$, $f\in L^2(\mathbb{T}^n)$ and $g=\widehat{\bf 1}_Yf$. Since $\sum_{k\in\mathbb{Z}^n}c_k<\infty$, there exists $K=K(\sigma)$ such that $\sum_{k\notin\mathbf{I}_K}c_k<\sigma$.
It follows from (5.2) that
$$
\begin{equation*}
\lim_{N\to\infty}\|W_N g-Wg\|=0, \quad\text{where }\ W_N=\sum_{k\in\mathbf{I}_N}\widehat {e^{i(k,\,\cdot\,)}}\operatorname{Conv}_{\Lambda_k}\!.
\end{equation*}
\notag
$$
Then
$$
\begin{equation}
\begin{aligned} \, \nonumber \|Wg-W_Kg\| &=\biggl\| \sum_{k\notin\mathbf{I}_K}e^{i(k,\,\cdot\,)}(\Lambda_k*g)\biggr\|\leqslant \sum_{k\notin\mathbf{I}_K}\|\Lambda_k*g\| \\ &\leqslant \sum_{k\notin\mathbf{I}_K}c_k\|g\|\leqslant\sigma\|f\|. \end{aligned}
\end{equation}
\tag{5.9}
$$
Note that
$$
\begin{equation*}
\begin{gathered} \, W_Kg =\sum_{k\in \mathbf{I}_K}\bigl(e^{i(k,a)}\Lambda_k\widehat {e^{i(k,\,\cdot\,)}}\bigr)*(e^{i(k,\cdot-a)}g)=L_{a,K}*g+\Delta, \\ L_{a,K}=\sum_{k\in\mathbf{I}_K}\Lambda_{a,k}, \qquad \Lambda_{a,k}=e^{i(k,a)}\Lambda_k\widehat {e^{i(k,\,\cdot\,)}}, \\ \Delta=W_Kg-L_{a,K}*g=\sum_{k\in\mathbf{I}_K}\Lambda_{a,k}*((e^{i(k,\cdot-a)}-1)g). \end{gathered}
\end{equation*}
\notag
$$
Since $\|\mathrm{Conv}_{\Lambda_{a,k}}\|=\|\Lambda_{a,k}\|_{\mathcal{AC}^*}=c_k$, using (4.8) we obtain
$$
\begin{equation}
\|\Delta\|\leqslant\sum_{k\in\mathbf{I}_K}c_k\|(e^{i(k,\cdot-a)}-1)g\| \leqslant\sum_{k\in\mathbf{I}_K}c_k\|k\|_1\varepsilon\|f\|\leqslant\varepsilon Kn\mathbf{c}\|f\|.
\end{equation}
\tag{5.10}
$$
It follows from (5.5) that $\|L_{a,N}*g-L_a*g\|\to0$ as $N\to\infty$, and thus Setting $\varepsilon_0={\sigma}/(Kn\mathbf{c})$ we arrive at the required result. Proof. It follows from Lemma 5.6 that
Therefore,
$$
\begin{equation*}
\|W\|^2_\mu= \int_{\mathbb{T}^n}\rho(L_a)\,\mu(da)=\frac{1}{(2\pi)^n}\int_{\mathbb{T}^n}\rho(L_a)\,da.
\end{equation*}
\notag
$$
The proof is complete. § 6. $\mu$-norm and the averaged trace6.1. The definition of $\mathbf{T}(W)$Let $W$ be a bounded operator on $L^2(\mathbb{T}^n)$. Set
$$
\begin{equation}
\mathbf{T}(W)=\lim_{M\to\infty}\sup_{\mathbf{I}\in\mathcal{P}^{n}_{M}}\mathbf{T}(\mathbf{I}, W), \qquad \mathbf{T}(\mathbf{I}, W)= \frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I},\,j\in\mathbb{Z}^n}|W_{l,j}|^2.
\end{equation}
\tag{6.1}
$$
Note that
$$
\begin{equation}
\mathbf{T}(\mathbf{I}, W)=\frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I},\,j\in\mathbb{Z}^n}W_{l,j}W^{*}_{j,l}=\frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I}}(WW^*)_{l,l}.
\end{equation}
\tag{6.2}
$$
Proof. Let $\sigma>0$. It follows from Lemma 5.5 and (6.1) that there exist $M=M(\sigma)\in\mathbb{N}$ and $\mathbf{I}\in\mathcal{P}^{n}_M$ such that
$$
\begin{equation*}
\rho(L_a)>\rho_{\mathbf{I}}(L_a)-\frac{\sigma}{2} \quad\text{for all }\ a\in\mathbb{T}^n\quad\text{and} \quad \mathbf{T}(\mathbf{I}, W)>\mathbf{T}(W)-\frac{\sigma}{2}.
\end{equation*}
\notag
$$
Moreover, by (5.6)
$$
\begin{equation*}
|w_l(a)|^2=\sum_{j,k\in\mathbb{Z}^n}W_{l,j}\overline W_{l,k}e^{i(k-j,a)} \quad\text{for all }\ l\in\mathbb{Z}^n, \quad a\in\mathbb{T}^n.
\end{equation*}
\notag
$$
Hence, taking Proposition 5.1 into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \|W\|^2_\mu &=\int_{\mathbb{T}^n}\rho(L_a)\,\mu(da)\geqslant\int_{\mathbb{T}^n} \frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I},\,j, k\in\mathbb{Z}^n}W_{l,j}\overline W_{l,k}e^{i(k-j,a)}\,\mu(da) - \frac{\sigma}{2} \\ &=\mathbf{T}(\mathbf{I}, W)-\frac{\sigma}{2}>\mathbf{T}(W)-\sigma, \end{aligned}
\end{equation*}
\notag
$$
so that, as $\sigma>0$ is arbitrary, we conclude that $\|W\|_{\mu}\geqslant \mathbf{T}(W)$.
The proof is complete. Proposition 6.2. Let $ F\in\mathrm{Aut}(\mathbb{T}^n, \mu)$, and let $g_0,\dots, g_K\in L^{\infty}(\mathbb{T}^n, \mu)$, where $K\in\mathbb{N}$. If $W=\widehat g_K U_F \widehat g_{K-1}\dotsb U_F \widehat g_0$, then Proof. Since $U_F^*=U_{F^{-1}}$ and $(\widehat g_k)^*=\widehat{\overline g}_k$ for $k=0,\dots,K$, it follows that
$$
\begin{equation*}
W^*=\widehat {\overline g}_0 U_{F^{-1}} \widehat {\overline g}_1 \dotsb U_{F^{-1}} \widehat {\overline g}_K
\end{equation*}
\notag
$$
and Set $W_K:=WW^{*}$. We show that for each $f\in L^{2}(\mathbb{T}^n, \mu)$ we have
We carry out the proof using induction on $K\in\mathbb{N}$. 1. The basis of induction. For $K=1$ we have
$$
\begin{equation*}
\begin{aligned} \, W_1(f)&=\widehat g_1 U_F \widehat g_0\widehat {\overline g}_0 U_{F^{-1}} \widehat {\overline g}_1(f) =\widehat g_1 U_F \widehat g_0\widehat {\overline g}_0\bigl((\overline g_1\circ F^{-1})(f\circ F^{-1})\bigr) \\ &=\widehat g_1 U_F(|g_0|^2(\overline g_1\circ F^{-1})(f\circ F^{-1})) =\widehat g_1(|g_0\circ F|^2 \overline g_1 f) \\ &=|g_1|^2\,|g_0\circ F|^2 f. \end{aligned}
\end{equation*}
\notag
$$
2. The induction step. Assume that (6.3) holds for $K$. Then for $K+1$ we have
$$
\begin{equation*}
\begin{aligned} \, W_{K+1}(f)&=\widehat g_{K+1} U_F W_K U_{F^{-1}} \widehat {\overline g}_{K+1}(f) \\ &=\widehat g_{K+1} U_F W_K \bigl(({\overline g}_{K+1}\circ F^{-1})(f\circ F^{-1})\bigr) \\ &=\widehat g_{K+1} U_F\bigl(|g_K|^2\,|g_{K-1}\circ F|^2\dotsb |g_0 \circ F^K|^2 ({\overline g}_{K+1}\circ F^{-1})(f\circ F^{-1}) \bigr) \\ &=\widehat g_{K+1}(|g_K\circ F|^2\,|g_{K-1}\circ F^2|^2\dotsb |g_0 \circ F^{K+1}|^2{\overline g}_{K+1}f) \\ &=|g_{K+1}|^2\,|g_K\circ F|^2|g_{K-1}\circ F^2|^2\dotsb |g_0 \circ F^{K+1}|^2 f, \end{aligned}
\end{equation*}
\notag
$$
where the third equality follows from the inductive assumption for $K$.
It follows from (6.3) that for each $k\in\mathbb{Z}^n$
$$
\begin{equation*}
\begin{aligned} \, (WW^{*})_{k,k} &=\langle WW^{*}(e^{i(k,\,\cdot\,)}),e^{i(k,\,\cdot\,)}\rangle \\ &=\int_{\mathbb{T}^n}|g_K|^2 \,|g_{K-1}\circ F|^2\dotsb |g_0 \circ F^K|^2\,d\mu=\|W\|_{\mu}^2 \end{aligned}
\end{equation*}
\notag
$$
(we prove the last equality below: see Proposition 8.1), so that, in view of (6.2) and (6.1) we obtain $\mathbf{T}(W)=\mathbf{T}(\mathbf{I},W)=\|W\|_{\mu}^2$.
The proof is complete. 6.2. Multiplication by a unitary operator Proof. Set $W'=WU$ and $W''=UW$. Since both $W$ and $U$ have the diagonal type, it follows that $W', W''\in\mathcal{DT}(\mathbb{T}^n)$.
Because $U^{-1}=U^*$, we have Taking (6.2) into account this proves the first equality. Now we prove the second. For each $l\in\mathbb{Z}^n$ we have
$$
\begin{equation*}
\sum_{j\in\mathbb{Z}^n}|W''_{l,j}|^2=\sum_{m,r,j\in\mathbb{Z}^n}U_{l,m}W_{m,j}\overline U_{l,r} \overline W_{r,j}=\sum_{m,r,j\in\mathbb{Z}^n}U_{l,m}U^{-1}_{r,l}W_{m,j}\overline W_{r,j}.
\end{equation*}
\notag
$$
Let $\mathbf{I}\in\mathcal{P}^n$. We represent $\mathbf{T}(\mathbf{I},W)$ as follows:
$$
\begin{equation*}
\mathbf{T}(\mathbf{I},W) =\frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I},\,m,r,j\in\mathbb{Z}^n} \delta_{r,l}\delta_{l,m}W_{m,j}\overline W_{r,j}.
\end{equation*}
\notag
$$
Let $\{c_k\}_{k\in\mathbb{Z}^n}$ and $\{d_k\}_{k\in\mathbb{Z}^n}$ be the majorizing sequences of $W$ and $U$, respectively. Then
$$
\begin{equation*}
\begin{aligned} \, |\mathbf{T}(\mathbf{I},W'')-\mathbf{T}(\mathbf{I},W)| &=\frac{1}{\#\mathbf{I}}\biggl|\sum_{l\in\mathbf{I},\,m,r,j\in\mathbb{Z}^n} (U^{-1}_{r,l}U_{l,m}-\delta_{r,l}\delta_{l,m})W_{m,j}\overline W_{r,j}\biggr| \\ &\leqslant \sum_{m,r,j\in\mathbb{Z}^n}\biggl|\frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I}} (U^{-1}_{r,l}U_{l,m}-\delta_{r,l}\delta_{l,m})\biggr|c_{m-j}c_{r-j}=\Delta, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, \Delta=\sum_{m,r\in\mathbb{Z}^n}\Gamma_{m,r}\widehat c_{m-r}, \\ \Gamma_{m,r}=\biggl|\frac{1}{\#\mathbf{I}} \sum_{l\in\mathbf{I}}(U^{-1}_{r,l}U_{l,m}-\delta_{r,l}\delta_{l,m})\biggr|, \qquad \widehat c_{p}=\sum_{j\in\mathbb{Z}^n}c_{p+j}c_{j}. \end{gathered}
\end{equation*}
\notag
$$
Note that $\sum_{p\in\mathbb{Z}^n}\widehat c_{p}=\mathbf{c}^2$ for $ \mathbf{c}=\sum_{k\in\mathbb{Z}^n}c_k$.
Let $\sigma>0$ be arbitrary. Since $\sum_{k\in\mathbb{Z}^n}d_k=\mathbf{d}<\infty$, there exists $M=M(\sigma)\in\mathbb{N}$ such that where $\mathbf{I}_M=I_M\times\dots\times I_M$ and $I_M=[-M,M]\cap\mathbb{Z}$. Now let $\mathbf{I}=I_1\times\dots\times I_n\in\mathcal{P}_{N}^{n}$, $ N>2M+1$. Set
$$
\begin{equation*}
\mathbf{I}_M^{+}=\{k\in\mathbb{Z}^n\mid(\mathbf{I}_M+k)\cap \mathbf{I}\ne\varnothing \}\quad\text{and} \quad \mathbf{I}_M^{-}=\{k\in\mathbb{Z}^n\mid (\mathbf{I}_M+k)\subset \mathbf{I}\}.
\end{equation*}
\notag
$$
Let $e_{j}=\sum_{m\in\mathbb{Z}^n}d_m\widehat c_{m-j}$, $j\in\mathbb{Z}^n$. Then $\sum_{j\in\mathbb{Z}^n} e_{j}=\mathbf{d}\mathbf{c}^2$.
Note that $\Delta=\Delta_1+\Delta_2+\Delta_3$, where
$$
\begin{equation*}
\begin{aligned} \, \Delta_1 &=\sum_{m\in\mathbb{Z}^n,\, r\notin \mathbf{I}_M^{+}}\Gamma_{m,r}\widehat c_{m-r}, \\ \Delta_2 &=\sum_{m\in\mathbb{Z}^n,\, r\in \mathbf{I}_M^{-}}\Gamma_{m,r}\widehat c_{m-r} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Delta_3 =\sum_{m\in\mathbb{Z}^n,\, r\in \mathbf{I}_M^{+}\setminus \mathbf{I}_M^{-}}\Gamma_{m,r}\widehat c_{m-r}.
\end{equation*}
\notag
$$
We estimate $\Delta_1$ first:
$$
\begin{equation*}
\Delta_1\leqslant \frac{1}{\#\mathbf{I}}\sum_{m\in\mathbb{Z}^n,\, r\notin \mathbf{I}_M^+}\sum_{l\in\mathbf{I}}d_{r-l}d_{m-l}\widehat c_{m-r} =\frac{1}{\#\mathbf{I}}\sum_{r\notin \mathbf{I}_M^+,\, l\in\mathbf{I}}d_{r-l}e_{r-l}\leqslant \mathbf{d}\mathbf{c}^2\sigma.
\end{equation*}
\notag
$$
For an estimate of $\Delta_2$ note that $\sum_{l\in\mathbb{Z}^n}U^{-1}_{n,l}U_{l,m}=\delta_{n, m}=\sum_{l\in\mathbb{Z}^n}\delta_{n, l}\delta_{l, m}$. Then
$$
\begin{equation*}
\begin{aligned} \, \Delta_2&=\sum_{m\in\mathbb{Z}^n,\, r\in\mathbf{I}_M^-}\biggl|\frac{1}{\#\mathbf{I}} \sum_{l\in\mathbf{I}}(U^{-1}_{r,l}U_{l,m}-\delta_{r,l}\delta_{l,m})\biggr|\widehat c_{m-r} \\ &=\frac{1}{\#\mathbf{I}}\sum_{m\in\mathbb{Z}^n, \,r\in\mathbf{I}_M^-} \biggl|\sum_{l\in\mathbb{Z}^n \setminus\mathbf{I}}U^{-1}_{r,l}U_{l,m}\biggr|\widehat c_{m-r} \\ &\leqslant\frac{1}{\#\mathbf{I}}\sum_{m\in\mathbb{Z}^n,\, r\in\mathbf{I}_M^-,\, l\in\mathbb{Z}^n\setminus\mathbf{I}}d_{r-l}d_{m-l}\widehat c_{m-r} \\ &=\frac{1}{\#\mathbf{I}}\sum_{r\in\mathbf{I}_M^-,\, l\in\mathbb{Z}^n\setminus\mathbf{I}}d_{r-l}e_{r-l} \leqslant\sum_{m\in\mathbb{Z}^n\setminus\mathbf{I}_M}d_me_m \leqslant\mathbf{d}\mathbf{c}^2\sigma. \end{aligned}
\end{equation*}
\notag
$$
Finally, we estimate $\Delta_3$:
$$
\begin{equation*}
\begin{aligned} \, \Delta_3&\leqslant \frac{1}{\#\mathbf{I}}\sum_{m\in\mathbb{Z}^n,\, r\in \mathbf{I}_M^+\setminus\mathbf{I}_M^-}\sum_{l\in\mathbf{I}}(d_{r-l}d_{m-l} +\delta_{r,l}\delta_{l,m})\widehat c_{m-r} \\ &\leqslant \frac{1}{\#\mathbf{I}}\sum_{r\in \mathbf{I}_M^+\setminus\mathbf{I}_M^-,\, l\in\mathbf{I}}(d_{r-l}e_{r-l}+\delta_{r,l}\widehat c_{l-r}) \\ &\leqslant \frac{1}{\#\mathbf{I}}\sum_{r\in \mathbf{I}_M^+\setminus\mathbf{I}_M^-}(\mathbf{d}\mathbf{c}^2+\mathbf{c}^2)\leqslant \frac{n2^{n+1}M}{N}(\mathbf{d}^2+1)\mathbf{c}^2, \end{aligned}
\end{equation*}
\notag
$$
where the last inequality follows from the relation
$$
\begin{equation*}
\mathbf{I}_M^+\setminus\mathbf{I}_M^-=\bigcup_{q=1}^{n}H_q,
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\begin{gathered} \, H_q=\{k=(k_1,\dots,k_n)\in\mathbf{I}_{M}^{+}\mid (I_M+k_q)\nsubseteq I_q \}, \\ \frac{|H_q|}{\#\mathbf{I}}\leqslant\frac{4M}{\# I_q}\prod_{j=1,\,j\ne q}^{n}\biggl(1+\frac{2M}{\# I_j}\biggr)\leqslant\frac{2^{n+1}M}{N} \end{gathered}
\end{equation*}
\notag
$$
for $q=1,\dots,n$.
We have thus proved that for each $\sigma>0$ there exists a positive integer $M=M(\sigma)$ such that the following inequality holds for all $N>2M+1$ and $\mathbf{I}\in\mathcal{P}^{n}_{N}$:
$$
\begin{equation*}
|\mathbf{T}(\mathbf{I},W'')-\mathbf{T}(\mathbf{I},W)|\leqslant 2\mathbf{d}\mathbf{c}^2\sigma+\frac{n2^{n+1}M}{N}(\mathbf{d}^2+1)\mathbf{c}^2.
\end{equation*}
\notag
$$
Taking here the supremum over $\mathbf{I}\in \mathcal{P}^{n}_{N}$ and the limit as $N\to\infty$ we obtain
$$
\begin{equation*}
|\mathbf{T}(W'')-\mathbf{T}(W)|\leqslant 2\mathbf{d}\mathbf{c}^2\sigma
\end{equation*}
\notag
$$
for any $\sigma>0$. Hence $\mathbf{T}( W'')=\mathbf{T}(W)$.
Proposition 6.3 is proved. § 7. Regular operators7.1. Nets in metric spacesIn this subsection we present some of the main definitions and statements concerning nets in metric spaces (see [6], § 1.9, (i), [11], § I.7, and [12], § 1.6). A partially ordered set $(\Sigma,\leqslant)$ is said to be directed if for any $\sigma_1,\sigma_2\in \Sigma$ there exists $\sigma_3\in\Sigma$ such that $\sigma_1\leqslant \sigma_3$ and $\sigma_2\leqslant \sigma_3$. A map $f\colon \Sigma\to X$ of the directed set $\Sigma$ to a set $X$ is called a net in $X$. In what follows, in place of $f$ we write $\{x_\sigma\}_{\sigma\in \Sigma}$, where $x_\sigma=f(\sigma)$. We say that a net $\{x_\sigma\}_{\sigma\in \Sigma}$ in a metric space $(X,\rho)$ converges towards $x\in X$ if for each $\varepsilon>0$ there exists an index $\sigma_0$ such that $\rho(x_\sigma,x)<\varepsilon$ for all $\sigma\in \Sigma$ with $\sigma_0\leqslant \sigma$. The notation: $\lim_{\sigma}x_\sigma=x$. We introduce an order relation $\preceq$ on $\mathbb{R}_{>0}$ as follows:
$$
\begin{equation*}
\varepsilon_1\preceq\varepsilon_2 \quad\text{if } \varepsilon_1\geqslant\varepsilon_2, \quad \varepsilon_1,\varepsilon_2\in\mathbb{R}_{>0}.
\end{equation*}
\notag
$$
We denote the limit of a net $\{x_\varepsilon\}_\varepsilon\colon (\mathbb{R}_{>0},\preceq)\to (X,\rho)$ in the metric space $(X,\rho)$ (if it exists) by $\lim_{\varepsilon\searrow0}x_\varepsilon$. For example, in § 2, item 10, $\vartheta(x)$ is the limit of the net $\bigl\{\|W \widehat{\bf 1}_{B_\varepsilon(x)}\|^2\bigr\}_{\varepsilon}$. A net $\{x_\sigma\}_{\sigma\in \Sigma}$ in the metric space $(X,\rho)$ is said to be Cauchy if for each $\varepsilon>0$ there exists $\sigma_0\in \Sigma$ such that $\rho(x_{\sigma'},x_{\sigma''})<\varepsilon$ for $\sigma_0\leqslant \sigma'$ and $\sigma_0\leqslant \sigma''$. If $\{x_\sigma\}_{\sigma\in \Sigma}$ is a bounded net in $\mathbb{R}$, then the net $\{\sup_{\xi\geqslant \sigma}x_{\xi}\}_{\sigma\in \Sigma}$ has a limit equal to $\inf_{\sigma\in \Sigma}\sup_{\xi\geqslant \sigma}x_{\xi}$. It is called the upper limit of $\{x_\sigma\}_{\sigma\in \Sigma}$ and denoted by $\limsup_{\sigma}x_\sigma$. Note that if $\lim_{\sigma}x_\sigma=x$, then $\limsup_{\sigma}x_\sigma=x$. Lemma 7.1 (for instance, see [11], Lemma I.7.5). Each Cauchy net in a complete metric space has a limit. Lemma 7.2 (see [11], Lemma I.7.6). Let $\Sigma$ and $ \Gamma$ be directed sets, let $X$ be a complete metric space, and assume that a map $h\colon \Sigma\times \Gamma\to X$ satisfies two conditions: Then the limits exist and coincide.Corollary 7.1. Let $(E,\|\,{\cdot}\,\|)$ be a Banach space and $S$ be a nonempty set. Consider a map $a\colon S\times\mathbb{Z}^n\to E$ such that $\|a(s,k)\|\leqslant b_k$ for all $s\in S$ and $ k\in\mathbb{Z}^n$. Assume that the $n$-fold series $\sum_{k\in\mathbb{Z}^n}b_k$ is convergent. Then the series of functions converge uniformly on $S$. Moreover, if $S$ is a directed set and for each $k\in\mathbb{Z}^n$ the limit $\lim_{s}a(s,k)=a_k$ exists, then the multiple series $\sum_{k\in\mathbb{Z}^n}a_k$ is absolutely convergent, and furthermore, 7.2. A partial order on $\mathcal{P}^n$We define a partial order relation $\leqslant$ on the set $\mathcal{P}^n$ (see § 4.1.5) by setting
$$
\begin{equation*}
I_1\times\dots\times I_n\leqslant J_1\times\dots\times J_n \quad \text{if }\ \#I_k\leqslant \#J_k \quad\text{for each}\ k=1,\dots,n.
\end{equation*}
\notag
$$
It is clear that $(\mathcal{P}^n,\leqslant)$ is a directed set. Lemma 7.3. Let $\{x_{\mathbf{I}}\}_{\mathbf{I}\in\mathcal{P}^n}$ be a net in $\mathbb{C}$. Then the following hold. 7.3. The definition of a regular operatorDefinition 7.1. We call $W\in\mathcal{DT}(\mathbb{T}^n)$ a regular operator if the following limit exists for any $m,k\in\mathbb{Z}^n$: Let $\mathcal{R}(\mathbb{T}^n)$ denote the set of regular operators. Note that
$$
\begin{equation*}
\omega_{\mathbf{I},0,0}=\mathbf{T}(\mathbf{I},W), \quad\omega_{0,0}=\mathbf{T}(W) \quad\text{for }\ W\in\mathcal{R}(\mathbb{T}^n).
\end{equation*}
\notag
$$
Proof. Since the measure $\#$ on $\mathbb{Z}^n$ is translation invariant, for each $\mathbf{I}\in\mathcal{P}^n$ we have $\omega_{\mathbf{I},m,k}=\overline\omega_{\mathbf{I}+m,-m,-k}$. This yields the required result. Lemma 7.5. Let $W\in\mathcal{R}(\mathbb{T}^n)$. Then for each $m\in\mathbb{Z}^n$ the series $\sum_{k\in\mathbb{Z}^n}\omega_{m,k}$ is absolutely convergent, and moreover, Proof. Let $m\in\mathbb{Z}^n$. From the relations
$$
\begin{equation*}
|\omega_{\mathbf{I},m,k}|\leqslant\frac{1}{\# \mathbf{I}}\sum_{l\in\mathbf{I},\,j\in\mathbb{Z}^n}c_{l+m-j}c_{l-j-k} =\sum_{j\in\mathbb{Z}^n}c_{j+m}c_{j-k}, \qquad\sum_{k,j\in\mathbb{Z}^n}c_{j+m}c_{j-k}=\mathbf{c}^2
\end{equation*}
\notag
$$
and Corollary 7.1 we obtain that the series $\sum_{k\in\mathbb{Z}^n}|\omega_{\mathbf{I},m,k}|$ and $\sum_{k\in\mathbb{Z}^n}\omega_{\mathbf{I},m,k}$ are uniformly convergent on $\mathcal{P}^n$; furthermore, and
The lemma is proved. Let $W\in\mathcal R(\mathbb{T}^n)$, $\mathbf{I}\in\mathcal{P}^n$ and $m\in\mathbb{Z}^n$. For each point $a\in\mathbb{T}^n$ set
$$
\begin{equation}
v_{\mathbf{I},m}(a)= \sum_{l\in \mathbf{I}} \frac{w_{l+m}(a) \overline w_l(a)}{\# \mathbf{I}}
\end{equation}
\tag{7.2}
$$
and
$$
\begin{equation}
v_m(a) = \sum_{k\in\mathbb{Z}^n} \omega_{m,k} e^{i(m+k, a)},
\end{equation}
\tag{7.3}
$$
where the functions $w_l$ are defined in (5.6). Note that $v_{\mathbf{I},m}\in\mathcal{AC}(\mathbb{T}^n)$ and $v_{m}\in\mathcal{AC}(\mathbb{T}^n)$, because $w_l\in \mathcal{AC}(\mathbb{T}^n)$ for each $l\in\mathbb{Z}^n$ and by Lemma 7.5. Proof. Let $m\in\mathbb{Z}^n$, $\mathbf{I}\in\mathcal{P}^n$ and $a\in\mathbb{T}^n$. It follows from (7.2) and (5.6) that
$$
\begin{equation*}
v_{\mathbf{I},m}(a)=\frac1{\#\mathbf{I}} \sum_{l\in \mathbf{I},\,k,j\in\mathbb{Z}^n}W_{l+m,l+m-k+j} \overline W_{l,l+j} e^{i(k, a)}=\sum_{k\in\mathbb{Z}^n}\omega_{\mathbf{I},m,k-m} e^{i(k, a)}.
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
\begin{aligned} \, \lim_{\mathbf{I}}\|v_{\mathbf{I},m}-v_m\|_{\mathcal{AC}} &=\lim_{\mathbf{I}}\sum_{k\in\mathbb{Z}^n}|\omega_{\mathbf{I},m,k-m}-\omega_{m,k-m}| \\ &=\sum_{k\in\mathbb{Z}^n}\lim_{\mathbf{I}}|\omega_{\mathbf{I},m,k}- \omega_{m,k}|=0. \end{aligned}
\end{equation*}
\notag
$$
The proof is complete, Note that
$$
\begin{equation}
| v_{\mathbf{I},m}(a)|\leqslant\mathbf{c}^2\quad\text{and} \quad | v_{m}(a)|\leqslant\mathbf{c}^2
\end{equation}
\tag{7.4}
$$
for all $\mathbf{I}\in\mathcal{P}^n$, $m\in\mathbb{Z}^n$ and $a\in\mathbb{T}^n$. 7.4. Examples of regular operatorsLemma 7.7. If $g\in\mathcal{AC}(\mathbb{T}^n)$, then $\widehat g\in\mathcal R(\mathbb{T}^n)$, and moreover, Proof. Let $\{g_k\}_{k\in\mathbb{Z}^n}$ and $\{\mathbf{g}_k\}_{k\in\mathbb{Z}^n}$ be the sequences of Fourier coefficients of $g$ and $|g|^2$, respectively. Note that
$$
\begin{equation*}
\mathbf{g}_k=\langle g,ge^{i(k,\,\cdot\,)}\rangle=\sum_{j\in\mathbb{Z}^n}g_j\overline g_{j-k} \quad\text{for each }\ k\in\mathbb{Z}^n.
\end{equation*}
\notag
$$
Then, taking arbitrary $m,k\in\mathbb{Z}^n$, for each $\mathbf{I}\in\mathcal P^n$ we obtain
$$
\begin{equation*}
\omega_{\mathbf{I},m,k}=\frac{1}{\# \mathbf{I}} \sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n}g_{l+m-j}\overline g_{l-j-k} =\sum_{j\in\mathbb{Z}^n}g_{j+m}\overline g_{j-k}=\mathbf{g}_{m+k}.
\end{equation*}
\notag
$$
Hence there exists a limit $\omega_{m,k}=\lim_{\mathbf{I}}\omega_{\mathbf{I},m,k}=\mathbf{g}_{m+k}$. This means that the operator $\widehat g$ is regular. Moreover, by (7.3) for each point $a\in\mathbb{T}^n$.
The proof is complete. Let $\lambda=\sum\lambda_ke_k^*\in(\mathcal{AC}(\mathbb{T}^n))^*$ and $W=\operatorname{Conv}_\lambda$. Since $W_{j,k}=\delta_{j,k}\lambda_k$, it follows that
$$
\begin{equation}
\omega_{\mathbf{I},m,k}=\frac{1}{\# \mathbf{I}} \sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n} \delta_{l+m,j}\delta_{l,j+k}\lambda_j \overline \lambda_{j+k}=\frac{\delta_{m,-k}}{\# \mathbf{I}} \sum_{l\in \mathbf{I}}\lambda_{l+m}\overline\lambda_l
\end{equation}
\tag{7.5}
$$
for all $\mathbf{I}\in\mathcal P^n$ and $m,k\in\mathbb{Z}^n$. Lemma 7.8. For each point $x\in\mathbb{T}^n$ the operator $\operatorname{Conv}_{\delta_x}$ is in the class $\mathcal R(\mathbb{T}^n)$; moreover, Proof. Since $(\delta_x)_k=\delta_x(e^{-i(k,\,\cdot\,)})=e^{-i(k,x)}$ for each $k\,{\in}\,\mathbb{Z}^n$, taking (7.5) (for ${\lambda\,{=}\,\delta_x}$) into account we obtain $\omega_{\mathbf{I},m,k}=\delta_{m,-k}e^{-i(m,x)}$ for all $\mathbf{I}\in\mathcal P^n$ and $m,k\in\mathbb{Z}^n$. Hence the limit
$$
\begin{equation*}
\omega_{m,k}=\lim_{\mathbf{I}}\omega_{\mathbf{I},m,k}=\delta_{m,-k}e^{-i(m,x)}
\end{equation*}
\notag
$$
exists, that is, in accordance with Definition 7.1, $\operatorname{Conv}_{\delta_x}\in\mathcal R(\mathbb{T}^n)$. Now, $v_m(a)=e^{-i(m,x)}$ for each $a\in\mathbb{T}^n$ by (7.3).
The proof is complete. In the next lemma we consider a convolution operator of special form in the case of the circle ($n=1$). Lemma 7.9. Let $t\in\mathbb{R}$ and let $ \lambda_k=e^{itk^2}$ for $k\in\mathbb{Z}$ and $\lambda=\sum_{k\in\mathbb{Z}}\lambda_ke^*_k$. Then $\operatorname{Conv}_\lambda\in\mathcal R(\mathbb{T})$, and moreover, Proof. It follows from (7.5) that for all $I\in\mathcal P$ and $m,k\in\mathbb{Z}$. Hence for all $m,k\in\mathbb{Z}$ the limit exists. This yields the required result. 7.5. The closedness of $\mathcal{R}( {\mathbb{T}}^n)$ with respect to the norm $\|\,{\cdot}\,\|_{\mathcal{DT}}$If $W\in\mathcal{R}(\mathbb{T}^n)$, then for each $\lambda\in\mathbb{C}$ the operator $W'=\lambda W$ is regular because
$$
\begin{equation*}
\omega'_{\mathbf{I},m,k}=|\lambda|^2\omega_{\mathbf{I},m,k}\quad\text{and} \quad \omega'_{m,k}=|\lambda|^2\omega_{m,k} \quad\text{for all }\ m,k\in\mathbb{Z}^n, \quad \mathbf{I}\in\mathcal{P}^n,
\end{equation*}
\notag
$$
where the quantities $\omega'_{\mathbf{I},m,k}$ and $\omega'_{m,k}$ are defined in (7.1). Hence the regular operators make up a cone in $\mathcal{DT}(\mathbb{T}^n)$. Lemma 7.10. The cone $\mathcal{R}(\mathbb{T}^n)$ is closed in the Banach space $\mathcal{DT}(\mathbb{T}^n)$. Proof. Assume that a sequence of operators $\{W_p\}_{p\in\mathbb{N}}$ in $\mathcal{R}(\mathbb{T}^n)$ converges to an operator $W\in\mathcal{DT}(\mathbb{T}^n)$ in the norm $\|\,{\cdot}\,\|_{\mathcal{DT}}$. Fix arbitrary $m,k\in\mathbb{Z}^n$. If we can prove that
$$
\begin{equation}
\lim_{p\to\infty}\sup_{\mathbf{I}\in\mathcal{P}^n}| (\omega_p)_{\mathbf{I},m,k} - \omega_{\mathbf{I},m,k}|=0,
\end{equation}
\tag{7.6}
$$
then $W$ turns out to be regular. For suppose that (7.6) holds, that is, the sequence $\{(\omega_p)_{\mathbf{I},m,k}\}_{p\in\mathbb{N}}$ converges to $\omega_{\mathbf{I},m,k}$ uniformly on $\mathcal{P}^n$. For each $p\in\mathbb{N}$ the limit $\lim_{\mathbf{I}}(\omega_p)_{\mathbf{I},m,k}=(\omega_p)_{m,k}$ exists, and therefore, by Lemma 7.2 the repeated limits $\lim_{p\to\infty}(\omega_p)_{m,k}$ and $\lim_{\mathbf{I}}\omega_{\mathbf{I},m,k}$ exist and coincide. Thus, for any $m$ and $k$ the net $\{\omega_{\mathbf{I},m,k}\}_{\mathbf{I}\in\mathcal{P}^n}$ has a limit, which means that $W$ is a regular operator.
Now we prove (7.6). Consider some $\varepsilon>0$. Since
$$
\begin{equation*}
\|W_p-W\|_{\mathcal{DT}}\xrightarrow[p\to\infty]{}0\quad\text{and} \quad \bigl|\|W_p\|_{\mathcal{DT}}-\|W\|_{\mathcal{DT}}\bigr|\xrightarrow[p\to\infty]{}0,
\end{equation*}
\notag
$$
there exists $N=N(\varepsilon)\in\mathbb{N}$ such that if $p>N$, then
$$
\begin{equation}
\|W_p-W\|_{\mathcal{DT}}<\varepsilon\quad\text{and} \quad \bigl|\|W_p\|_{\mathcal{DT}}-\|W\|_{\mathcal{DT}}\bigr|<\varepsilon.
\end{equation}
\tag{7.7}
$$
It follows from (7.1) that for each $\mathbf{I}\in\mathcal P^n$
$$
\begin{equation}
\begin{gathered} \, \begin{split} |(\omega_p)_{\mathbf{I},m,k} - \omega_{\mathbf{I},m,k}| &=\frac1{\# \mathbf{I}}\biggl| \sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n} ((W_p)_{l+m,j} (\overline W_p)_{l,j+k}- W_{l+m,j} \overline W_{l,j+k})\biggr| \\ &\leqslant \frac1{\# \mathbf{I}}(\Sigma_1 + \Sigma_2), \end{split} \\ \nonumber \Sigma_1=\sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n} |(W_p)_{l+m,j} ( (\overline W_p)_{l,j+k} - \overline W_{l,j+k})|, \\ \nonumber \Sigma_2=\sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n} |( (W_p)_{l+m,j} - W_{l+m,j}) \overline W_{l,j+k}|. \end{gathered}
\end{equation}
\tag{7.8}
$$
Set
$$
\begin{equation*}
(c_p)_k=\sup_{j\in\mathbb{Z}^n} |(W_p)_{k+j,j}|, \qquad c_k=\sup_{j\in\mathbb{Z}^n} |W_{k+j,j}|
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
d_k=\sup_{j\in\mathbb{Z}^n} |(W_p)_{k+j,j} - W_{k+j,j}|,
\end{equation*}
\notag
$$
so that $\{(c_p)_k\}$, $\{c_k\}$ and$\{d_k\}$ are the majorizing sequences of $W_p$, $W$ and $W_p-W$, respectively. Now, taking (7.7) into account we obtain
$$
\begin{equation*}
\sum_{k\in\mathbb{Z}^n}d_k=\|W_p-W\|_{\mathcal{DT}}<\varepsilon\quad\text{and} \quad \sum_{k\in\mathbb{Z}^n}(c_p)_k=\|W_p\|_{\mathcal{DT}}<\varepsilon +\mathbf{c},
\end{equation*}
\notag
$$
where $p>N$ and $\mathbf{c}=\|W\|_{\mathcal{DT}}$. Therefore,
$$
\begin{equation*}
\Sigma_1\leqslant \sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n} (c_p)_{l+m-j} d_{l-j-k}\leqslant \# \mathbf{I} (\varepsilon +\mathbf{c}) \varepsilon
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\Sigma_2\leqslant \sum_{l\in \mathbf{I},\,j\in\mathbb{Z}^n} d_{l+m-j} c_{l-j-k}\leqslant \# \mathbf{I} \mathbf{c} \varepsilon.
\end{equation*}
\notag
$$
In view of (7.8) this yields
$$
\begin{equation*}
| (\omega_p)_{\mathbf{I},m,n} - \omega_{\mathbf{I},m,n} |<(2\mathbf{c}+\varepsilon)\varepsilon \quad\text{for all }\ p>N(\varepsilon), \qquad \mathbf{I}\in\mathcal{P}^n.
\end{equation*}
\notag
$$
Thus (7.6) holds.
Lemma 7.10 is proved 7.6. The $\mu$-norm of a regular operator Proof. Let $a\in\mathbb{T}^n$. It follows from (5.8) and (7.2) that $\rho_{\mathbf{I}}(L_a)=v_{\mathbf{I},0}(a)$ for each ${\mathbf{I}\in\mathcal{P}^n}$. Moreover, by Lemma 7.6 Hence $v_0(a)=\rho(L_a)$.
For each $\mathbf{I}\in\mathcal P^n$
$$
\begin{equation*}
\begin{aligned} \, \biggl|\int_{\mathbb{T}^n}v_{\mathbf{I},0}(a)\,\mu(da) -\int_{\mathbb{T}^n}v_{0}(a)\,\mu(da)\biggr| &\leqslant \int_{\mathbb{T}^n}|v_{\mathbf{I},0}(a)-v_{0}(a)|\,\mu(da) \\ &\leqslant \|v_{\mathbf{I},0}-v_0\|_{C}\leqslant \|v_{\mathbf{I},0}-v_0\|_{\mathcal{AC}}. \end{aligned}
\end{equation*}
\notag
$$
Then using Proposition 5.1 we obtain
$$
\begin{equation*}
\begin{aligned} \, \|W\|_{\mu}^2 &=\int_{\mathbb{T}^n}v_{0}(a)\,\mu(da) =\lim_{\mathbf{I}}\int_{\mathbb{T}^n}v_{\mathbf{I},0}(a)\,\mu(da) \\ &=\lim_{\mathbf{I}}\int_{\mathbb{T}^n}\sum_{k\in\mathbb{Z}^n}\omega_{\mathbf{I},0,k} e^{i(k, a)}\,\mu(da) =\lim_{\mathbf{I}}\omega_{\mathbf{I},0,0}=\omega_{0,0}=\mathbf{T}(W). \end{aligned}
\end{equation*}
\notag
$$
The proof is complete. Proposition 6.3 and Lemma 7.9 have the following consequence. Corollary 7.2. Let $W,U\in\mathcal{DT}(\mathbb{T}^n)$, where $U$ is a unitary operator and both $W$ and $WU$ are regular. Then § 8. Measure associated with an operator8.1. Transition measuresWe give the definition of a transition measure and state a lemma which is used in what follows (see [5], vol. 2, § 10.7). Let $(\mathcal X_1,\mathcal B_1)$ and $(\mathcal X_2,\mathcal B_2)$ be arbitrary measurable spaces. A function $P(\,\cdot\,{,}\,\cdot\,)$: $\mathcal X_1\times\mathcal B_2\to\mathbb{R}$ is called a transition measure for this pair of spaces if the following conditions are satisfied:
A transition measure $P(\,\cdot\,{,}\,\cdot\,)$ is called a transition probability if for each $x\in\mathcal X_1$ the measure $ P(x,\,\cdot\,)$ is a probability measure. Theorem 8.1 (see [5], Theorem 10.7.2). Let $ P(\,\cdot\,{,}\,\cdot\,)$ be a transition probability for the spaces $(\mathcal X_1,\mathcal B_1)$ and $(\mathcal X_2,\mathcal B_2)$, and let $\mu$ be a probability measure on $\mathcal B_1$. Then there exists a probability measure $\nu$ on $(\mathcal X_1\times\mathcal X_2,\mathcal B_1\otimes\mathcal B_2)$ such that In addition, if $f=f(x_1,x_2)$ is a $\nu$-integrable function, then for $\mu$-almost all $x_1\in\mathcal X_1$ the function $x_2\mapsto f(x_1,x_2)$ is integrable with respect to $P(x_1,\,\cdot\,)$, the function $\displaystyle x_1\mapsto\int_{\mathcal X_2}f(x_1,x_2)\,P(x_1,dx_2)$ is integrable with respect to $\mu$, and 8.2. A Koopman operatorLet $(\mathcal{X}, \mathcal{B}, \mu)$ be a probability space. Proposition 8.1. Let $F\in\mathrm{Aut}(\mathcal{X}, \mu)$ and $g_0,\dots, g_K\in L^{\infty}(\mathcal{X}, \mu)$, where $K\in \mathbb{N}$. Then Proof. Set We have to prove that $\|W\|_{\mu}=\|g\|$. Consider several cases.
(a) If $g_k=\mathbf{1}_{Y_k}$ and $Y_k\in\mathcal{B}$ for each $k=0,\dots,K$, then (8.1) follows from § 2, items 2 and 4 (see also the proof of Corollary 2.5 in [28]). (b) Now let $g_k$, $k=0,\dots,K$, be simple $\mathcal B$-measurable functions. Then there exists a partition $\{Y_1,\dots, Y_J\}$ of $\mathcal{X}$ such that $g_k\,{=}\sum_{j=1}^{J}g_{k, j}\mathbf{1}_{Y_j}$. Applying (2.1) we obtain
$$
\begin{equation*}
W=\sum_{j_0,\dots,j_K} g_{K,j_{K}} g_{K-1,j_{K-1}} \dotsb g_{0,j_{0}} \widehat{\bf 1}_{Y_{j_{K}}} \widehat{\bf 1}_{F^{-1}(Y_{j_{K-1}})} \dotsb \widehat{\bf 1}_{F^{-K}(Y_{j_{0}})} U_F^K.
\end{equation*}
\notag
$$
We set $Y_{j_K,\dots,j_0} :=Y_{j_K} \cap F^{-1}(Y_{j_{K-1}}) \cap \dots \cap F^{-K}(Y_{j_0})$. Since the $ Y_{j_K,\dots,j_0}$ make up a partition of $\mathcal{X}$, using properties (2.2) and (2.3) of the seminorm $\|\,{\cdot}\,\|_{\mu}$ we obtain
$$
\begin{equation*}
\begin{aligned} \, \|W\|_{\mu}^2 &=\|WU_{F^{-K}}\|_{\mu}^2=\biggl\|\sum_{j_0,\dots,j_K} g_{K,j_K} g_{K-1,j_{K-1}} \dotsb g_{0,j_0} \widehat{\bf 1}_{Y_{j_K,\dots,j_0}}\biggr\|_{\mu}^2 \\ &=\sum_{j_0,\dots,j_K} |g_{K,j_K}|^2\, |g_{K-1,j_{K-1}}|^2 \dotsb |g_{0,j_0}|^2\,\| \widehat{\bf 1}_{Y_{j_K,\dots,j_0}}\|_{\mu}^2 \\ &=\sum_{j_0,\dots,j_K} |g_{K,j_K}|^2 \,|g_{K-1,j_{K-1}}|^2 \dotsb |g_{0,j_0}|^2 \mu({Y_{j_K,\dots,j_0}}). \end{aligned}
\end{equation*}
\notag
$$
On the other hand, since $g=\sum_{j_0,\dots,j_K} g_{K,j_K} g_{K-1,j_{K-1}} \dotsb g_{0,j_0}\mathbf{1}_{Y_{j_K,\dots,j_0}}$ and the system of functions $\bigl\{\mathbf{1}_{Y_{j_K,\dots,j_0}}\bigr\}$ in $L^2(\mathcal{X}, \mu)$ is orthogonal, we have
$$
\begin{equation*}
\begin{aligned} \, \|g\|^2&= \sum_{j_0,\dots,j_K} |g_{K,j_K}|^2\, |g_{K-1,j_{K-1}}|^2 \dotsb |g_{0,j_0}|^2\,\|\mathbf{1}_{Y_{j_K,\dots,j_0}}\|^2 \\ &=\sum_{j_0,\dots,j_K} |g_{K,j_K}|^2 \,|g_{K-1,j_{K-1}}|^2 \dotsb |g_{0,j_0}|^2 \mu({Y_{j_K,\dots,j_0}}). \end{aligned}
\end{equation*}
\notag
$$
Thus, $\|W\|_{\mu}^2=\|g\|^2$.
(c) Consider the general case: $\{g_k\}\subset L^{\infty}(\mathcal{X}, \mu)$. For each $k=0, \dots, K$ there exists a sequence of simple $\mathcal B$-measurable functions $\{\varphi_{k,m}\}_{m\in\mathbb{N}}$ such that $\varphi_{k,m}\xrightarrow[]{L^{\infty}}g_k$ as $m\to\infty$. Then
$$
\begin{equation*}
\lim_{m\to\infty}\|\varphi_{k,m}\circ F^{K-k}-g_k\circ F^{K-k}\|_\infty=0\quad\text{and} \quad \lim_{m\to\infty}\|\widehat {\varphi}_{k,m}-\widehat g_k\|=0
\end{equation*}
\notag
$$
for all $k=0,\dots,K$. Since the spaces $L^{\infty}(\mathcal{X}, \mu)$ and $\mathcal L(\mathcal H)$ (the set of bounded operators on $\mathcal H=L^2(\mathcal X,\mu)$) are Banach algebras, we have
$$
\begin{equation*}
\begin{gathered} \, \lim_{m\to\infty}\bigl\|\varphi_{K,m}(\varphi_{K-1,m}\circ F)\dotsb (\varphi _{0,m} \circ F^K)-g_K (g_{K-1}\circ F)\dotsb (g_0 \circ F^K)\bigr\|_\infty=0, \\ \lim_{m\to\infty}\bigl\|\widehat \varphi_{K,m} U_F \widehat \varphi_{K-1,m}\dotsb U_F \widehat \varphi_{0,m}-\widehat g_K U_F \widehat g_{K-1}\dotsb U_F \widehat g_0\bigr\|=0. \end{gathered}
\end{equation*}
\notag
$$
Finally, as the operator norm majorizes the $\mu$-norm (see (1.4)), taking part (b) of this proof into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \|\widehat g_K U_F \widehat g_{K-1}\dotsb U_F \widehat g_0\|_{\mu}^2 &=\lim_{m\to\infty}\|\widehat \varphi_{K,m} U_F \widehat \varphi_{K-1,m}\dotsb U_F \widehat \varphi_{0,m}\|^2_\mu \\ &=\lim_{m\to\infty}\bigl\|\varphi_{K,m}(\varphi_{K-1,m}\circ F)\dotsb (\varphi _{0,m} \circ F^K)\bigr\|^2 \\ &=\bigl\|g_K (g_{K-1}\circ F)\dotsb (g_0 \circ F^K)\bigr\|^2. \end{aligned}
\end{equation*}
\notag
$$
Proposition 8.1 is proved. 8.3. The measure associated with a Koopman operatorConsider a probability space $(\mathcal{X},\mathcal{B},\mu)$. Let $\delta(x,\,\cdot\,)$ be a Dirac measure at $x\in\mathcal{X}$, that is,
$$
\begin{equation*}
\delta(x,B)=\mathbf{1}_B(x)= \begin{cases} 1,& x\in B, \\ 0,& x\notin B, \end{cases} \qquad B\in\mathcal{B}.
\end{equation*}
\notag
$$
Let $F\in\mathrm{Aut}(\mathcal X,\mu)$ and let $U=U_F$. Consider the transition measure $\mu_{U}(\,\cdot\,{,}\,\cdot\,)$: $\mathcal X\times\mathcal B\to\mathbb{R}_+$ defined by
$$
\begin{equation*}
\mu_U(x,B):=\delta(F^{-1}(x),B)=\mathbf{1}_{F(B)}(x) \quad\text{for }\ x\in\mathcal X, \quad B\in\mathcal B.
\end{equation*}
\notag
$$
Definition 8.1. The transition measure $\mu_U(\,\cdot\,{,}\,\cdot\,)$ is called the measure associated with the Koopman operator $U=U_F$. Lemma 8.1. Let $F\in\mathrm{Aut}(\mathcal{X},\mu)$ and $g_0,\dots,g_K\in L^{\infty}(\mathcal X,\mu)$. Then Proof. Let $I$ denote the repeated integral on the right-hand side of the required equality. Then we have
$$
\begin{equation*}
\begin{aligned} \, I&=\int_{\mathcal X}|g_0(x_0)|^2\int_{\mathcal X}\dotsi\int_{\mathcal X}|g_K(x_K)|^2\,\delta(F^{-1}(x_{K-1},dx_K)\dotsb\mu(dx_0) \\ &=\int_{\mathcal X}|g_0(x_0)|^2\int_{\mathcal X}\dotsi\int_{\mathcal X}|g_{K-1}(x_{K-1})|^2\,|g_K\circ F^{-1}(x_{K-1})|^2 \\ &\qquad\times \delta(F^{-1}(x_{K-2},dx_{K-1})\dotsb\mu(dx_0) \\ &=\dots=\int_{\mathcal X}|g_0(x_0)|^2\,|g_1\circ F^{-1}(x_0)|^2\dotsb|g_K\circ F^{-K}(x_{0})|^2\,\mu(dx_0). \end{aligned}
\end{equation*}
\notag
$$
Making the substitution $x=F^{-K}(x_0)$ in the last integral and bearing in mind that $F$ preserves the measure $\mu$ we see that $I$ coincides with the integral on the right-hand side of (8.1).
The proof is complete. 8.4. The measure associated with a regular operator Lemma 8.2. Let $W\in\mathcal{R}(\mathbb{T}^n)$ (see Definition 7.1) and let $a\in\mathbb{T}^n$. To each $\varepsilon>0$ assign a function $f_{a,\varepsilon}\in L^2(\mathbb{T}^n)$ and a functional $F_{a,\varepsilon}\in (C(\mathbb{T}^n))^*$ such that Proof. For brevity set
Note that for each $\varepsilon>0$
$$
\begin{equation}
\|F_\varepsilon\|_{C^*}=\int_{\mathbb{T}^n}|L_a*f_\varepsilon|^2\,d\mu\leqslant\|\mathrm{Conv}_{L_a}\|^2\leqslant\mathbf{c}^2,
\end{equation}
\tag{8.5}
$$
so that the net $\{F_\varepsilon\}_\varepsilon$ (see § 7.1) is bounded in the strong topology of $(C(\mathbb{T}^n))^*$. In addition, the span of the functions $E=\{e^{-i(m,\,\cdot\,)}\mid m\in\mathbb{Z}^n\}$ is dense in $C(\mathbb{T}^n)$. Hence, in accordance with the criterion of weal-$*$ convergence, for the existence of the weak-$*$ limit (8.3) it is sufficient to show that
$$
\begin{equation}
\lim_{\varepsilon\searrow0}\xi_{\varepsilon, m}=v_m(a)e^{-i(m,a)} \quad\text{for each }\ m\in\mathbb{Z}^n.
\end{equation}
\tag{8.6}
$$
Now we present the proof of (8.6).
Let $\ f_{\varepsilon}=\sum_k f_ke^{i(k,\,\cdot\,)}$, $f_k=(f_\varepsilon)_k$, $k\in\mathbb{Z}^n$. Then $ L_a*f_{\varepsilon}=\sum_k w_k(a)f_ke^{i(k,\,\cdot\,)}$, and therefore
$$
\begin{equation*}
\xi_{\varepsilon, m}=\langle e^{-i(m,\,\cdot\,)}(L_a*f_{\varepsilon}), L_a*f_{\varepsilon}\rangle=\sum_{k\in\mathbb{Z}^n}w_{k+m}(a)\overline w_k(a)f_{k+m}\overline f_k.
\end{equation*}
\notag
$$
Let $\mathbf{I}=\mathbf{I}_B$, $B\in\mathbb{N}$ (see § 4.1.5). Set
$$
\begin{equation}
\widetilde\xi_{\varepsilon, m}=\sum_{k\in\mathbb{Z}^n}\sum_{l',l''\in\mathbf{I}}w_{k+m}(a)\overline w_k(a)\frac{f_{k+m+l'}\overline f_{k+l''}}{(2B+1)^{2n}}e^{i(l'-l'',a)}.
\end{equation}
\tag{8.7}
$$
Using (5.7), the Cauchy-Schwarz-Bunyakovskii inequality and (4.8) we obtain
$$
\begin{equation*}
\begin{aligned} \, &|\xi_{\varepsilon, m}-\widetilde\xi_{\varepsilon, m}| \leqslant \sum_{k\in\mathbb{Z}^n}\sum_{l',l''\in\mathbf{I}}|w_{k+m}(a)\overline w_k(a)|\frac{|f_{k+m}\overline f_k-f_{k+m+l'}\overline f_{k+l''}e^{i(l'-l'',a)}|}{(2B+1)^{2n}} \\ &\qquad\leqslant\sum_{k\in\mathbb{Z}^n}\sum_{l',l''\in\mathbf{I}}\mathbf{c}^2 \frac{|f_{k+m}-f_{k+m+l'}e^{i(l',a)}|\cdot|f_k|+|f_{k+m+l'}| \cdot|f_k-f_{k+l''}e^{i(l'',a)}|}{(2B+1)^{2n}} \\ &\qquad\leqslant\frac{2\mathbf{c}^2}{(2B+1)^n}\sum_{l\in\mathbf{I}} \|f_{\varepsilon}\|\,{\cdot}\,\|f_{\varepsilon}-e^{-i(l,\cdot-a)}f_{\varepsilon}\| \leqslant \frac{2\mathbf{c}^2}{(2B+1)^n}\|f_{\varepsilon}\|^2\varepsilon \sum_{l\in\mathbf{I}}\|l\|_1. \end{aligned}
\end{equation*}
\notag
$$
Since $\|l\|_1\leqslant nB$ for $l\in\mathbf{I}$, we arrive at the inequality
$$
\begin{equation}
|\xi_{\varepsilon, m}-\widetilde\xi_{\varepsilon, m}|\leqslant 2\mathbf{c}^2\|f_{\varepsilon}\|^2nB\varepsilon.
\end{equation}
\tag{8.8}
$$
In (8.7) we make the substitution $l=l'$, $u=k+l'$, $ s=l'-l''$. Then taking (7.2) into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \widetilde\xi_{\varepsilon, m} &=\sum_{s\in2\mathbf{I}}\sum_{\substack{l\in\mathbf{I}\\l\in(\mathbf{I}+s)}} \sum_{u\in\mathbb{Z}^n}w_{u+m-l}(a)\overline w_{u-l}(a)\frac{f_{u+m}\overline f_{u-s}}{(2B+1)^{2n}}e^{i(s,a)} \\ &=\sum_{s\in\mathbf{J}}\sum_{u\in\mathbb{Z}^n}v_{\mathbf{I}_{u,s}, m}(a)b_sf_{u+m}\overline f_{u-s}e^{i(s,a)}, \end{aligned}
\end{equation*}
\notag
$$
where, as in the proof of Lemma 4.4,
$$
\begin{equation*}
\mathbf{J}=J\times\dots\times J,\qquad J=[-2B,2B]\cap\mathbb{Z},\quad\text{and}\quad\qquad\mathbf{I}_{u,s}= I_{u_1,s_1}\times\dots\times I_{u_n,s_n},
\end{equation*}
\notag
$$
$$
\begin{equation*}
I_{u_j,s_j}=\begin{cases} [u_j+s_j-B, u_j+B]\cap\mathbb{Z} &\text{for }\ s_j\geq 0, \\ [u_j-B, u_j+s_j+B]\cap\mathbb{Z} &\text{for }\ s_j < 0,
\end{cases} \qquad \mathbf{b}(s)=\frac{\# \mathbf{I}_{u,s}}{(\#\mathbf{I})^2}.
\end{equation*}
\notag
$$
Recall that $\sum_{s\in\mathbf{J}}\mathbf{b}(s)=1$ (see (4.17)).
Let $\sigma>0$. It follows from Lemma 7.6 that there exists $Q_m=Q_m(\sigma)\in\mathbb{N}$ such that
$$
\begin{equation}
|v_{\mathbf{C},m}(a)-v_{m}(a)|<\sigma \quad\text{for each }\ \mathbf{C}\in\mathcal{P}^n_{Q_m}.
\end{equation}
\tag{8.9}
$$
Now let $2B+1>Q_m$. We set $\mathbf{S}=S\times\dots\times S$ and $ S=[-(2B+1-Q_m), 2B+1-Q_m]\cap\mathbb{Z}$. Then $\widetilde\xi_{\varepsilon, m}=\Sigma_1+\Sigma_2+\Sigma_2$, where
$$
\begin{equation*}
\begin{aligned} \, \Sigma_1&=\sum_{s\in\mathbf{J},\,u\in\mathbb{Z}^n}v_{m}(a)\mathbf{b}(s)f_{u+m}\overline f_{u-s}e^{i(s,a)}, \\ \Sigma_2&=\sum_{s\in\mathbf{S},\,u\in\mathbb{Z}^n}(v_{\mathbf{I}_{u,s}, m}(a)-v_m(a))\mathbf{b}(s)f_{u+m}\overline f_{u-s}e^{i(s,a)}, \\ \Sigma_3&=\sum_{s\in\mathbf{J}\setminus\mathbf{S},\,u\in\mathbb{Z}^n}(v_{\mathbf{I}_{u,s}, m}(a)-v_m(a))\mathbf{b}(s)f_{u+m}\overline f_{u-s}e^{i(s,a)}. \end{aligned}
\end{equation*}
\notag
$$
Using (7.4) and (4.10) we obtain
$$
\begin{equation}
\begin{aligned} \, \nonumber &\bigl|\Sigma_1-v_m(a)e^{-i(m,a)}\|f_\varepsilon\|^2\bigr| \\ \nonumber &\qquad=\biggl|v_{m}(a)e^{-i(m,a)}\sum_{s\in\mathbf{J}}\mathbf{b}(s)\sum_{u\in\mathbb{Z}^n} (f_{u+m}\overline f_{u-s}e^{i(m+s,a)}-\|f_{\varepsilon}\|^2)\biggr| \\ \nonumber &\qquad\leqslant\mathbf{c}^2\varepsilon\|f_{\varepsilon}\|^2\sum_{s\in\mathbf{J}}\mathbf{b}(s) (\|m\|_1+\|s\|_1) \\ &\qquad\leqslant\mathbf{c}^2\varepsilon\|f_{\varepsilon}\|^2(\|m\|_1+2Bn). \end{aligned}
\end{equation}
\tag{8.10}
$$
If $s\in\mathbf{S}$, then $\mathbf{I}_{u,s}\in\mathcal{P}^n_{Q_m}$ for each $u\in\mathbb{Z}^n$, so taking (8.9) into account we find that
$$
\begin{equation}
|\Sigma_2|\leqslant \sum_{s\in\mathbf{S},\,u\in\mathbb{Z}^n}\sigma\mathbf{b}(s)|f_{u+m}|\, |f_{u-s}|\leqslant\sigma\|f_\varepsilon\|^2.
\end{equation}
\tag{8.11}
$$
Since
$$
\begin{equation*}
\sum_{s\in\mathbf{J}\setminus\mathbf{S}}\mathbf{b}(s)\leqslant\frac{2Q_m(Q_m-1)n}{(2B+1)^2}
\end{equation*}
\notag
$$
(which follows by replacing $M'$ by $Q_m$ in (4.19)), it follows that
$$
\begin{equation}
\begin{aligned} \, \nonumber |\Sigma_3| &\leqslant2\mathbf{c}^2\sum_{s\in\mathbf{J}\setminus\mathbf{S},\,u\in\mathbb{Z}^n}\mathbf{b}(s)|f_{u+m}|\,| f_{u-s}| \\ &\leqslant2\mathbf{c}^2\sum_{s\in\mathbf{J}\setminus\mathbf{S}}\mathbf{b}(s)\|f_{\varepsilon}\|^2\leqslant 4n\mathbf{c}^2\frac{Q_m(Q_m-1)}{(2B+1)^2}\|f_{\varepsilon}\|^2. \end{aligned}
\end{equation}
\tag{8.12}
$$
Since $\|f_{\varepsilon}\|=1$, taking (8.8) and (8.10)–(8.12) into account we obtain the relation
$$
\begin{equation*}
|\xi_{\varepsilon, m}-e^{-i(m,a)}v_{m}(a)|\leqslant \mathbf{c}^2(4nB+\|m\|_1)\varepsilon+\sigma+4n\mathbf{c}^2\frac{Q_m(\sigma)(Q_m(\sigma)-1)}{(2B+1)^2},
\end{equation*}
\notag
$$
which holds for all $\sigma>0$, $\varepsilon>0$ and $B>(Q_m(\sigma)-1)/2$. Now (8.6) is proved.
The resulting functional $F_a$ is independent of the family of functions $f_\varepsilon$, $\varepsilon>0$, satisfying (8.2) because the quantities $F_a(e^{-i(m,\,\cdot\,)})=v_m(a)e^{-i(m,a)}$, $m\in\mathbb{Z}^n$, are independent of this family. Finally, (8.4) is a consequence of (8.5) and the definition of weak-$*$ convergence. Lemma 8.2 is proved. As before, let $\mathcal{B}(\mathbb{T}^n)$ be the Borel $\sigma$-algebra of subsets of $\mathbb{T}^n$, and let $\mu$ be the normalized Lebesgue measure on $(\mathbb{T}^n,\mathcal B(\mathbb{T}^n))$. Set $\mathcal{M}(\mathbb{T}^n)$ to be the class of complex Borel measures. It is known (for instance, see [5], vol. 1, Theorem 4.6.1) that $\mathcal{M}(\mathbb{T}^n)$ is a Banach space with the norm $\|\,{\cdot}\,\|$ defined by $\|\nu\|=|\nu|(\mathbb{T}^n)$, where $|\nu|$ is the total variation of $\nu\in \mathcal{M}(\mathbb{T}^n)$. Since $\mathbb{T}^n$ is a compact Hausdorff space, by Riesz’s theorem on the representation of a linear functional (for instance, see [11], Theorem IV.6.3) there is an isometric isomorphism between $C^{*}(\mathbb{T}^n)$ and $\mathcal{M}(\mathbb{T}^n)$ such that two elements $F\in C^{*}(\mathbb{T}^n)$ and $\nu\in \mathcal{M}(\mathbb{T}^n)$ corresponding one to the other are related by
$$
\begin{equation}
F(\varphi)=\int_{\mathbb{T}^n}\varphi(x)\,\nu(dx), \qquad \varphi\in C(\mathbb{T}^n).
\end{equation}
\tag{8.13}
$$
Let $W\in\mathcal R(\mathbb{T}^n)$. By Lemma 8.2, for each point $a\in\mathbb{T}^n$ there exists a unique functional $F_a$ satisfying (8.3). For each functional $F_a$, $a\in\mathbb{T}^n$, there is a measure $\nu_a\in\mathcal M(\mathbb{T}^n)$ such that $F_a$ and $\nu_a$ are related by (8.13). Each measure $\nu_a$, $a\in\mathbb{T}^n$, is nonnegative because $F_{a,\varepsilon}(\varphi)\geqslant0$ for all $\varphi\in C(\mathbb{T}^n),\varphi\geqslant0$ and all $\varepsilon>0$. Thus we have obtained a function
$$
\begin{equation}
\mu_W(\,\cdot\,{,}\,\cdot\,)\colon \mathbb{T}^n\times\mathcal{B}(\mathbb{T}^n)\to \mathbb{R}_+\quad\text{such that} \quad \mu_W(a,B):=\nu_a(B).
\end{equation}
\tag{8.14}
$$
The next lemma shows that this is a transition measure. Lemma 8.3. For each set $B\in\mathcal{B}(\mathbb{T}^n)$ the function $a\mapsto \mu_W (a,B)$ is Borel. Proof. For each $a\in\mathbb{T}^n$ and any function $\varphi\in C(\mathbb{T}^n)$ set $H_{\varphi}(a):=F_a(\varphi)$. Note that the functions $H_{e^{-i(m,\,\cdot\,)}}$, $m\in\mathbb{Z}^n$, are continuous on $\mathbb{T}^n$ because $H_{e^{-i(m,\,\cdot\,)}}(a)=v_m(a)e^{-i(m,a)}$ for all $a\in\mathbb{T}^n$ and $m\in\mathbb{Z}^n$.
Let $\varphi\in C(\mathbb{T}^n)$. Then there exists a sequence of functions $\varphi_k$ such that
$$
\begin{equation*}
\lim_{k\to\infty}\|\varphi_k-\varphi\|_{C}=0, \qquad \varphi_k\in\mathrm{span}\{e^{i(m,\,\cdot\,)}\mid m\in\mathbb{Z}^n\}, \quad k\in\mathbb{N}.
\end{equation*}
\notag
$$
Then $H_{\varphi_{k}}(a)\to H_{\varphi}(a)$ as $k\to\infty$ for each $a\in\mathbb{T}^n$. Since each function $H_{\varphi_{k}}$ is continuous, $H_{\varphi}$ is a Borel function.
Let $F$ be a closed set in the metric space $(\mathbb{T}^n,\mathrm{dist})$ (see (4.1)). For $x\in\mathbb{T}^n$ and ${k\in\mathbb{N}}$ set
$$
\begin{equation}
f_k(x)=\frac{\mathrm{dist}(x,\mathbb{T}^n\setminus G_k)}{\mathrm{dist}(x,\mathbb{T}^n\setminus G_k)+\mathrm{dist}(x,F)}\quad\text{and} \quad G_k=\biggl\{y\in\mathbb{T}^n\Bigm| \mathrm{dist}(y,F)<\frac{1}{k}\biggr\},
\end{equation}
\tag{8.15}
$$
where $\mathrm{dist}(x,A):=\inf\{\mathrm{dist}(x,y)\mid y\in A\}$ is the distance from $x$ to $A$.
The following assertions are easy to verify:
Then from Lebesgue’s dominated convergence theorem we obtain
$$
\begin{equation*}
H_{f_k}(a)=\int_{\mathbb{T}^n}f_k(x)\, \mu_W (a,dx)\to \mu_W (a,F) \quad\text{as }\ k\to\infty
\end{equation*}
\notag
$$
for each $a\in\mathbb{T}^n$. Since the $H_{f_k}$, $k\in\mathbb{N}$, are Borel functions, $ \mu_W (\,\cdot\,,F)$ is too.
Now look at the set
$$
\begin{equation*}
\mathcal{E}=\bigl\{B\in\mathcal{B}(\mathbb{T}^n)\mid\, \text{the function }\ \mu_W (\,\cdot\,,B)\text{ is Borel}\bigr\}
\end{equation*}
\notag
$$
and let $\mathcal{F}$ denote the set of closed subsets of $\mathbb{T}^n$. Since $\mathcal{F}$ and $\mathcal{E}$ are $\pi$- and $\lambda$-systems, respectively, and we have $\mathcal{F}\subset\mathcal{E}$, the minimal $\sigma$-algebra containing $\mathcal{F}$ coincides with $\mathcal{E}$ (see [25], Ch. 2, § 2, Definition 2 and Theorem 2), that is, $\mathcal{B}(\mathbb{T}^n)=\mathcal{E}$.
Lemma 8.3 is proved. Definition 8.2. We call the transition measure $ \mu_W (\,\cdot\,{,}\,\cdot\,)$ in (8.14) the measure associated with the regular operator $W$. With each measure $\nu\in \mathcal{M}(\mathbb{T}^n)$ we associate the (multiple) Fourier series
$$
\begin{equation*}
\nu\sim\sum_{k\in\mathbb{Z}^n}\nu_ke^{i(k,x)}, \qquad \nu_k=\int_{\mathbb{T}^n}e^{-i(k,x)}\,\nu(dx), \quad k\in\mathbb{Z}^n
\end{equation*}
\notag
$$
(for instance, see [27], Ch. VII). By Lemma 8.2 the transition measure $ \mu_W (\,\cdot\,{,}\,\cdot\,)$ associated with the regular operator $W$ has the following property: for each point $a\in\mathbb{T}^n$
$$
\begin{equation}
\mu_W (a,\,\cdot\,)\sim\sum_{m\in\mathbb{Z}^n}(v_m(a)e^{-i(m,a)})e^{i(m,x)}.
\end{equation}
\tag{8.16}
$$
Examples. 1. Let $g\in\mathcal{AC}(\mathbb{T}^n)$. By Lemma 7.7 the operator $\widehat g$ is regular. In view of (8.16), for each $a\in\mathbb{T}^n$ and any $m\in\mathbb{Z}^n$ we have $(\mu_{\widehat g}(a,\,\cdot\,))_m=e^{-i(m,a)}|g(a)|^2$. Therefore, where $\delta(a,\,\cdot\,)$ is the Dirac measure at $a$. 2. Let $x\in\mathbb{T}^n$. If $W\!=\!\operatorname{Conv}_{\delta_x}$, then $W\!\in\!\mathcal R(\mathbb{T}^n)$ by Lemma 7.8 and $(\mu_W(a,\,\cdot\,))_m=e^{-i(m,x+a)}$ for all $a\in\mathbb{T}^n$ and $m\in\mathbb{Z}^n$. Hence
$$
\begin{equation*}
\mu_W(a,\,\cdot\,)=\delta(x+a,\,\cdot\,)=\delta(F_x(a),\,\cdot\,),
\end{equation*}
\notag
$$
where $F_x$ is the automorphism defined in (4.4). The next lemma establishes a connection between the measure associated with a regular operator and the $\mu$-norm of this operator. Remark 8.1. Since $\operatorname{Conv}_{\delta_x}=U_{F_{-x}}$, to the convolution operator $\operatorname{Conv}_{\delta_x}$ we can assign the transition measure $\delta(F^{-1}_{-x}(\cdot),\,\cdot\,)$ associated with the Koopman operator $U_{F_{-x}}$. Since $F^{-1}_{-x}=F_x$, the transition measures mentioned in Definitions 8.1 and 8.2 coincide in accordance with example 2. Lemma 8.4. Let $W\in\mathcal{R}(\mathbb{T}^n)$ and let $ \mu_W (\,\cdot\,{,}\,\cdot\,)$ be the transition measure associated with $W$. Then Proof. It follows from (8.16) that
$$
\begin{equation*}
\mu_W (a,\mathbb{T}^n)=\int_{\mathbb{T}^n}\, \mu_W (a,dx)=v_0(a)=\rho(L_a).
\end{equation*}
\notag
$$
It remains to use Proposition 5.1. The proof is complete. § 9. The operators $W\widehat g$, $\widehat g W$ and $\widehat g_2 W \widehat g_1$Recall that in § 7.2 we introduced the partial order relation $\leqslant$ on $\mathcal{P}^n$, which transformed $\mathcal{P}^n$ into a directed set. As before, we denote the limit of a net $\{x_{\mathbf{I}}\}_{\mathbf{I}\in\mathcal{P}^n}$ in the metric space $X$ by $\lim_{\mathbf{I}}x_{\mathbf{I}}$. 9.1. The operator $W\widehat g$ Lemma 9.1. Let $W\in\mathcal{R}(\mathbb{T}^n)$ and $g\in \mathcal{AC}(\mathbb{T}^n)$. Then $\widetilde W:=W\widehat g$ is a regular operator and Proof. Since $W\in\mathcal{DT}(\mathbb{T}^n)$ and $\widehat{g}\in\mathcal{DT}(\mathbb{T}^n)$, it follows that $\widetilde W \in \mathcal{DT}(\mathbb{T}^n)$. Let $\widetilde \omega_{\mathbf{I},m,k}$ be the quantities (7.1) for the operator $\widetilde W$. We fix some $m,k\in\mathbb{Z}^n$ and show that where the $\mathbf{g}_q$, $q\in\mathbb{Z}^n$, are the Fourier coefficients of $|g|^2$.
Since
$$
\begin{equation*}
\mathbf{g}_q=\int_{\mathbb{T}^n}|g(x)|^2 e^{-(q,x)}\,\mu(dx)=\langle g, ge^{i(q,\,\cdot\,)}\rangle=\sum_{p\in\mathbb{Z}^n}g_p\overline g_{p-q}, \qquad g_p=\langle g, e^{i(p,\,\cdot\,)}\rangle,
\end{equation*}
\notag
$$
by the definition of $\widetilde \omega_{\mathbf{I},m,k}$ we have
$$
\begin{equation*}
\begin{aligned} \, \widetilde\omega_{\mathbf{I},m,k} &= \frac1{\# \mathbf{I}} \sum_{r,s,j\in\mathbb{Z}^n,\,l\in \mathbf{I}} W_{l+m,r} g_{r-j} \overline W_{l,s} \overline g_{s-j-k} \\ &= \frac1{\# \mathbf{I}} \sum_{r,q\in\mathbb{Z}^n,\,l\in \mathbf{I}}W_{l+m,r}\overline W_{l,r+k-q} \mathbf{g}_q \\ &=\sum_{q\in\mathbb{Z}^n}\mathbf{g}_q\omega_{\mathbf{I},m,k-q}=\sum_{q\in\mathbb{Z}^n} f_{\mathbf{I},q}, \quad\text{where}\ f_{\mathbf{I},q}:=\mathbf{g}_q\omega_{\mathbf{I},m,k-q}. \end{aligned}
\end{equation*}
\notag
$$
For each $q\in\mathbb{Z}^n$ and any $\mathbf{I}\in\mathcal P^n$ we obtain
$$
\begin{equation*}
|f_{\mathbf{I},q}|\leqslant |\mathbf{g}_q|\frac{1}{\# \mathbf{I}}\sum_{r\in\mathbb{Z}^n, l\in\mathbf{I}}c_{l+m-r}c_{l-r+q-k} =\sum_{s\in\mathbb{Z}^n}|\mathbf{g}_q| c_{s+m}c_{s+q-k}:=h_q.
\end{equation*}
\notag
$$
Moreover,
$$
\begin{equation*}
\sum_{q\in\mathbb{Z}^n}h_q\leqslant\mathbf{c}^2\mathbf{g}<\infty \quad\text{for }\ \mathbf{g}=\sum_{q\in\mathbb{Z}^n}|\mathbf{g}_q|<\infty.
\end{equation*}
\notag
$$
Hence, by Corollary 7.1 the multiple series $ \sum_{q}f_{\mathbf{I},q}$ converges uniformly on $\mathcal{P}^n$ and (as the operator $W$ is regular) the limit
$$
\begin{equation*}
\widetilde{\omega}_{m,k}=\lim_{\mathbf{I}} \widetilde\omega_{\mathbf{I},m,k}= \lim_{\mathbf{I}}\sum_{q\in\mathbb{Z}^n}f_{\mathbf{I},q} =\sum_{q\in\mathbb{Z}^n}\mathbf{g}_q\lim_{\mathbf{I}}\omega_{\mathbf{I},m,k-q} =\sum_{q\in\mathbb{Z}^n}\mathbf{g}_q\omega_{m,k-q}
\end{equation*}
\notag
$$
exists, so that $\widetilde {W}\in\mathcal{R}(\mathbb{T}^n)$. Then for each $m\in\mathbb{Z}^n$ we obtain the function ${\widetilde{v}_m\in\mathcal{AC}(\mathbb{T}^n)}$ defined by (7.3), and
$$
\begin{equation*}
\begin{aligned} \, \widetilde{v}_m(a) &=\sum_{k\in\mathbb{Z}^n} \sum_{q\in\mathbb{Z}^n}\mathbf{g}_q\omega_{m,k-q} e^{i(m+k, a)} \\ &=\sum_{q\in\mathbb{Z}^n}\mathbf{g}_qe^{i(q,a)}\sum_{k\in\mathbb{Z}^n} \omega_{m,p} e^{i(m+p, a)} =|g(a)|^2v_m(a) \end{aligned}
\end{equation*}
\notag
$$
for each point $a\in\mathbb{T}^n$.
The proof is complete. Proposition 9.1. Let $W\in\mathcal{R}(\mathbb{T}^n)$ and $g\in \mathcal{AC}(\mathbb{T}^n)$. If $ \mu_W (\,\cdot\,{,}\,\cdot\,)$ and $\mu_{\widetilde W} (\,\cdot\,{,}\,\cdot\,)$ are the measures associated with the operators $W$ and $\widetilde W=W\widehat g$, respectively (see Definition 8.2), then Proof. It follows from Lemma 9.1 and (8.16) that for each $a\in\mathbb{T}^n$ the measures $|g(a)|^2 \mu_W (a,\,\cdot\,)$ and $\mu_{\widetilde W} (a,\,\cdot\,)$ have the same Fourier coefficients and thus coincide.
To prove (9.2) we have to use Lemma 8.4. The proof is complete. Consider a regular operator $W$ with $\mathcal{DT}$-norm $\mathbf{c}$, and let $\mu_0$ denote the measure with density $v_0$ with respect to $\mu$. If $f\in L^1(\mathbb{T}^n, \mu)$, then using (7.4) and bearing in mind that $v_0$ is nonnegative we obtain
$$
\begin{equation}
\int_{\mathbb{T}^n}|f(a)|\,\mu_0(da)=\int_{\mathbb{T}^n}|f(a)|v_0(a)\,\mu(da)\leqslant \mathbf{c}^2\int_{\mathbb{T}^n}|f(a)|\,\mu(da),
\end{equation}
\tag{9.3}
$$
so that $L^p(\mathbb{T}^n, \mu)\subset L^p(\mathbb{T}^n, \mu_0)$ for $1\leqslant p<\infty$, and moreover, $\|\,{\cdot}\,\|_{L^p(\mu_0)}\leqslant \mathbf{c}^2\|\,{\cdot}\,\|_{L^p(\mu)}$; in particular,
$$
\begin{equation}
\|f\|_{L^2(\mu_0)}\leqslant \mathbf{c}^2\|f\|, \qquad f\in L^2(\mathbb{T}^n,\mu).
\end{equation}
\tag{9.4}
$$
Proposition 9.2. Assume that $W\in\mathcal{R}(\mathbb{T}^n)$ and $g\in L^{\infty}(\mathbb{T}^n)$. Then equality (9.2) holds. Proof. For each $N\in\mathbb{N}$ and any point $x\in\mathbb{T}^n$ set $S_N(x):=\sum_{k\in\mathbf{I}_N}g_ke^{i(k,x)}$, where $\{g_k\}$ are the Fourier coefficients of $g$. It is obvious that $S_N\in\mathcal{AC}(\mathbb{T}^n)$ and $S_N\xrightarrow[N\to\infty]{L^2}g$. We claim the following equalities:
$$
\begin{equation}
\|W\widehat g\|^2_\mu=\lim_{N\to\infty}\|W\widehat S_N\|^2_\mu=\int_{\mathbb{T}^n}\!\int_{\mathbb{T}^n}|g(a)|^2\, \mu_W (a,dx)\,\mu(da).
\end{equation}
\tag{9.5}
$$
In fact,
$$
\begin{equation*}
\bigl|\|W\widehat S_N\|_\mu-\|W\widehat g\|_\mu\bigr|\leqslant \|W(\widehat S_N-\widehat g)\|_\mu\leqslant\|W\|\,\|\widehat S_N-\widehat g\|_\mu
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\|\widehat S_N-\widehat g\|_\mu=\| S_N- g\|\xrightarrow[N\to\infty]{}0,
\end{equation*}
\notag
$$
which yields the first equality in (9.5). Now we prove the second.
From Proposition 9.1, the definition of $\mu_0$ and estimate (9.4) we obtain
$$
\begin{equation*}
\begin{aligned} \, \lim_{N\to\infty}\|W\widehat S_N\|^2_\mu &=\lim_{N\to\infty}\int_{\mathbb{T}^n}|S_N(a)|^2 \mu_W (a,\mathbb{T}^n)\,\mu(da) \\ &=\lim_{N\to\infty}\int_{\mathbb{T}^n}|S_N(a)|^2v_0(a)\,\mu(da) =\lim_{N\to\infty}\|S_N\|^2_{L^2(\mu_0)} \\ &=\|g\|^2_{L^2(\mu_0)} =\int_{\mathbb{T}^n}\!\int_{\mathbb{T}^n}|g(a)|^2\, \mu_W (a,dx)\,\mu(da). \end{aligned}
\end{equation*}
\notag
$$
The proposition is proved. 9.2. The operator $\widehat g W$Lemma 9.2. Let $W\in\mathcal{R}(\mathbb{T}^n)$ and $g\in \mathcal{AC}(\mathbb{T}^n)$. Then $\widetilde W:=\widehat g W$ is a regular operator and if $ \mu_W (\,\cdot\,{,}\,\cdot\,)$ and $ \mu_{\widetilde W} (\,\cdot\,{,}\,\cdot\,)$ are the measures associated with $W$ and $\widetilde W$, respectively, then Proof. Since $W$, $\widehat{g}\in\mathcal{DT}(\mathbb{T}^n)$, we have $\widetilde W \in \mathcal{DT}(\mathbb{T}^n)$. Let $\widetilde \omega_{\mathbf{I},m,k}$ denote the quantities (7.1) for the operator $\widetilde W$. We show that for all $m,k\in\mathbb{Z}^n$ where $\{g_q\}_{q\in\mathbb{Z}^n}$ is the sequence of Fourier coefficients of $g$.
By the definition of $\widetilde \omega_{\mathbf{I},m,k}$ we have
$$
\begin{equation}
\begin{aligned} \, \nonumber \widetilde \omega_{\mathbf{I},m,k} &=\frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I},\,j,r,s\in\mathbb{Z}^n} g_{l+m-r}W_{r,j}\overline{g}_{l-s}\overline{W}_{s,j+k} \\ \nonumber &=\frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I},\,j,r,s\in\mathbb{Z}^n} g_{m-r}\overline{g}_{-s}W_{r+l,j}\overline{W}_{s+l,j+k} \\ \nonumber &=\frac{1}{\#\mathbf{I}}\sum_{j,r,s\in\mathbb{Z}^n,\,l\in(\mathbf{I}+s)} g_{m-r}\overline{g}_{-s}W_{r-s+l,j}\overline{W}_{l,j+k} \\ &=\sum_{r,s\in\mathbb{Z}^n}g_{m-r}\overline{g}_{-s}\omega_{\mathbf{I}+s, r-s, k}=\sum_{r,s\in\mathbb{Z}^n}f_{\mathbf{I},r,s}, \end{aligned}
\end{equation}
\tag{9.7}
$$
where $ f_{\mathbf{I},r,s} :=g_{m-r}\overline{g}_{-s}\omega_{\mathbf{I}+s,r-s,k}$. Set $\mathbf{c}_g:=\|g\|_{\mathcal{AC}}=\sum_{j\in\mathbb{Z}^n}|g_j|$.
For arbitrary $r,s\in\mathbb{Z}^n$ we have
$$
\begin{equation*}
|\omega_{\mathbf{I}+s,r-s,k}|\leqslant \frac{1}{\#\mathbf{I}}\sum_{j\in\mathbb{Z}^n,\,l\in(\mathbf{I}+s)}c_{r-s+l-j}c_{l-j-k} =\frac{1}{\#\mathbf{I}}\sum_{l\in\mathbf{I}}\sum_{j\in\mathbb{Z}^n}c_{r+l-j}c_{l+s-j-k},
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
| f_{\mathbf{I},r,s} |\leqslant|g_{m-r}||g_{-s}|\sum_{j\in\mathbb{Z}^n}c_{r+j}c_{s+j-k}:=h_{r,s}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\sum_{r,s\in\mathbb{Z}^n}| f_{\mathbf{I},r,s} | \leqslant\sum_{r,s\in\mathbb{Z}^n}h_{r,s}\leqslant\mathbf{c}_{g}^2\mathbf{c}^2<\infty.
\end{equation*}
\notag
$$
Hence the series (9.7) converges uniformly on $\mathcal{P}^n$. In addition, if $r$ and $s$ are fixed elements of $\mathbb{Z}^n$, then (since $W$ is regular) the following limits exist:
$$
\begin{equation*}
\lim_{\mathbf{I}}\omega_{\mathbf{I}+s,r-s,k} =\lim_{\mathbf{I}}\omega_{\mathbf{I},r-s,k}=\omega_{r-s,k}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\lim_{\mathbf{I}} f_{\mathbf{I},r,s} =g_{m-r}\overline{g}_{-s}\omega_{r-s,k}.
\end{equation*}
\notag
$$
Therefore, by Corollary 7.1 the limit
$$
\begin{equation*}
\widetilde{\omega}_{m,k}=\lim_{\mathbf{I}}\widetilde{\omega}_{\mathbf{I},m,k} =\lim_{\mathbf{I}}\sum_{r,s\in\mathbb{Z}^n} f_{\mathbf{I},r,s}=\sum_{r,s\in\mathbb{Z}^n}\lim_{\mathbf{I}} f_{\mathbf{I},r,s}=\sum_{r,s\in\mathbb{Z}^n}g_{m-r}\overline{g}_{-s}\omega_{r-s,k}
\end{equation*}
\notag
$$
exists, that is, $\widetilde W\in\mathcal{R}(\mathbb{T}^n)$.
Now we prove (9.6). Let $a\in\mathbb{T}^n$. Since the Fourier series of $ \mu_{\widetilde W} (a,\,\cdot\,)$ has the form $\sum_{m\in\mathbb{Z}^n}(\widetilde v_m(a)e^{-i(m,a)})e^{i(m,x)}$ (see (8.16)), taking (7.3) into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \mu_{\widetilde W} (a,\mathbb{T}^n) &=\widetilde{v}_0(a)=\sum_{k\in\mathbb{Z}^n}\widetilde{\omega}_{0,k}e^{i(k,a)} \\ &=\sum_{r,s\in\mathbb{Z}^n}g_{-r}\overline{g}_{-s}\sum_{k\in\mathbb{Z}^n}\omega_{r-s,k}e^{i(k,a)} =\sum_{r,s\in\mathbb{Z}^n}g_{-r}\overline{g}_{-s}v_{r-s}(a)e^{i(s-r,a)} \\ &=\sum_{r,s\in\mathbb{Z}^n}g_{r}\overline{g}_{r-s}v_{-s}(a)e^{i(s,a)}. \end{aligned}
\end{equation*}
\notag
$$
Since the multiple series $\sum_{s,r\in\mathbb{Z}^n}g_r\overline g_{r-s}e^{i(s,x)}$ converges absolutely to $|g(x)|^2$ at each point $x\in\mathbb{T}^n$, we have
$$
\begin{equation*}
\begin{aligned} \, \int_{\mathbb{T}^n}|g(x)|^2\, \mu_W (a,dx) &=\sum_{s,r\in\mathbb{Z}^n}g_r\overline g_{r-s}\int_{\mathbb{T}^n}e^{i(s,x)}\, \mu_W (a,dx) \\ &=\sum_{r,s\in\mathbb{Z}^n}g_r\overline g_{r-s}v_{-s}(a)e^{i(s,a)}= \mu_{\widetilde W} (a,\mathbb{T}^n). \end{aligned}
\end{equation*}
\notag
$$
Lemma 9.2 is proved. Using Lemmas 8.4 and 9.2 we obtain the following. Proposition 9.3. Let $W\in\mathcal{R}(\mathbb{T}^n)$ and $g\in\mathcal{AC}(\mathbb{T}^n)$. If $ \mu_W (\,\cdot\,{,}\,\cdot\,)$ is the measure associated with $W$, then 9.3. The operator $\widehat g_2 W \widehat g_1$From Proposition 9.1 and Lemma 9.2 we obtain the following. Proposition 9.4. Let $W\in\mathcal{R}(\mathbb{T}^n)$ and $g_1,g_2\in \mathcal{AC}(\mathbb{T}^n)$. Then the operator ${\widetilde W:=\widehat g_2 W\widehat g_1}$ is regular and § 10. Markov operatorConsider the probability space $(\mathcal{X},\mathcal{B},\mu)$. Set
$$
\begin{equation*}
\begin{gathered} \, \mathcal N_1:=\{U_F\mid F\in\operatorname{Aut}(\mathcal X,\mu)\}, \\ \mathcal N_2:=\{U\in \mathcal R(\mathcal X)\mid U \text{ is a unitary operator}\}, \quad\text{provided that }\ \mathcal{X}=\mathbb{T}^n, \end{gathered}
\end{equation*}
\notag
$$
and Lemma 10.1. Let $\mu_U(\,\cdot\,{,}\,\cdot\,)$ be the transition measure associated with an operator $U\in\mathcal N$ (see Definitions 8.1 and 8.2). Then that is, $\mu_U(\,\cdot\,{,}\,\cdot\,)$ is a transition probability. Proof. Consider two cases.
(a) If $U=U_F$, where $F\in\operatorname{Aut}(\mathcal X,\mu)$, then for each point $x\in\mathcal X$
$$
\begin{equation*}
\mu_U(x,\mathcal X):=\delta(F^{-1}(x),\mathcal X)={\bf 1}_{F(\mathcal X)}(x)={\bf 1}_{\mathcal X}(x)=1.
\end{equation*}
\notag
$$
(b) Let $U$ be a unitary operator on $L^2(\mathbb{T}^n)$ such that $U\in\mathcal R(\mathbb{T}^n)$. Using Proposition 9.4 for $g={\bf 1}_X$, $X\in\mathcal B(\mathbb{T}^n)$, we obtain
$$
\begin{equation*}
\int_{\mathbb{T}^n}{\bf 1}_X(a)\mu_U (a,\mathbb{T}^n)\,\mu(da)=\|U\widehat {\bf 1}_X\|_{\mu}^2=\int_{\mathbb{T}^n}{\bf 1}_X(a)\,\mu(da)
\end{equation*}
\notag
$$
for each set $X\in\mathcal B(\mathbb{T}^n)$ (the second equality follows from (1.7) and § 2, item 1). Hence $\mu_U(a,\mathbb{T}^n)=1$ for $\mu$-almost all $a\in\mathbb{T}^n$. Moreover, $\mu_U(\,\cdot\,,\mathbb{T}^n)=v_0$ by (8.16). Since $v_0$ is a continuous function on $\mathbb{T}^n$, it follows that $\mu_U(a,\mathbb{T}^n)=1$ for $a\in\mathbb{T}^n$.
The lemma is proved. Let $U\in\mathcal N$. Since $\mu_U(\,\cdot\,{,}\,\cdot\,)$ is a transition probability, by Theorem 8.1 there exists a probability measure $\nu$ on the measurable space $(\mathcal X\times\mathcal X,\mathcal B\otimes\mathcal B)$ such that
$$
\begin{equation}
\nu(B_1\times B_2)=\int_{B_1}\mu_U(x,B_2)\,\mu(dx), \qquad B_1,B_2\in\mathcal B.
\end{equation}
\tag{10.1}
$$
Set $\mathcal{Z}_\mu:=\{C\subset\mathcal{X}\mid \exists\, D\in\mathcal{B} \colon C\subset D\text{ and }\ \mu(D)=0\}$, and let $\mathcal{B}_\mu$ denote the completion of the $\sigma$-algebra $\mathcal{B}$ with respect to $\mu$, so that $\mathcal{B}_\mu$ is the set of subsets of the form $X\cup C$, where $X\in\mathcal{B}$ and $C\in\mathcal{Z}_\mu$. The extension of $\mu$ to the $\sigma$-algebra $\mathcal{B}_\mu$ is defined by $\overline{\mu}(X\cup C)=\mu(X)$. We define the coordinate maps $p_j\colon\mathcal{X}\times\mathcal{X}\to \mathcal{X}$ by
$$
\begin{equation*}
p_j(x_1,x_2)=x_j, \qquad x_j\in\mathcal{X}, \quad j=1,2.
\end{equation*}
\notag
$$
Lemma 10.2. Let $U\in\mathcal N$ and let $\nu$ be the measure on $(\mathcal X\times\mathcal X,\mathcal B\otimes\mathcal B)$ defined by (10.1). Then for each $\mu$-integrable function $f\colon\mathcal X\to\mathbb{C}$ the functions $f\circ p_1$ and $f\circ p_2$ are $\nu$-integrable and Proof. First let $f$ be a $\mathcal B$-measurable function. It is obvious that $f\circ p_1$ and $f\circ p_2$ are $\mathcal{B}\otimes \mathcal{B}$-measurable. As concerns their integrability and equalities (10.2), we must only verify these in the case when $f$ is nonnegative. However, if $f\geqslant0$, then there exists a sequence of simple nonnegative $\mathcal B$-measurable functions ${f_m}$ such that $f_m(x)\uparrow f(x)$ as $ m\to\infty$ for each $x\in\mathcal{X}$. So we limit ourselves to the indicators of sets in the $\sigma$-algebra $\mathcal B$, that is, we assume below that $f=\mathbf{1}_X$, where $X\in\mathcal{B}$. Then it is straightforward that $f\circ p_1$ and $f\circ p_2$ are $\nu$-integrable functions because they are the indicators of the sets $p_1^{-1}(X)$ and $p_2^{-1}(X)$, respectively, and $\nu$ is a probability measure. Moreover, now (10.2) writes as
We consider two cases separately. (a) If $U=U_F$, $F\in\operatorname{Aut}(\mathcal X,\mu)$, then
$$
\begin{equation*}
\nu(B_1\times B_2)=\int_{B_1}\delta(F^{-1}(x),B_2)\,\mu(dx)=\int_{B_1}{\bf 1}_{F(B_2)}(x)\,\mu(dx)=\mu(B_1\cap F(B_2))
\end{equation*}
\notag
$$
for all $B_1,B_2\in\mathcal B$. Then $\nu(X\times \mathcal X)=\mu(X)=\mu(F(X))=\nu(\mathcal X\times X)$, as required.
(b) Now let $(\mathcal X,\mathcal B)=(\mathbb{T}^n,\mathcal B(\mathbb{T}^n))$, and let $U$ be a regular unitary operator. By Lemma 10.1 $\mu_{U}(\,\cdot\,{,}\,\cdot\,)$ is a transition probability, so
$$
\begin{equation*}
\nu(p_1^{-1}(X))=\int_{X}\mu_U(x,\mathbb{T}^n)\,\mu(dx)=\int_{X}\mu(dx)=\mu(X).
\end{equation*}
\notag
$$
We have established the first equality in (10.3) in case $\mathrm{(b)}$. Now we prove the second.
Note first that if $g\in\mathcal{AC}(\mathcal{\mathbb{T}}^n)$, then from Proposition 9.3 and Corollary 7.2 we obtain
$$
\begin{equation*}
\int_{\mathbb{T}^n}\!\int_{\mathbb{T}^n}|g(x)|^2\,\mu_U(a,dx)\,\mu(da)=\|\widehat g U\|_{\mu}^2=\|\widehat g\|_{\mu}^2=\|g\|^2,
\end{equation*}
\notag
$$
which means that $\|g\circ p_2\|_{L^2(\nu)}=\|g\|$. We prove the same equality for a continuous function $g$.
If $g\in C(\mathbb{T}^n)$, then there exists a sequence of functions $g_m$ such that $g_m\in\mathrm{span}\{e^{i(k,\,\cdot\,)}\mid k\in\mathbb{Z}^n\}$ and $\|g_m-g\|_C\to0$. Therefore, $\|g_m\circ p_2-g\circ p_2\|_C\to0$, so that, since $\nu$ and $\mu$ are probability measures,
$$
\begin{equation*}
\|g\circ p_2\|_{L^2(\nu)}=\lim_{m\to\infty}\|g_m\circ p_2\|_{L^2(\nu)}=\lim_{m\to\infty}\|g_m\|=\|g\|.
\end{equation*}
\notag
$$
Now let $X$ be a closed set in $\mathbb{T}^n$. Then there exists a sequence of continuous functions $f_m$ such that $0\leqslant f_m(x)\leqslant 1$ for all $m\in\mathbb{N}$ and $x\in\mathbb{T}^n$, and ${f_m(x)\to\mathbf{1}_X(x)}$ as $m\to\infty$ at each point $x\in\mathbb{T}^n$ (such functions can be defined by (8.15)). The same holds if we replace the $f_m$ by $f_m\circ p_2$ and $\mathbf{1}_X$ by $\mathbf{1}_{p_2^{-1}(X)}$. Using Lebesgue’s dominated convergence theorem we obtain
$$
\begin{equation*}
\begin{aligned} \, \mu(X)&=\lim_{m\to\infty}\int_{\mathbb{T}^n}f_m(x)\,\mu(dx) =\lim_{m\to\infty}\|f_m\|^2=\lim_{m\to\infty}\|f_m\circ p_2\|^2_{L^2(\nu)} \\ &=\lim_{m\to\infty}\int_{\mathbb{T}^n\times\mathbb{T}^n}f_m\circ p_2\,d\nu=\nu(p_2^{-1}(X)). \end{aligned}
\end{equation*}
\notag
$$
Since $\mathcal{B}$ is the minimal $\sigma$-algebra containing all closed sets, the second equality in (10.3) holds for each $X\in\mathcal{B}$. Thus we have established (10.3) in case $\mathrm{(b)}$.
We can generalize our proof to $\mu$-integrable functions $f$. As above, it is sufficient to consider the indicators of $\mu$-measurable sets. So we assume that $f=\mathbf{1}_E$, where ${E\in\mathcal{B}_\mu}$. We can find sets $X\in\mathcal{B}$ and $C\in\mathcal{Z}_\mu$ such that $E=X\cup C$ and $C\subset D$ for some $D\in\mathcal{B}$ such that $\mu(D)=0$. Then $p_j^{-1}(E)=p_j^{-1}(X)\cup p_j^{-1}(C)$, $ p_j^{-1}(C)\subset p_j^{-1}(D)$ and $p_j^{-1}(X),p_j^{-1}(D)\in \mathcal{B}\otimes \mathcal{B}$ for $j=1, 2$. Hence, taking (10.3) into account we obtain
$$
\begin{equation*}
p_j^{-1}(C)\in\mathcal{Z}_\nu\quad\text{and} \quad p_j^{-1}(E)\in (\mathcal{B}\otimes \mathcal{B})_\nu, \qquad \overline\nu\circ p_1^{-1}(E)=\overline\mu(E)=\overline\nu\circ p_2^{-1}(E).
\end{equation*}
\notag
$$
Lemma 10.2 is proved. Lemma 10.3. If $U\in\mathcal N$, then the following results hold for each $\mu$-integrable function $f$. (a) For $\mu$-almost all points $x_1$ the function is integrable with respect to the measure $\mu_U(x_1,\,\cdot\,)$, the function is integrable with respect to $\mu$, and
$$
\begin{equation}
\int_{\mathcal{X}}\!\int_{\mathcal{X}}f(x_2)\,\mu_U(x_1,dx_2)\,\mu(dx_1)=\int_{\mathcal{X}}f\,d\mu.
\end{equation}
\tag{10.4}
$$
(b) If $f=0$ ($\mu$-almost everywhere), then Proof. Part $\mathrm{(a)}$ follows from Lemma 10.2 and Theorem 8.1.
We prove $\mathrm{(b)}$. Set
$$
\begin{equation*}
\widetilde f(x_1):=\int_{\mathcal{X}}f(x_2)\,\mu_U(x_1,dx_2), \qquad x_1\in\mathcal{X}.
\end{equation*}
\notag
$$
There exists a set $X\in\mathcal{B}$ such that $f(x)=0$ for $x\in X$ and $\mu(\mathcal{X}\setminus X)=0$. Then we set $X'=\mathcal{X}\setminus X$, and we have
$$
\begin{equation*}
\int_{\mathcal{X}}\mu_U(x_1,X')\,\mu(dx_1)=\nu(\mathcal{X}\times X')=\nu(p_2^{-1}(X'))=\mu(X')=0.
\end{equation*}
\notag
$$
Now, since $\mu_U(\,\cdot\,,X')$ is nonnegative, $\mu_U(x_1,X')=0$ for $\mu$-almost all $x_1\in\mathcal{X}$, and therefore
$$
\begin{equation*}
\widetilde f(x_1)=\int_{X}f(x_2)\,\mu_U(x_1,dx_2)+\int_{X'}f(x_2)\,\mu_U(x_1,dx_2)=0 \quad\text{for }\ \mu\text{-almost all }\ x_1.
\end{equation*}
\notag
$$
The proof is complete. Theorem 10.1. Let $U\in\mathcal N$. It follows from Lemma 10.3 that the operator is well defined. It is bounded and Moreover, $T_U$ is a Markov operator, that is, Proof. Property $\mathrm{(a)}$ is obvious.
Property $\mathrm{(b)}$ follows from Lemma 10.1 because $T_U\mathbf{1}_{\mathcal{X}}=\mu_U(\,\cdot\,,\mathcal X)$. If $f\in L^1(\mathcal{X},\mu)$, then property $\mathrm{(c)}$ follows from (10.4). Property (10.6) follows from $\mathrm{(b)}$ and the relations
$$
\begin{equation*}
\begin{aligned} \, \|T_Uf\|_1&=\int_{\mathcal{X}}\biggl|\int_{\mathcal{X}}f(x_2)\,\mu_U(x_1,dx_2)\biggr|\, \mu(dx_1)\leqslant\int_{\mathcal{X}}\!\int_{\mathcal{X}}|f(x_2)|\,\mu_U(x_1,dx_2)\,\mu(dx_1) \\ &=\int_{\mathcal{X}}T_U|f|\,d\mu=\int_{\mathcal{X}}|f|\,d\mu=\|f\|_1. \end{aligned}
\end{equation*}
\notag
$$
The proof is complete. Remark 10.1. If $U=U_F$ for $F\in\operatorname{Aut}(\mathcal X,\mu)$, then for each function $f\in L^1(\mathcal X,\mu)$ § 11. Entropy of a unitary operator11.1. Preliminary constructionsI. Let $(\mathcal{X},\mathcal{B},\mu)$ be a probability space. For any finite system of partitions $\xi_1,\dots,\xi_m$ (where $m\in\mathbb{N}$) set
$$
\begin{equation*}
\bigvee_{s=1}^{m}\xi_s:=\bigl\{X_1\cap\dots\cap X_m\mid X_1\in\xi_1,\dots,X_m\in\xi_m\bigr\}.
\end{equation*}
\notag
$$
With each partition $\xi$ we associate a quantity $h_{\mu}(\xi)$: If $F$ is an endomorphism of $(\mathcal{X},\mathcal{B},\mu)$, then the quantity $h_\mu(F,\xi,m)$ introduced in § 1 has the following representation:
$$
\begin{equation}
h_{\mu}(F,\xi,m)=h_{\mu}\biggl(\bigvee_{i=0}^{m-1}F^{-i}\xi\biggr),
\end{equation}
\tag{11.2}
$$
where $F^{-i}\xi:=\{F^{-i}X\mid X\in\xi\}$, $i=0,1,2,\dots$ . It is known that $h_\mu$ has the following properties: II. We introduce the following objects. $\bullet$ For each positive integer $m$ we let $\mathcal{B}^{m}$ denote the minimal $\sigma$-algebra in $\mathcal{X}^{m}$ generated by the ‘rectangles’ of the form
$$
\begin{equation*}
X_1\times\dots\times X_{m}, \quad\text{where }\ X_s\in\mathcal{B}, \quad s=1,\dots,m.
\end{equation*}
\notag
$$
$\bullet$ Let $\mathcal{X}^{\infty}$ denote the set of sequences $x=(x_1,x_2,x_3,\dots)$ such that $x_s\in\mathcal{X}$ for ${s\in\mathbb{N}}$. We define the coordinate maps $p_s\colon \mathcal{X}^{\infty}\to\mathcal{X}$ by
$$
\begin{equation*}
p_s(x)=x_s, \quad\text{where }\ s\in\mathbb{N},\quad x\in\mathcal{X}^{\infty}.
\end{equation*}
\notag
$$
We call the set $C_m(B)=\{x\in\mathcal{X}^{\infty}\mid (x_1,\dots,x_m)\in B\}$ the cylinder with base ${B\in\mathcal{B}^{m}}$. We define the cylindrical $\sigma$-algebra $\mathcal{B}^{\infty}$ to be the minimal $\sigma$-algebra containing all cylinders. III. Let $U\in\mathcal N$, where, as in § 10, the class $\mathcal N$ consists of the Koopman operators and regular unitary operators. Recall that for $U$ we have defined the transition measure $\mu_U$ (see Definitions 8.1 and 8.2), which is a transition probability by Lemma 10.1. On the basis of Theorem 8.1, for each $m\in\mathbb{N}$ we can introduce a probability measure $\nu_{m}$ on $(\mathcal{X}^{m},\mathcal{B}^{m})$ such that
$$
\begin{equation*}
\begin{aligned} \, \nu_{m}\biggl(\prod_{s=1}^{m}X_s\biggr) &= \int_{X_1}\!\int_{X_2}\cdots\int_{X_{m-1}}\!\int_{X_{m}}\mu_U(x_{m-1}, dx_{m}) \\ &\qquad\times\mu_U(x_{m-2},dx_{m-1})\dotsb\mu_U(x_1,dx_2)\,\mu(dx_1), \end{aligned}
\end{equation*}
\notag
$$
where $X_s\in\mathcal{B}$, $s=1,\dots,m$ (the measure $\nu_1$ coincides with $\mu$, and $\nu_2$ coincides with the measure $\nu$ satisfying (10.1)). It is clear that the measures $\nu_{m}$, $m\in\mathbb{N}$, satisfy the condition of compatibility: $\nu_{m+1}(B\times\mathcal{X})=\nu_{m}(B)$ for each $B\in\mathcal{B}^m$. For any finite set $\mathbf{G}=(g_1,\dots,g_m)$ of functions $g_s\in L^{\infty}(\mathcal{X},\mu)$, $s=1,\dots,m$, where $m\geqslant1$, we set
$$
\begin{equation*}
\mathcal{I}_U(\mathbf{G}):=\int_{\mathcal{X}}g_1(x_1)\!\int_{\mathcal{X}}g_2(x_2) \cdots\!\int_{\mathcal{X}}g_m(x_m)\,\mu_U(x_{m-1},dx_{m})\dotsb\mu_U(x_1,dx_2)\,\mu(dx_1).
\end{equation*}
\notag
$$
Note that $\displaystyle \mathcal{I}_U(\mathbf{G})=\int_{\mathcal{X}^{m}}g\,d\nu_{m}$, where $g(x_1,\dots,x_m)=g_1(x_1)\dotsb g_m(x_m)$. Examples. 1. Let $U\,{=}\,U_F$, where $F\,{\in}\,\operatorname{Aut}(\mathcal X,\mu)$. Recall that $\mu_U(\,\cdot\,{,}\,\cdot\,)\,{=}\,\delta(F^{-1}(\cdot),\,\cdot\,)$ in this case. It follows from Lemma 8.1 that for all functions $g_0,\dots,g_K\in L^\infty(\mathcal X,\mu)$ and each $K\in\mathbb{N}$. 2. If $U\in\mathcal R(\mathbb{T}^n)$, then by Proposition 9.4 for each pair of functions $g_1,g_2\in\mathcal{AC}(\mathbb{T}^n)$.We return to the general case when $U\in\mathcal N$. From the definition of the Markov operator $T_U$ (see (10.5)) we easily obtain the following formula:
$$
\begin{equation*}
\mathcal{I}_U(\mathbf{G})=\int_{\mathcal{X}}\widehat g_1T_U\widehat g_2T_U\dotsb T_U\widehat g_{m-1}T_U\widehat g_m (\mathbf{1}_{\mathcal{X}})\,d\mu.
\end{equation*}
\notag
$$
It implies that
$$
\begin{equation}
\nu_{m}\biggl(\prod_{s=1}^{m}X_s\biggr)=\int_{\mathcal{X}} \widehat{\bf 1}_{X_1}T_U \widehat{\bf 1}_{X_2}T_U\dotsb T_U \widehat{\bf 1}_{X_{m-1}}T_U \widehat{\bf 1}_{X_m} (\mathbf{1}_{\mathcal{X}})\,d\mu.
\end{equation}
\tag{11.5}
$$
By Ionescu-Tulcea’s theorem (see [5], vol. 2, Corollary 10.7.4, or [25], Ch. 2, § 9, Theorem 2) there exists a unique probability measure $\nu_{\infty}$ on the measurable space $(\mathcal{X}^{\infty},\mathcal{B}^{\infty})$ such that
$$
\begin{equation*}
\nu_{\infty}(C_m(B))=\nu_{m}(B), \qquad B\in\mathcal{B}^{m}, \quad m\in\mathbb{N}.
\end{equation*}
\notag
$$
We claim that for all $m\in\mathbb{N}$ and $B\in\mathcal{B}^{m}$ we have
$$
\begin{equation}
\nu_{\infty}(C_m(\mathcal{X}\times B))=\nu_{\infty}(C_m(B)).
\end{equation}
\tag{11.6}
$$
In fact, let $X_1,\dots,X_m\in\mathcal{B}$. Using (11.5) and taking the equality $ \widehat{\bf 1}_{\mathcal{X}}=\mathrm{id}$ into account we obtain
$$
\begin{equation*}
\begin{aligned} \, \nu_{m+1}\biggl(\mathcal{X}\times\prod_{s=1}^{m}X_s\biggr) &=\int_{\mathcal{X}}T_U \widehat{\bf 1}_{X_1}T_U\dotsb T_U \widehat{\bf 1}_{X_{m-1}}T_U \widehat{\bf 1}_{X_m} (\mathbf{1}_{\mathcal{X}})\,d\mu \\ &=\int_{\mathcal{X}} \widehat{\bf 1}_{X_1}T_U\dotsb T_U \widehat{\bf 1}_{X_{m-1}}T_U \widehat{\bf 1}_{X_m} (\mathbf{1}_{\mathcal{X}})\,d\mu =\nu_{m}\biggl(\prod_{s=1}^{m}X_s\biggr) \end{aligned}
\end{equation*}
\notag
$$
(the second equality holds by property $\mathrm{(c)}$ of $T_U$: see Theorem 10.1). Hence (11.6) holds for $B=X_1\times\dots\times X_m$ and therefore for all $B\in\mathcal{B}^{m}$. Note that it follows from (11.6) that
$$
\begin{equation*}
\nu_{\infty}(p_s^{-1}(X))=\mu(X), \qquad s\in\mathbb{N}, \quad X\in\mathcal{B}.
\end{equation*}
\notag
$$
Therefore, considering an arbitrary partition $\chi\!=\!\{\mkern-1mu X_1,\mkern-1mu\dots,\mkern-1mu X_J\mkern-1mu\}$ of the space $(\mkern-1mu\mathcal{X},\mkern-1mu\mathcal{B},\mkern-1mu\mu\mkern-1mu)$, we see that for each $s \in \mathbb{N}$ the system of sets $p_s^{-1}\chi := \{p_s^{-1}(X_1),\dots,p_s^{-1}(X_J)\}$ is a partition of $(\mathcal{X}^{\infty},\mathcal{B}^{\infty},\nu_{\infty})$. IV. In the space $\mathcal{X}^{\infty}$ of infinite sequences we can introduce the shift transformation
$$
\begin{equation}
Q\colon \mathcal{X}^{\infty}\to \mathcal{X}^{\infty}, \qquad Qx=x', \quad \text{where }\ x'_s=x_{s+1}, \quad s\in\mathbb{N}.
\end{equation}
\tag{11.7}
$$
It follows from (11.6) that $Q$ is an endomorphism of the probability space $(\mathcal{X}^{\infty},\mathcal{B}^{\infty},\nu_{\infty})$. 11.2. The definition of the entropy of a unitary operatorIn this subsection we consider a probability space $(\mathcal{X},\mathcal{B},\mu)$ and a unitary operator $U\in\mathcal N$. For arbitrary integers $m,J\in\mathbb{N}$ let $\mathcal{S}_{m,J}$ denote the family of maps $\sigma$: ${\{1,\dots,m\}\to\{1,\dots,J\}}$. Let $\chi=\{X_1,\dots,X_J\}$ be a partition of $(\mathcal{X},\mathcal{B},\mu)$, and let $\sigma\in \mathcal{S}_{m,J}$. Set
$$
\begin{equation*}
\mathbf{G}_{\sigma}=\mathbf{G}_{\sigma}(\chi) =(\mathbf{1}_{X_{\sigma(1)}},\dots,\mathbf{1}_{X_{\sigma(m)}}).
\end{equation*}
\notag
$$
Note that $\mathcal{I}_U(\mathbf{G}_{\sigma})=\nu_{m}(X_{\sigma(1)}\times\dots\times X_{\sigma(m)})$, and therefore
$$
\begin{equation}
\mathcal{I}_U(\mathbf{G}_{\sigma})=\nu_\infty \biggl(\bigcap_{s=1}^{m}p_s^{-1}(X_{\sigma(s)})\biggr).
\end{equation}
\tag{11.8}
$$
For each $m\in\mathbb{N}$ and any partition $\chi=\{X_1,\dots,X_J\}$ of the space $\mathcal{X}$ set
$$
\begin{equation*}
\mathfrak{h}(U,\chi,m)=-\sum_{\sigma\in\mathcal{S}_{m,J}}\mathcal{I}_U(\mathbf{G}_{\sigma})\log \mathcal{I}_U(\mathbf{G}_{\sigma}), \qquad \mathbf{G}_{\sigma}=\mathbf{G}_{\sigma}(\chi).
\end{equation*}
\notag
$$
It follows from (11.8) that $\mathfrak{h}(U,\chi,m)=h_{\nu_{\infty}}\bigl(\bigvee_{s=1}^{m}p_s^{-1}\chi\bigr)$, where the function $h_{\nu_\infty}(\cdot)$ is defined by (11.1) for the probability space $(\mathcal X^\infty,\mathcal B^\infty,\nu_\infty)$. Moreover, since for each nonnegative integer $i$ we have $p_1\circ Q^i=p_{i+1}$, where $Q$ is defined in (11.7), it follows that
$$
\begin{equation*}
h_{\nu_{\infty}}\biggl(\bigvee_{s=1}^{m}p_s^{-1}\chi\biggr) =h_{\nu_{\infty}}\biggl(\bigvee_{i=0}^{m-1}Q^{-i}(p_1^{-1}\chi)\biggr),
\end{equation*}
\notag
$$
so that, in view of (11.2),
$$
\begin{equation}
\mathfrak{h}(U,\chi,m)=h_{\nu_{\infty}}(Q,p_1^{-1}\chi,m).
\end{equation}
\tag{11.9}
$$
Equality (11.9) and properties (11.3) and (11.4) yield the following lemmas. Lemma 11.1. If $\chi$ is a partition of the space $\mathcal{X}$, then for all positive integers $m_1$ and $m_2$ Lemma 11.2. Let $m\in\mathbb{N}$ and let $\kappa$ and $\chi$ be partitions of the space $\mathcal{X}$. If $\kappa$ is a subpartition of $\chi$, then From Lemma 11.1 we see that there exists a finite nonnegative limit
$$
\begin{equation*}
\mathfrak{h}(U,\chi)=\lim_{m\to\infty}\frac{1}{m}\mathfrak{h}(U,\chi,m) =\inf_{m\geqslant1}\frac{1}{m}\mathfrak{h}(U,\chi,m).
\end{equation*}
\notag
$$
Definition 11.1. The entropy of a unitary operator $U\in\mathcal N$ is the quantity where the supremum is taken over all finite partitions $\chi$ of the space $(\mathcal{X},\mathcal{B},\mu)$. It follows from Lemma 11.2 that the function $\mathfrak{h}(U,\chi)$ tends to the supremum $\mathfrak{h}(U)$ as the partition $\chi$ is successively refined. The next lemma follows from the construction of the entropy $\mathfrak{h}$. Lemma 11.3. Let $U_1$ and $U_2$ be operators in $\mathcal N$. If $\mu_{U_1}(\,\cdot\,{,}\,\cdot\,)=\mu_{U_2}(\,\cdot\,{,}\,\cdot\,)$, then $\mathfrak{h}(U_1)=\mathfrak{h}(U_2)$. 11.3. Examples of calculations of entropy11.3.1. The entropy of a Koopman operatorLet $F\in\operatorname{Aut}(\mathcal X,\mu)$, $U=U_F$, and let $\chi=\{X_1,\dots,X_J\}$ be a partition of the space $\mathcal X$. It follows from the definition of $\mathcal I_U$ that
$$
\begin{equation*}
\mathcal I_U(\mathbf{G}_\sigma)=\mu\biggl(\bigcap_{i=0}^{m-1}F^{-i}(X_{\sigma(m-i)})\biggr)
\end{equation*}
\notag
$$
for all $m\in\mathbb{N}$ and $\sigma\in\mathcal S_{m,J}$. Hence $\mathfrak{h}(U,\chi,m)=h_{\mu}(F,\chi,m)$, and therefore where $h_\mu(F)$ is the Kolmogorov-Sinai entropy of the automorphism $F$. In particular, $\mathfrak{h}(U_{\operatorname{id}})=h_\mu(\operatorname{id})=0$. 11.3.2. The entropy of an operator $\widehat g$Let $g\in\mathcal{AC}(\mathbb{T}^n)$ and $|g|=1$. By Lemma 7.7 $\widehat g$ is a regular unitary operator. Moreover, by example 1 in § 8.4, $\mu_{\widehat g}(a,\,\cdot\,)=\delta(a,\,\cdot\,)$ at each point $a\in\mathbb{T}^n$, so that
$$
\begin{equation*}
\mathfrak{h}(\widehat g)=\mathfrak{h}(U_{\operatorname{id}})=h_\mu(\operatorname{id})=0
\end{equation*}
\notag
$$
by Lemma 11.3. 11.3.3. The entropy of the operator $\operatorname{Conv}_{\delta_x}$Let $x\in\mathbb{T}^n$. It follows from Lemma 7.8 and the equality $\operatorname{Conv}_{\delta_x}=U_{F_{-x}}$ that $\operatorname{Conv}_{\delta_x}$ is a regular unitary operator and
$$
\begin{equation*}
\mathfrak{h}(\operatorname{Conv}_{\delta_x})=h_\mu(F_{-x})=0.
\end{equation*}
\notag
$$
11.3.4. The entropy of a Schrödinger propagatorThe Schrödinger propagator $U$ of a free particle on the circle has the form
$$
\begin{equation}
U=\operatorname{Conv}_\lambda, \qquad \lambda=\sum_{k\in\mathbb{Z}}\lambda_ke^*_k, \quad \lambda_k=e^{itk^2},
\end{equation}
\tag{11.10}
$$
where $t\in\mathbb{R}$. By Lemma 7.9 $U$ is a regular unitary operator such that
$$
\begin{equation*}
v_m(a)=e^{itm^2}\delta_{tm,\pi\mathbb{Z}} \quad\text{for all }\ m\in\mathbb{Z}, \quad a\in\mathbb{T}.
\end{equation*}
\notag
$$
Proposition 11.1. The entropy of the operator (11.10) can be calculated as follows: where $(p,q)$ is the greatest common divisor of $p$ and $q$. Proof. We identify the circle $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$ and the half-closed interval $[0,2\pi)$. Consider three cases.
1) If $t/\pi \in\mathbb{R}\setminus\mathbb{Q}$, then for each $m\in\mathbb{Z}$ the function $v_m$ is identically equal to $\delta_{m,0}$. Therefore, taking (8.16) into account we obtain that for each $a\in \mathbb{T}$ the measure $\mu_U(a,\,\cdot\,)$ coincides with the normalized Lebesgue measure $\mu$ on $\mathcal B(\mathbb{T})$. Now we look at the partition
$$
\begin{equation}
\chi_J=\{X_1,\dots,X_J\}, \qquad X_j=\biggl[\frac{2\pi(j-1)}{J},\frac{2\pi j}{J}\biggr), \quad j=1,\dots,J,
\end{equation}
\tag{11.11}
$$
and for all $m\in\mathbb{N}$ and $\sigma\in\mathcal S_{m,J}$ consider the system of functions $\mathbf{G}_\sigma=\mathbf{G}_{\sigma}(\chi_J)$. Then
$$
\begin{equation*}
\mathcal I_U(\mathbf{G}_\sigma)= \frac1{(2\pi)^m} \int_{\mathbb{T}^m} {\bf 1}_{X_{\sigma(1)}}(x_1) \dotsb {\bf 1}_{X_{\sigma(m)}}(x_m)\, dx_m\dotsb dx_1 = J^{-m}.
\end{equation*}
\notag
$$
Therefore, $\mathfrak{h}(U,\chi_J,m)=m\log J$, so that $\mathfrak{h}(U,\chi_J)=\log J$ and thus $\mathfrak{h}(U)=\infty$.
2) Now let $t/\pi=p / q$, where $p\in\mathbb{Z}$, $ q\in\mathbb{N}$, $ (p,q)=1$ and either $p$ or $q$ is even. Then by (8.16), at each point $a\in\mathbb{T}$ the sequence of Fourier coefficients of the measure $\mu_U(a,\,\cdot\,)$ coincides with $\{e^{-ima}\delta_{m,q\mathbb{Z}}\}_{m\in\mathbb{Z}}$, and therefore
$$
\begin{equation*}
\mu_U(a,\,\cdot\,)=\frac{1}{q}\sum_{s=0}^{q-1}\delta\biggl(a+\frac{2\pi s}q,\,\cdot\,\biggr), \qquad a\in\mathbb{T}.
\end{equation*}
\notag
$$
Let $\xi$ be some partition of the space $(\mathbb{T},\mathcal B(\mathbb{T}),\mu)$ and let $T\colon \mathbb{T}\to\mathbb{T}$ be the rotation of the circle by $2\pi/q$: Consider the partition where $\chi_q$ is defined by (11.11) (for $J=q$).Then $\chi$ has the following properties:
Thus we can represent $\chi$ as follows:
$$
\begin{equation*}
\chi=\{X_{jk}\}_{1\leqslant j\leqslant J,\, 0\leqslant k\leqslant q-1}, \qquad X_{j,k}=T^{k}(X_{j,0}).
\end{equation*}
\notag
$$
Let $m\in\mathbb{N}$ and $\sigma\colon \{1,\dots,m\}\to \{1,\dots,J\}\times\{0,\dots,q-1\}$. We can represent $\sigma$ as the diagonal of two maps, $\sigma'\in\mathcal S_{m,J}$ and $\sigma''\in\mathcal S_{m,q}$:
$$
\begin{equation*}
\sigma(r)=(\sigma'(r),\sigma''(r)), \qquad \sigma'(r)\in \{1,\dots,J\}, \quad \sigma''(r)\in \{0,\dots,q-1\}
\end{equation*}
\notag
$$
for $r=1,\dots,m$. Then
$$
\begin{equation*}
\begin{aligned} \, \mathcal{I}_U(\mathbf{G}_\sigma) &=\int_{\mathbb{T}}\!\dotsi\!\int_{\mathbb{T}}{\bf 1}_{X_{\sigma(1)}}(x_1)\dotsb{\bf 1}_{X_{\sigma(m)}}(x_m)\,\mu_U(x_{m-1},dx_{m})\dotsb\mu(dx_1) \\ &=q^{1-m}\sum_{0\leqslant s_2,\dots,s_{m}\leqslant q-1}\mu\bigl(T^{-s_2-\dots-s_m}(X_{\sigma(m)})\cap\dots\cap T^{-s_2}(X_{\sigma(2)})\cap X_{\sigma(1)}\bigr) \\ &=\begin{cases} q^{1-m}\mu(X_{\sigma'(1),0}) & \text{if }\ \sigma'(m)=\dots=\sigma'(1), \\ 0& \text{otherwise}. \end{cases} \end{aligned}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\begin{aligned} \, \mathfrak h(U,\chi,m) &=- q^{m} \sum_{j=1}^J \frac{\mu(X_{j,0})}{q^{m-1}}\log \frac{\mu(X_{j,0})}{q^{m-1}} \\ &=- q \sum_{j=1}^J \mu(X_{j,0})\log \mu(X_{j,0})+q \sum_{j=1}^J \mu(X_{j,0})\log q^{m-1}. \end{aligned}
\end{equation*}
\notag
$$
Since $\sum_{j=1}^J\mu(X_{j,0})=1/q$, it follows that $\mathfrak h(U,\chi)=\log q$.
Thus, for each partition $\xi$ we have constructed a subpartition $\chi$ such that $\mathfrak h(U,\chi)=\log q$. Thus, applying Lemma 11.2 we obtain $\mathfrak h(U)=\log q$. 3) Finally, let $t/\pi=p / q$, where $p\in\mathbb{Z}$, $q\in\mathbb{N}$, $(p,q)=1$, and both $p$ and $q$ are odd. Then at each point $a\in\mathbb{T}$ the sequence of Fourier coefficients of the measure $\mu_U(a,\,\cdot\,)$ coincides with $\{e^{i\pi m/q}e^{-ima}\delta_{m,q\mathbb{Z}}\}_{m\in\mathbb{Z}}$ by (8.16), and we obtain
$$
\begin{equation*}
\mu_U(a,\,\cdot\,)=\frac{1}{q}\sum_{s=0}^{q-1}\delta \biggl(a+\frac{2\pi s}q-\frac\pi q,\,\cdot\,\biggr), \qquad a\in\mathbb{T}.
\end{equation*}
\notag
$$
The rest of the argument is similar to case 2).
Proposition 11.1 is proved.
Citation in format AMSBIB
\Bibitem{AfoTre22} |
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