Toeplitz operators on weighted Hardy spaces

Let $\sigma$ be a weight on $\mathbb Z^+$ such that the usual shift $S\colon(u_n)_{n\geq0}\mapsto(u_{n-1})_{n\geq0}$ (with the convention $u_{-1}=0$) and the backward shift $T\colon(u_n)_{n\geq0}\mapsto(u_{n+1})_{n\geq0}$ are bounded on the weighted Hilbert space $l_\sigma^2(\mathbb Z^+):=\{u=(u_n)_{n\geq0}\mid\sum_{n\geq0}|u_n|^2\sigma^2(n)<+\infty\}$. Set $\sigma_*(n)=1/\sigma(-n)$ for $n\leq0$, and set $l^2_{\sigma_*}(\mathbb Z^-):=\{v=(v_n)_{n\leq0}\mid\sum_{n\leq0}|u_n|^2\sigma^2_*(n)<+\infty\}$. The existence of pairs $(u,v)\in l_\sigma^2(\mathbb Z^+)\times l^2_{\sigma_*}(\mathbb Z^-)$ with $u\ne0$, $v\ne0$ and $u\ast v=0$ is discussed. Such pairs will be called nontrivial solutions of the equation $u\ast v=0$.
A bounded operator $U$ on $l^2_\sigma(\mathbb Z^+)$ is called a Toeplitz operator if $TUS=U$. The map $(u,v)\mapsto u\ast v$ is a continuous bilinear map from $l_\sigma^2(\mathbb Z^+)\times l^2_{\sigma_*}(\mathbb Z^-)$ into a Banach space that can be identified with the predual of the space $\mathcal T_\sigma$ of Toeplitz operators on $l^2_\sigma(\mathbb Z^+)$. These Toeplitz operators, their “Fourier transforms”, and their symbols are discussed in $\S\,2,3$. In $\S\,4$, by using the “Brown approximation method”, many examples of weights $\sigma$ on $\mathbb Z^+$ are given for which the equation $u\ast v=0$ has nontrivial solutions. In particular, it is shown that there exist nondecreasing weights on $\mathbb Z^+$ of arbitrarily slow growth such that $\rho(S)=\rho(T)=1$ and the equation $u\ast v=0$ has many nontrivial solutions. This result is somewhat surprising because, surely, the equation $u\ast v$ has only trivial solutions on $l^2(\mathbb Z^+)\times l^2(\mathbb Z^-)$.