On functions of quasi-Toeplitz matrices

Let $a(z)=\sum_{i\in\mathbb Z}a_iz^i$ be a complex-valued function, defined for $|z|=1$, such that $\sum_{i=-\infty}^{+\infty} |ia_i|<\infty$. Consider the semi-infinite Toeplitz matrix $T(a)=(t_{i,j})_{i,j\in\mathbb Z^+}$ associated with the symbol $a(z)$ such that $t_{i,j}=a_{j-i}$. A quasi-Toeplitz matrix associated with the symbol $a(z)$ is a matrix of the form $A=T(a)+E$ where $E=(e_{i,j})$, $\sum_{i,j\in\mathbb Z^+}|e_{i,j}|<\infty$, and is called a $\mathrm{QT}$-matrix. Given a function $f(x)$ and a $\mathrm{QT}$-matrix $M$, we provide conditions under which $f(M)$ is well defined and is a $\mathrm{QT}$-matrix. Moreover, we introduce a parametrization of $\mathrm{QT}$-matrices and algorithms for the computation of $f(M)$. We treat the case where $f(x)$ is given in terms of power series and the case where $f(x)$ is defined in terms of a Cauchy integral. This analysis is also applied to finite matrices which can be written as the sum of a Toeplitz matrix and a low rank correction.
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