On the exponential of semi-infinite quasi-Toeplitz matrices

Abstract

Let $a(z)={\sum }_{i\in \mathbb{Z}}{a}_i{z}^i$ be a complex valued function defined for $|z|=1$, such that ${\sum }_{i\in \mathbb{Z}}|i{a}_i|<\infty$, and let $E=(e_{i,j}{)}_{i,j\in {\mathbb{Z}}^{+}}$ be such that ${\sum }_{i,j\in {\mathbb{Z}}^{+}}|e_{i,j}|<\infty$. A semi-infinite quasi-Toeplitz matrix is a matrix of the kind $A=T(a)+E$, where $T(a)=(t_{i,j}{)}_{i,j\in {\mathbb{Z}}^{+}}$ is the semi-infinite Toeplitz matrix associated with the symbol $a(z)$, that is, $t_{i,j}=a_{j-i}$ for $i,j\in {\mathbb{Z}}^{+}$. We analyze theoretical and computational properties of the exponential of $A$. More specifically, it is shown that $\exp (A)=T(\exp (a))+F$ where $F=(f_{i,j}{)}_{i,j\in {\mathbb{Z}}^{+}}$ is such that ${\sum }_{i,j\in {\mathbb{Z}}^{+}}|f_{i,j}|$ is finite, i.e., $\exp (A)$ is a semi-infinite quasi-Toeplitz matrix as well, and an effective algorithm for its computation is given. These results can be extended from the function $\exp (z)$ to any function $f(z)$ satisfying mild conditions, and can be applied to finite quasi-Toeplitz matrices.

References in corpus (4)

• Semi-Infinite Quasi-Toeplitz Matrices with Applications to QBD Stochastic Processes
• Fast computation of the matrix exponential for a Toeplitz matrix
• On Functions of quasi Toeplitz matrices
• Decay properties for functions of matrices over $C^{\ast }$-algebras

Cited by in corpus (4)

• Semi-Infinite Quasi-Toeplitz Matrices with Applications to QBD Stochastic Processes
• On Functions of quasi Toeplitz matrices
• A computational framework for two-dimensional random walks with restarts
• Computing eigenvalues of semi-infinite quasi-Toeplitz matrices