Semi-Infinite Quasi-Toeplitz Matrices with Applications to QBD Stochastic Processes

Abstract

Denote by ${\mathcal{W}}_1$ the set of complex valued functions of the form $a(z)={\sum }_{i=-\infty }^{+\infty }{a}_i{z}^i$ which are continuous on the unit circle, and such that ${\sum }_{i=-\infty }^{+\infty }|i{a}_i|<\infty$. We call CQT matrix a quasi-Toeplitz matrix $A$, associated with a continuous symbol $a(z)\in {\mathcal{W}}_1$, of the form $A=T(a)+E$, where $T(a)=(t_{i,j}{)}_{i,j\in {\mathbb{Z}}^{+}}$ is the semi-infinite Toeplitz matrix such that $t_{i,j}=a_{j-i}$, for $i,j\in {\mathbb{Z}}^{+}$, and $E=(e_{i,j}{)}_{i,j\in {\mathbb{Z}}^{+}}$ is a semi-infinite matrix such that ${\sum }_{i,j=1}^{+\infty }|e_{i,j}|$ is finite. We prove that the class of CQT matrices is a Banach algebra with a suitable sub-multiplicative matrix norm $\Vert \cdot \Vert$. We introduce a finite representation of CQT matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind $A{X}^2+BX+C=0$, encountered in the solution of Quasi-Birth and Death (QBD) stochastic processes with a denumerable set of phases, is presented where $A,B,C$ are CQT matrices.

References in corpus (4)

• Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process
• On the exponential of semi-infinite quasi-Toeplitz matrices
• On Functions of quasi Toeplitz matrices
• Decay properties for functions of matrices over $C^{\ast }$-algebras

Cited by in corpus (5)

• On the exponential of semi-infinite quasi-Toeplitz matrices
• Solving quadratic matrix equations arising in random walks in the quarter plane
• On Functions of quasi Toeplitz matrices
• A computational framework for two-dimensional random walks with restarts
• Computing eigenvalues of semi-infinite quasi-Toeplitz matrices