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 $a(z)=\sum _{i=-\infty }^{+\infty }a_iz^i$ such that $\sum _{i=-\infty }^{+\infty }|ia_i|<\infty$. We call QT-matrix a quasi-Toeplitz matrix $A$, associated with a 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 QT-matrices is a Banach algebra with a suitable sub-multiplicative matrix norm. We introduce a finite representation of QT-matrices together with algorithms which implement elementary matrix operations. An application to solving quadratic matrix equations of the kind $AX^2+BX+C=0$, 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 QT-matrices.