Semi-infinite quasi-Toeplitz matrices with applications to QBD stochastic processes

The class of semi-infinite Analytically Quasi-Toeplitz (AQT) matrices is introduced. This class is formed by matrices which can be written in the form $A=T(a)+E$, where $T(a)=(t_{i,j})_{i,j\in\Z^+}$ is the semi-infinite Toeplitz matrix associated with the symbol $a(z)=\sum_{i=-\infty}^{+\infty}a_iz^i$, that is $t_{i,j}=a_{j-i}$, for $i,j\in\mathbb Z^+$, $E=(e_{i,j})_{i,j\in\Z^+}$ is a semi-infinite matrix such that $\sum_{i,j=1}^{+\infty}|e_{i,j}|$ is finite, and $a(z)$ is an analytic function over an annulus $\mathbb A(r,R)=\{z\in\mathbb C:\quad r<|z|<R\}$ for $r<1<R$. We prove that AQT matrices are closed under multiplication and inversion, moreover we define a matrix norm $\|\cdot\|$ such that $\|AB\|\le\|A\|\cdot\|B\|$ for any pair $A,B$ of AQT matrices. We introduce a finite representation of AQT matrices and 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.