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

摘要

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 }ert ia_iertinfty $. 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,jin mathbb{Z}^+}$ is the semi-infinite Toeplitz matrix such that $ t_{i,j}=a_{j-i}$, for $ i,jin mathbb{Z}^+$, and $ E=(e_{i,j})_{i,jin mathbb{Z}^+}$ is a semi-infinite matrix such that $ sum _{i,j=1}^{+infty }ert e_{i,j}ert$ 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.