General Solution of the Poisson Equation for Quasi-Birth-and-Death Processes

We consider the Poisson equation $(I-P)u=g$, where $P$ is the transition matrix of a quasi-birth-and-death process with infinitely many levels, $g$ is a given infinite dimensional vector, and $u$ is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix $P$ to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples.