General solution of the Poisson equation for Quasi-Birth-and-Death processes

We consider the Poisson equation $(I-P)\mathit{u}=\mathit{g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, $\mathit{g}$ is a given infinite dimensional vector and $\mathit{u}$ 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.