Inversion of an analytic operator function through Fredholm quotients and its application

Abstract

We characterize the inverse of an analytic Fredholm operator-valued function A(z) near an isolated singularity within a general Banach space framework. Our approach relies on the sequential factorization of A(z) via Fredholm quotient operators. By analyzing the properties of these quotient operators near an isolated singularity, we fully characterize the Laurent series expansion of the inverse of A(z) in terms of its Taylor coefficients around the singularity. These theoretical results are subsequently applied to characterize the solution of a general autoregressive law of motion in a Banach space.