The fundamental equations for inversion of operator pencils on Banach space

TL;DR

This paper establishes fundamental equations characterizing the resolvent of operator pencils on Banach spaces, extending previous results to cases with isolated essential singularities and applying to polynomial pencils.

Contribution

It introduces a comprehensive set of fundamental equations for the resolvent of operator pencils, including cases with essential singularities, and provides a closed-form resolvent expression.

Findings

Resolvent is analytic on an annulus if and only if fundamental equations are satisfied.

Extended fundamental equations to include isolated essential singularities.

Derived spectral separation properties for pencils with finite isolated singularities.

Abstract

We prove that the resolvent of a linear operator pencil is analytic on an open annulus if and only if the coefficients of the Laurent series satisfy a system of fundamental equations and are geometrically bounded. Our analysis extends earlier work on the fundamental equations to include the case where the resolvent has an isolated essential singularity. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. Finally we show that our results can also be applied to polynomial pencils.