Reductions of Operator Pencils

TL;DR

This paper investigates the reduction processes of operator pencils on Banach spaces, establishing their commutativity, and explores how these reductions and pivot operators help describe the generalized spectrum and related problems.

Contribution

It proves the commutativity of two natural reduction operations for operator pencils and links invertibility of pivot operators to regularity and spectral properties.

Findings

Two reduction operations commute under mild conditions.

Invertibility of all pivot operators characterizes regular pencils.

Reduction and pivot operators describe the generalized spectrum.

Abstract

We study problems associated with an operator pencil, i.e., a pair of operators on Banach spaces. Two natural problems to consider are linear constrained differential equations and the description of the generalized spectrum. The main tool to tackle either of those problems is the reduction of the pencil. There are two kinds of natural reduction operations associated to a pencil, which are conjugate to each other. Our main result is that those two kinds of reductions commute, under some mild assumptions that we investigate thoroughly. Each reduction exhibits moreover a pivot operator. The invertibility of all the pivot operators of all possible successive reductions corresponds to the notion of regular pencil in the finite dimensional case, and to the inf-sup condition for saddle point problems on Hilbert spaces. Finally, we show how to use the reduction and the pivot operators to describe the generalized spectrum of the pencil.