Local spectrum of a family of operators

TL;DR

This paper extends the concepts of local spectrum and resolvent set from single operators to families of operators on Banach spaces, introducing new definitions and generalizing known results.

Contribution

It introduces the local spectrum, resolvent set, and single-valued extension property for families of operators, expanding classical operator theory to a broader context.

Findings

Defined local spectrum and resolvent set for families of operators

Extended classical results to the case of operator families

Established properties analogous to single operators

Abstract

Starting from the classic definitions of local resolvent set and spectrum of a linear bounded operator on a Banach space, we introduce the local resolvent set and spectrum, the local space and the single-valued extention property of a family of linear bounded operators on a Banach space. Keeping the analogy with the classic case, we extend some the known results from the case of a linear bouded operator to the case of a family of linear bounded operators on a Banach space.