Spectrum of a family of operators

TL;DR

This paper extends the classical concepts of resolvent set and spectrum from individual operators to families of operators on Banach spaces, including asymptotic cases, providing new theoretical insights.

Contribution

It introduces the spectrum and resolvent set for families of operators and adapts classical results to asymptotic scenarios, broadening spectral theory.

Findings

Defined spectrum and resolvent set for operator families

Extended classical spectral results to asymptotic cases

Provided theoretical framework for future research

Abstract

Having as start point the classic definitions of resolvent set and spectrum of a linear bounded operator on a Banach space, we introduce the resolvent set and spectrum of a family of linear bounded operators on a Banach space. In addition, we present some results which adapt to asymptotic case the classic results.