Decomposition of the (n, ϵ) -pseudospectrum of an element of a Banach algebra

Article Type

Research Article

Publication Title

Advances in Operator Theory

Abstract

Let A be a complex Banach algebra with unit. For an integer n≥ 0 and ϵ> 0 , the (n, ϵ) -pseudospectrum of a∈ A is defined by Λn,ϵ(A,a):={λ∈C:(λ-a)is not invertible inAor‖(λ-a)-2n‖1/2n≥1ϵ}.Let p∈ A be a nontrivial idempotent. Then pAp= { pbp: b∈ A} is a Banach subalgebra of A with unit p, known as a reduced Banach algebra. Suppose ap= pa. We study the relationship of Λn,ϵ(A, a) and Λn,ϵ(pAp, pa). We extend this by considering first a finite family, and then an at most countable family of idempotents satisfying some conditions. We establish that under suitable assumptions, the (n, ϵ) -pseudospectrum of a can be decomposed into the union of the (n, ϵ) -pseudospectra of some elements in reduced Banach algebras.

First Page

248

Last Page

260

DOI

10.1007/s43036-019-00016-x

Publication Date

2-1-2020

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