Pseudospectra of elements of reduced Banach algebras (Pre-published version)

Abstract

Let A be a Banach algebra with identity 1 and p ∈ A be a non-trivial idempotent. Then q = 1−p is also an idempotent. The subalgebras pAp and qAq are Banach algebras, called reduced Banach algebras, with identities p and q respectively. For a ∈ A and ε > 0, we examine the relationship between the ε-pseudospectrum Λε(A, a) of a ∈ A, and ε-pseudospectra of pap ∈ pAp and qaq ∈ qAq. We also extend this study by considering a finite number of idempotents p1, · · · , pn, as well as an arbitrary family of idempotents satisfying certain conditions.