Pseudospectra of elements of reduced banach algebras

Let A be a Banach algebra with identity 1 and p ∈ A be a nontrivial 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 2 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.