Pseudospectra of elements of reduced Banach algebras

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