Pseudospectra of elements of reduced Banach algebras II

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. Let x ∈ A be such that pxp = xp, and ε > 0. We examine the relationship between the spectrum of x ∈ A, σ(A, x), and the spectra of pxp ∈ pAp, σ(pAp, pxp) and qxq ∈ qAq, σ(qAq, qxq). Similarly, we examine the relationship betweeen the ε-pseudospectrum of x ∈ A, Λε(A, x) and ε-pseudospectra of pxp ∈ pAp, Λε(pAp, pxp) and of qxq ∈ qAq, Λε(qAq, qxq)