Perturbation theory for the socle in Banach algebras (Q2414795)

Let \(A\) be a semisimple unital complex Banach algebra. The author proves that an element \(x\in A\) belongs to the socle of \(A\) (i.e., the sum of all minimal left ideals of \(A\)) if and only if \(\widehat{\sigma}(x+a)\setminus\widehat{\sigma}(a)\) is finite for each \(a\in A\), where \(\widehat{\sigma}\) stands for the polynomially convex hull of the spectrum.