Problem of descent spectrum equality

Abstract

Let ${\mathcal {B}}(X)$ be the algebra of all bounded operators acting on an infinite dimensional complex Banach space X. We say that an operator $T \in {\mathcal {B}}(X)$ satisfies the problem of descent spectrum equality, if the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. In this paper we are interested in the problem of descent spectrum equality. Specifically, the problem is to consider the following question: let $T \in {\mathcal {B}}(X)$ such that $\sigma (T)$ has non empty interior, under which condition on T does $\sigma _{desc}(T)=\sigma _{desc}(T, {\mathcal {B}}(X))$?