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 $\sigma_{desc}(T)=\sigma_{desc}(T, \mathcal{B}(X))$ ?