Problem of Descent Spectrum Equality

TL;DR

This paper investigates conditions under which the descent spectrum of an operator on an infinite dimensional Banach space matches its spectrum as an element of the algebra of bounded operators, focusing on operators with spectra having non-empty interior.

Contribution

It characterizes when the descent spectrum equality holds for operators with spectra of non-empty interior, advancing understanding of spectral properties in Banach space operator theory.

Findings

Identifies conditions for descent spectrum equality when the spectrum has non-empty interior.

Provides criteria linking spectral properties of operators to their descent spectrum.

Enhances theoretical understanding of spectral behavior in infinite-dimensional settings.

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