Essential Descent Spectrum Equality

Abstract

A bounded operator $T$ in a Banach space $X$ is said to satisfy the essential descent spectrum equality, if the descent spectrum of $T$ as an operator on $X$ coincides with the essential descent spectrum of $T$. In this note, we give some conditions under which the equality ${\sigma }_{desc}(T)={\sigma }_{desc}^e(T)$ holds for a single operator $T$.