Ascent spectrum and essential ascent spectrum
Volume 187 / 2008
Studia Mathematica 187 (2008), 59-73
MSC: 47A53, 47A55.
DOI: 10.4064/sm187-1-3
Abstract
We study the essential ascent and the related essential ascent spectrum of an operator on a Banach space. We show that a Banach space $X$ has finite dimension if and only if the essential ascent of every operator on $X$ is finite. We also focus on the stability of the essential ascent spectrum under perturbations, and we prove that an operator $F$ on $X$ has some finite rank power if and only if $\sigma_{{\rm asc}}^{{\rm e}} (T+F)=\sigma_{{\rm asc}}^{{\rm e}} (T)$ for every operator $T$ commuting with $F$. The quasi-nilpotent part, the analytic core and the single-valued extension property are also analyzed for operators with finite essential ascent.
