Ascent spectrum and essential ascent spectrum

O. Bel Hadj Fredj¹ ; M. Burgos² ; M. Oudghiri³

¹ UFR de Mathématiques, UMR-CNRS 8524 Université de Lille 1 59655 Villeneuve d'Ascq, France
² Departamento de Análisis Matemático Universidad de Granada 18071 Granada, Spain
³ Faculté des Sciences Université d'Oujda Oujda, Maroc

Studia Mathematica, Tome 187 (2008) no. 1, pp. 59-73

Résumé

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.

Détail

DOI : 10.4064/sm187-1-3

Keywords: study essential ascent related essential ascent spectrum operator banach space banach space has finite dimension only essential ascent every operator finite focus stability essential ascent spectrum under perturbations prove operator has finite rank power only sigma asc sigma asc every operator commuting quasi nilpotent part analytic core single valued extension property analyzed operators finite essential ascent