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 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm187-1-3,
author = {O. Bel Hadj Fredj and M. Burgos and M. Oudghiri},
title = {Ascent spectrum and essential ascent spectrum},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {59-73},
zbl = {1160.47007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm187-1-3}
}
O. Bel Hadj Fredj; M. Burgos; M. Oudghiri. Ascent spectrum and essential ascent spectrum. Studia Mathematica, Tome 187 (2008) pp. 59-73. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm187-1-3/