2008
DOI: 10.4064/sm187-1-3
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Ascent spectrum and essential ascent spectrum

Abstract: 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 σ e asc (T + F ) = σ e asc (T ) for every operator T commuting with F . The quasi-nilpotent part, the analytic cor… Show more

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Cited by 22 publications
(4 citation statements)
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“…Directly from Theorem 3.1 we obtain the next corollary which generalizes the corresponding results of [8,12,13]. The TUD spectrum of an operator is not necessarily non-empty.…”
Section: Corollary 33 Let T ∈ B(x) Then T Is Meromorphic If and Onmentioning
confidence: 50%
See 1 more Smart Citation
“…Directly from Theorem 3.1 we obtain the next corollary which generalizes the corresponding results of [8,12,13]. The TUD spectrum of an operator is not necessarily non-empty.…”
Section: Corollary 33 Let T ∈ B(x) Then T Is Meromorphic If and Onmentioning
confidence: 50%
“…Several authors in [8,12,13] discuss the left Drazin spectrum, essential left Drazin spectrum, descent spectrum and essential descent spectrum of an operator. They get some characterizations of operators whose spectra of these kinds are at most countable.…”
Section: Corollary 33 Let T ∈ B(x) Then T Is Meromorphic If and Onmentioning
confidence: 99%
“…It is worth noting that the space of bounded linear operators B(X) is a particular case of the Banach algebra \scrA and Asc(B(X)) and Dsc(B(X)) are regularities on B(X). For more information, see [14,5,20]. Throughout our work, under suitable optimal hypothesis, we will prove that conditions (C 1 ) and (C 2 ) are approved properties for the ascent and descent spectra.…”
Section: Introduction and Main Resultsmentioning
confidence: 81%
“…Recently, stability problems of operators under perturbation have attracted many researchers and undergone important contributions, see for instance [2,13,14,4,6,7,8,12,15,18,20]. Recall that, in [10], authors have proved nice relations between the left Browder spectrum and the left invertible spectrum (respectively, between the right The concept of Definition 1.3 is truly suitable since it comprises for example the left (right) spectrum, the left (right) approximate point spectrum, the Harte (the union of the left and right) spectrum.…”
Section: Introduction and Main Resultsmentioning
confidence: 99%