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Classes of operators satisfying a-Weyl's theorem

100%
Aiena P.
Studia Mathematica
|
2005
|
tom 169
|
nr 2
105-122
EN
In this article Weyl's theorem and a-Weyl's theorem on Banach spaces are related to an important property which has a leading role in local spectral theory: the single-valued extension theory. We show that if T has SVEP then Weyl's theorem and a-Weyl's theorem for T* are equivalent, and analogously, if T* has SVEP then Weyl's theorem and a-Weyl's theorem for T are equivalent. From this result we deduce that a-Weyl's theorem holds for classes of operators for which the quasi-nilpotent part H₀(λI - T) is equal to $ker(λI - T)^{p}$ for some p ∈ ℕ and every λ ∈ ℂ, and for algebraically paranormal operators on Hilbert spaces. We also improve recent results established by Curto and Han, Han and Lee, and Oudghiri.
2
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Generalized Weyl's theorem and quasi-affinity

64%
Aiena P., Berkani M.
Studia Mathematica
|
2010
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tom 198
|
nr 2
105-120
EN
A bounded operator T ∈ L(X) acting on a Banach space X is said to satisfy generalized Weyl's theorem if the complement in the spectrum of the B-Weyl spectrum is the set of all eigenvalues which are isolated points of the spectrum. We prove that generalized Weyl's theorem holds for several classes of operators, extending previous results of Istrăţescu and Curto-Han. We also consider the preservation of generalized Weyl's theorem between two operators T ∈ L(X), S ∈ L(Y) intertwined or asymptotically intertwined by a quasi-affinity A ∈ L(X,Y).
3
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Polaroid type operators under perturbations

64%
Aiena P., Aponte E.
Studia Mathematica
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2013
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tom 214
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nr 2
121-136
EN
A bounded operator T defined on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. The "polaroid" condition is related to the conditions of being left polaroid, right polaroid, or a-polaroid. In this paper we explore all these conditions under commuting perturbations K. As a consequence, we give a general framework from which we obtain, and also extend, recent results concerning Weyl type theorems (generalized or not) for T + K, where K is an algebraic or a quasi-nilpotent operator commuting with T.
4
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On inessential and improjective operators.

64%
Aiena P., González M.
Studia Mathematica
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1998
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tom 131
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nr 3
271-287
EN
We give several characterizations of the improjective operators, introduced by Tarafdar, and we characterize the inessential operators among the improjective operators. It is an interesting problem whether both classes of operators coincide in general. A positive answer would provide, for example, an intrinsic characterization of the inessential operators. We give several equivalent formulations of this problem and we show that the inessential operators acting between certain pairs of Banach spaces coincide with the improjective operators.
5
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On generalized a-Browder's theorem

64%
Aiena P., Miller T.
Studia Mathematica
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2007
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tom 180
|
nr 3
285-300
EN
We characterize the bounded linear operators T satisfying generalized a-Browder's theorem, or generalized a-Weyl's theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H₀(λI - T) as λ belongs to certain sets of ℂ. In the last part we give a general framework in which generalized a-Weyl's theorem follows for several classes of operators.
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