Single-valued extension property at the points of the approximate point spectrum

Aiena, Pietro; Rosas, Ennis

doi:10.1016/s0022-247x(02)00708-4

Cited by 29 publications

(32 citation statements)

References 6 publications

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“…A local version of this property which dates back to Finch [7] has been recently investigated in the local spectral theory and Fredholm theory by many authors (see [1], [2], [3], and the references contained therein). Recall that a bounded linear operator T on a complex Banach space X is said to have the single-valued extension property at a point λ 0 ∈ C if for every open disc U centered at λ 0 the only analytic function φ : U → X that satisfies the equation…”

Section: Resultsmentioning

confidence: 99%

The single-valued extension property for bilateral operator weighted shifts

Bourhim

Chidume

2004

Proc. Amer. Math. Soc.

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Section: Resultsmentioning

confidence: 99%

The single-valued extension property for bilateral operator weighted shifts

Bourhim

Chidume

2004

Proc. Amer. Math. Soc.

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“…This property plays a crucial role in local spectral theory (see the recent monograph of Laursen and Neumann [26]). We shall consider a local version of this property, which has been studied in recent papers [3], [4], [6], and previously by Finch [20] and Mbekhta [29].…”

Section: Obviously P 00 (T ) ⊆ π 00 (T ) For Every T ∈ L(x) (1)mentioning

confidence: 99%

“…Furthermore, σ a (T ) does not cluster at λ ⇒ T has SVEP at λ, and σ s (T ) does not cluster at λ ⇒ T * has SVEP at λ (see [6]). Let us consider the quasi-nilpotent part of T , i.e.…”

Section: Obviously P 00 (T ) ⊆ π 00 (T ) For Every T ∈ L(x) (1)mentioning

confidence: 99%

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Classes of operators satisfying a-Weyl's theorem

Aiena

2005

Studia Math.

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Abstract. 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 H 0 (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. Notation and terminology.We begin with some standard notations in Fredholm theory. Throughout this note by L(X) we denote the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space X. For every T ∈ L(X) we denote by α(T ) and β(T ) the dimension of the kernel ker T and the codimension of the range T (X), respectively. The class of upper semi-Fredholm operators is defined by (T ) < ∞ and T (X) is closed}, whilst the class of lower semi-Fredholm operators is defined byFor a linear operator T the ascent p := p(T ) is defined as the smallest nonnegative integer p such that ker T p = ker T p+1 . If such an integer does not exist we put p(T ) = ∞. Analogously, the descent q := q(T ) is defined as the smallest nonnegative integer q such that T q (X) = T q+1 (X), and if such an integer does not exist we put q(T ) = ∞. A classical result states that if 2000 Mathematics Subject Classification: Primary 47A10, 47A11; Secondary 47A53, 47A55.

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“…Note that H 0 (T ) and K(T ) generally are not closed and above become equivalences if we assume that λI − T ∈ Φ ± (X), see [4], [6].…”

Section: Introduction and Definitionsmentioning

confidence: 99%

Generalized Browder’s Theorem and SVEP

Aiena

García

2007

MedJM

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A bounded operator T ∈ L(X), X a Banach space, is said to verify generalized Browder's theorem if the set of all spectral points that do not belong to the B-Weyl's spectrum coincides with the set of all poles of the resolvent of T , while T is said to verify generalized Weyl's theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder's theorem, or generalized 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 subsets of C. In the last part we give a general framework for which generalized Weyl's theorem follows for several classes of operators. Mathematics Subject Classification (2000). Primary 47A10, 47A11; Secondary 47A53, 47A55.

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Single-valued extension property at the points of the approximate point spectrum

Cited by 29 publications

References 6 publications

The single-valued extension property for bilateral operator weighted shifts

The single-valued extension property for bilateral operator weighted shifts

Classes of operators satisfying a-Weyl's theorem

Generalized Browder’s Theorem and SVEP

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