On the isolated points of the surjective spectrum of a bounded operator

Мbekhta, Mostafa; Oudghiri, Mourad

doi:10.1090/s0002-9939-08-09549-x

Cited by 11 publications

(11 citation statements)

References 12 publications

(8 reference statements)

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“…(ii) ⇒ (i) The condition X = H 0 (λI − T ) + K(λI − T ) is equivalent to the inclusion λ ∈ iso σ s (T ), see [22,Theorem 5]. Hence ∆ g a (T ) ⊆ iso σ s (T ) and from Theorem 3.4 it immediately follows that T satisfies property (gb).…”

Section: Theorem 38 For An Operators T ∈ L(x) the Following Statements Are Equivalentmentioning

confidence: 94%

Property (gb) through local spectral theory

Aiena¹,

Guillén²,

Peña³

2014

Mathematical Proceedings of the Royal Irish Academy

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Property (gb) for a bounded linear operator T ∈ L(X) on a Banach space X means that the points λ of the approximate point spectrum for which λI − T is upper semi B-Weyl are exactly the poles of the resolvent. In this paper we shall give several characterizations of property (gb). These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gb) holds for large classes of operators and prove the stability of property (gb) under some commuting perturbations.1991 Mathematics Reviews Primary 47A10, 47A11. Secondary 47A53, 47A55.

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Section: Theorem 38 For An Operators T ∈ L(x) the Following Statements Are Equivalentmentioning

confidence: 94%

Property (gb) through local spectral theory

Aiena¹,

Guillén²,

Peña³

2014

Mathematical Proceedings of the Royal Irish Academy

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“…(1) T has the SVEP at 0, We mention that the equivalence between (2) and (4) has recently been established in [10] without any restriction on T .…”

Section: It Is Well Known That T (K(t )) = K(t ) ⊆ Co(t ) and That Nementioning

confidence: 95%

Ascent spectrum and essential ascent spectrum

2008

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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 core and the single-valued extension property are also analyzed for operators with finite essential ascent.1. Introduction. Throughout this paper X will be an infinite-dimensional complex Banach space. We will denote by L (X) the algebra of all operators on X, and by F (X) and K (X) its ideals of finite rank and compact operators on X, respectively. For an operator T ∈ L (X), write T * for its adjoint, N(T ) for its kernel and R(T ) for its range. Also, denote by σ(T ), σ ap (T ) and σ su (T ) its spectrum, approximate point spectrum and surjective spectrum, respectively. An operator T ∈ L (X) is upper semiFredholm (respectively lower semi-Fredholm) if R(T ) is closed and dim N(T ) (respectively codim R(T )) is finite. If T is upper or lower semi-Fredholm, then T is called semi-Fredholm. The index of such an operator is given by ind(T ) = dim N(T ) − codim R(T ), and when it is finite we say that T is Fredholm. Recall that for T ∈ L (X), the ascent, a(T ), and the descent, d(T ), are defined by

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“…Proof. Suppose that 0 is an isolated point in σ ap (T ), then T has the SVEP at 0 and by [13,Proposition 9. ], H 0 (T ) and K(T ) are closed subspaces of X with K(T ) = X, H 0 (T ) = {0} and K(T ) ∩ H 0 (T ) = {0}.…”

Section: Left and Right Generalized Drazin Invertible Operators And Tmentioning

confidence: 99%

Left and right generalized Drazin invertible operators and local spectral theory

Benharrat

Hocine

Messirdi

2019

Proyecciones (Antofagasta)

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In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverse.2010 Mathematics Subject Classification. 47A10.

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On the isolated points of the surjective spectrum of a bounded operator

Cited by 11 publications

References 12 publications

Property (gb) through local spectral theory

Property (gb) through local spectral theory

Ascent spectrum and essential ascent spectrum

Left and right generalized Drazin invertible operators and local spectral theory

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