Spectra originating from semi-B-Fredholm theory and commuting perturbations

Qingping Zeng; Qiaofen Jiang; Huaijie Zhong

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Displaying similar documents to “Spectra originating from semi-B-Fredholm theory and commuting perturbations”

B-Fredholm and Drazin invertible operators through localized SVEP

M. Amouch, H. Zguitti (2011)

Mathematica Bohemica

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Let $X$ be a Banach space and $T$ be a bounded linear operator on $X$ . We denote by $S (T)$ the set of all complex $λ \in ℂ$ such that $T$ does not have the single-valued extension property at $λ$ . In this note we prove equality up to $S (T)$ between the left Drazin spectrum, the upper semi-B-Fredholm spectrum and the semi-essential approximate point spectrum. As applications, we investigate generalized Weyl’s theorem for operator matrices and multiplier operators.

Single valued extension property and generalized Weyl’s theorem

M. Berkani, N. Castro, S. V. Djordjević (2006)

Mathematica Bohemica

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Let $T$ be an operator acting on a Banach space $X$ , let $σ (T)$ and $σ_{B W} (T)$ be respectively the spectrum and the B-Weyl spectrum of $T$ . We say that $T$ satisfies the generalized Weyl’s theorem if $σ_{B W} (T) = σ (T) ∖ E (T)$ , where $E (T)$ is the set of all isolated eigenvalues of $T$ . The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$ , extending an earlier result of J. K. Finch...

Ascent spectrum and essential ascent spectrum

O. Bel Hadj Fredj, M. Burgos, M. Oudghiri (2008)

Studia Mathematica

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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 $σ_{a s c}^{e} (T + F) = σ_{a s c}^{e} (T)$ for every operator T commuting with F. The quasi-nilpotent part, the analytic core and the single-valued...

Extended Weyl type theorems

M. Berkani, H. Zariouh (2009)

Mathematica Bohemica

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An operator $T$ acting on a Banach space $X$ possesses property $(gw)$ if $σ_{a} (T) ∖ σ_{{SBF}_{+}^{-}} (T) = E (T),$ where $σ_{a} (T)$ is the approximate point spectrum of $T$ , $σ_{{SBF}_{+}^{-}} (T)$ is the essential semi-B-Fredholm spectrum of $T$ and $E (T)$ is the set of all isolated eigenvalues of $T .$ In this paper we introduce and study two new properties $(b)$ and $(gb)$ in connection with Weyl type theorems, which are analogous respectively to Browder’s theorem and generalized Browder’s theorem. Among other, we prove that if $T$ is a bounded linear operator acting on a Banach space...

On the essential approximate point spectrum II

V. Rakočević (1984)

Matematički Vesnik

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Restriction of an operator to the range of its powers

M. Berkani (2000)

Studia Mathematica

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Let T be a bounded linear operator acting on a Banach space X. For each integer n, define $T_{n}$ to be the restriction of T to $R (T^{n})$ viewed as a map from $R (T^{n})$ into $R (T^{n})$ . In [1] and [2] we have characterized operators T such that for a given integer n, the operator $T_{n}$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where $T_{n}$ belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with...

A characterization of some subsets of S-essential spectra of a multivalued linear operator

Teresa Álvarez, Aymen Ammar, Aref Jeribi (2014)

Colloquium Mathematicae

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We characterize some S-essential spectra of a closed linear relation in terms of certain linear relations of semi-Fredholm type.

A note on the $a$ -Browder’s and $a$ -Weyl’s theorems

M. Amouch, H. Zguitti (2008)

Mathematica Bohemica

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Let $T$ be a Banach space operator. In this paper we characterize $a$ -Browder’s theorem for $T$ by the localized single valued extension property. Also, we characterize $a$ -Weyl’s theorem under the condition $E^{a} (T) = π^{a} (T),$ where $E^{a} (T)$ is the set of all eigenvalues of $T$ which are isolated in the approximate point spectrum and $π^{a} (T)$ is the set of all left poles of $T .$ Some applications are also given.

Ascent, descent and roots of Fredholm operators

Bertram Yood (2003)

Studia Mathematica

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Let T be a Fredholm operator on a Banach space. Say T is rootless if there is no bounded linear operator S and no positive integer m ≥ 2 such that $S^{m} = T$ . Criteria and examples of rootlessness are given. This leads to a study of ascent and descent whether finite or infinite for T with examples having infinite ascent and descent.