Displaying similar documents to “A note on preservation of spectra for two given operators”

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

Qingping Zeng, Qiaofen Jiang, Huaijie Zhong (2013)

Studia Mathematica

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Burgos, Kaidi, Mbekhta and Oudghiri [J. Operator Theory 56 (2006)] provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator F is of power finite rank if and only if σ d s c ( T + F ) = σ d s c ( T ) for every operator T commuting with F. Later, several authors extended this result to the essential descent spectrum, left Drazin spectrum and left essential Drazin spectrum. In this paper, using the theory of operators with eventual topological uniform descent and the technique used...

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.

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...

A note on the index of B -Fredholm operators

M. Berkani, Dagmar Medková (2004)

Mathematica Bohemica

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From Corollary 3.5 in [Berkani, M; Sarih, M.; Studia Math. 148 (2001), 251–257] we know that if S , T are commuting B -Fredholm operators acting on a Banach space X , then S T is a B -Fredholm operator. In this note we show that in general we do not have error ( S T ) = error ( S ) + error ( T ) , contrarily to what has been announced in Theorem 3.2 in [Berkani, M; Proc. Amer. Math. Soc. 130 (2002), 1717–1723]. However, if there exist U , V ∈ L ( X ) such that S , T , U , V are commuting and U S + V T = I , then error ( S T ) = error ( S ) + error ( T ) , where error stands for the index of a B -Fredholm...

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 ∈ 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.

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...