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A note on preservation of spectra for two given operators

Carlos Carpintero, Alexander Gutiérrez, Ennis Rosas, José Sanabria (2020)

Mathematica Bohemica

We study the relationships between the spectra derived from Fredholm theory corresponding to two given bounded linear operators acting on the same space. The main goal of this paper is to obtain sufficient conditions for which the spectra derived from Fredholm theory and other parts of the spectra corresponding to two given operators are preserved. As an application of our results, we give conditions for which the above mentioned spectra corresponding to two multiplication operators acting on the...

A note on the index of B -Fredholm operators

M. Berkani, Dagmar Medková (2004)

Mathematica Bohemica

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 operator.

A perturbation characterization of compactness of self-adjoint operators

Heydar Radjavi, Ping-Kwan Tam, Kok-Keong Tan (2003)

Studia Mathematica

A characterization of compactness of a given self-adjoint bounded operator A on a separable infinite-dimensional Hilbert space is established in terms of the spectrum of perturbations. An example is presented to show that without separability, the perturbation condition, which is always necessary, is not sufficient. For non-separable spaces, another condition on the self-adjoint operator A, which is necessary and sufficient for the perturbation, is given.

A perturbation problem in the presence of affine symmetries

Tullio Valent (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

An approach to a local analysis of solutions of a perturbation problem is proposed when the unperturbed operator has affine symmetries. The main result is a local theorem on existence, uniqueness, and analytic dependence on a parameter.

A-Browder-type theorems for direct sums of operators

Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)

Mathematica Bohemica

We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties ( SBaw ) , ( SBab ) , ( SBw ) and ( SBb ) are not preserved under direct sums of operators. However, we prove that if S and T are bounded linear operators acting on Banach spaces and having the property ( SBab ) , then S T has the property ( SBab ) if and only if σ SBF + - ( S T ) = σ SBF + - ( S ) σ SBF + - ( T ) , where σ SBF + - ( T ) is the upper semi-B-Weyl spectrum of T . We obtain analogous preservation results for the properties ( SBaw ) , ( SBb ) and ( SBw ) with...

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