Bulletin of the Belgian Mathematical Society - Simon Stevin

Property $(aw)$ and perturbations

M. H.M. Rashid

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Abstract

A bounded linear operator $T\in\mathbf{L}(\mathbb{X})$ acting on a Banach space satisfies property $(aw)$, a variant of Weyl's theorem, if the complement in the spectrum $\sigma(T)$ of the Weyl spectrum $\sigma_w(T)$ is the set of all isolated points of the approximate-point spectrum which are eigenvalues of finite multiplicity. In this article we consider the preservation of property $(aw)$ under a finite rank perturbation commuting with $T$, whenever $T$ is polaroid, or $T$ has analytical core $K(T-\lambda_0 I)=\{0\}$ for some $\lambda_0\in \mathbb{C}$. The preservation of property $(aw)$ is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations or under Riesz perturbations. The theory is exemplified in the case of some special classes of operators.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1 (2013), 1-18.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1366306710

Digital Object Identifier
doi:10.36045/bbms/1366306710

Mathematical Reviews number (MathSciNet)
MR2907609

Zentralblatt MATH identifier
06186904

Subjects
Primary: 47A13: Several-variable operator theory (spectral, Fredholm, etc.) 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]

Keywords
Weyl's theorem Weyl spectrum Polaroid operators Property $(w)$ Property $(aw)$

Citation

Rashid, M. H.M. Property $(aw)$ and perturbations. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 1, 1--18. doi:10.36045/bbms/1366306710. https://projecteuclid.org/euclid.bbms/1366306710


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