Bulletin of the Belgian Mathematical Society - Simon Stevin

Property $(aw)$ and perturbations

M. H.M. Rashid

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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

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

First available in Project Euclid: 18 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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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