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Zbl 1221.47009
Amouch, Mohamed
Polaroid operators with SVEP and perturbations of property $(gw)$. (English)
[J] Mediterr. J. Math. 6, No. 4, 461-470 (2009). ISSN 1660-5446; ISSN 1660-5454/e

A bounded linear operator $T$ on a Banach space $X$ belongs to the class $PS(X)$ if it is a polaroid operator with the single-valued extension property. It satisfies the $(gw)$ property if the set of its isolated eigenvalues coincides with $\sigma_a(T)\setminus \sigma _{SBF^+_-}(T)$, where $\sigma_a(T)=\{\lambda \in \mathbb{C} \mid T-\lambda I$ is not bounded from below\} and $\sigma _{SBF^+_-}(T)=\{\lambda \in \mathbb{C} \mid \text{ind}( T-\lambda I)>0\}$. The author proves that, if $T\in PS(X)$ and $G$ satisfies some additional assumptions (e.g., it is algebraic or quasinilpotent and commutes with $T$), then $f(T^*+G^*)$ satisfies the $(gw)$ property for any analytic function $f$ on a neighbourhood of $\sigma (T+G)$. He also adds some corollaries and examples concerning different versions of Weyl's theorem as well as other properties of operators.
[Dorota Gabor (Toruń)]

MSC 2000:: *47A10 Spectrum and resolvent of linear operators
47A11 Local spectral properties
47A53 (Semi-)Fredholm operators; index theories

Keywords: bounded linear operator; spectrum; polaroid operator; single-valued extension property; property $(gw)$

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