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Property \((w)\) and perturbations. II. (English) Zbl 1171.47010

The property \((w)\) is a variant of Wely’s theorem introduced by V.Rakočević [Mat.Vesn.37, 423–426 (1985; Zbl 0596.47001)]. The present paper is continuation of [P.Aiena and M.T.Biondi, J. Math.Anal.Appl.336, No.1, 683–692 (2007; Zbl 1182.47012)].
The author is concerned with “the stability of property \((w)\), a variant of Weyl’s theorem for a bounded operator \(T\) acting on a Banach space, under finite-dimensional perturbations \(K\) commuting with \(T\). A counterexample shows that property \((w)\), in general, is not preserved under finite-dimensional perturbations commuting with \(T\), also under the assumption that \(T\) is \(a\)-isoloid.”
For the third part of this series, see [P.Aiena,M.T.Biondi and F.Villafañe, J. Math.Anal.Appl.353, No.1, 205–214 (2009; Zbl 1171.47011), reviewed below.]

MSC:

47A53 (Semi-) Fredholm operators; index theories
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
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