Polaroid operators, SVEP and perturbed Browder, Weyl theorems. (English) Zbl 1196.47030

Summary: A Banach space operator \(T\in B(\mathcal X)\) is polaroid, \(T\in\mathcal P\), if the isolated points of the spectrum of \(T\) are poles of the resolvent of \(T\). Let \(\mathcal{PS}\) denote the class of operators in \(\mathcal P\) which have the single-valued extension property (SVEP). It is proved that, if \(\mathcal T\) is polynomially \(\mathcal{PS}\) and \(A\in B(\mathcal X)\) is an algebraic operator which commutes with \(T\), then \(f(T+A)\) satisfies Weyl’s theorem and \(f(T^*+A^*)\) satisfies \(a\)-Weyl’s theorem for every \(f\) which is holomorphic on a neighbourhood of \(\sigma(T+A)\).


47B47 Commutators, derivations, elementary operators, etc.
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
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