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On the Equivalence of Weyl Theorem and Generalized Weyl Theorem

Research paper by M. Berkani

Indexed on: 30 Mar '06Published on: 30 Mar '06Published in: Acta Mathematica Sinica, English Series



Abstract

We know that an operator T acting on a Banach space satisfying generalized Weyl’s theorem also satisfies Weyl’s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl’s theorem, then it also satisfies generalized Weyl’s theorem. We give also a similar result for the equivalence of a–Weyl’s theorem and generalized a–Weyl’s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.