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