On the Equivalence of Weyl Theorem and Generalized Weyl Theorem

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.