On the Equivalence of Weyl Theorem and Generalized Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyls theorem also satisfies Weyls theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyls theorem, then it also satisfies generalized Weyls theorem. We give also a similar result for the equivalence of aWeyls theorem and generalized aWeyls theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators