A Note on *-paranormal Operators

In this note we show that if either T or T^* is totally *-paranormalthen Weyls theorem holds for f(T) for every f , and also a-Weyls theorem holds for f(T) if T is totally *-paranormal. We prove that if eitherT or T^* is *-paranormal then the spectral mapping theorem holds for theWeyl spectrum and for the essential approximate point spectrum.