A NOTE ON ∗-PARANORMAL OPERATORS AND RELATED CLASSES OF OPERATORS

Abstract

We shall show that the Riesz idempotent <TEX>$E_{\lambda}$</TEX> of every *-paranormal operator T on a complex Hilbert space H with respect to each isolated point <TEX>${\lambda}$</TEX> of its spectrum <TEX>${\sigma}(T)$</TEX> is self-adjoint and satisfies <TEX>$E_{\lambda}\mathcal{H}=ker(T-{\lambda})= ker(T-{\lambda})^*$</TEX>. Moreover, Weyl's theorem holds for *-paranormal operators and more general for operators T satisfying the norm condition <TEX>$||Tx||^n{\leq}||T^nx||\,||x||^{n-1}$</TEX> for all <TEX>$x{\in}\mathcal{H}$</TEX>. Finally, for this more general class of operators we find a sufficient condition such that <TEX>$E_{\lambda}\mathcal{H}=ker(T-{\lambda})= ker(T-{\lambda})^*$</TEX> holds.