SPECTRAL MAPPING THEOREM AND WEYL'S THEOREM FOR (m, n)-PARANORMAL OPERATORS

Abstract

In  the present paper, we prove spectral mapping theorem for $(m,n)$-paranormal operator $T$ on a separable Hilbert space, that is, $f(\sigma_{w}(T))= \sigma_{w}(f(T))$ when $f$ is an analytic function on some open neighborhood of $\sigma$(T). We also show that for $(m,n)$-paranormal operator $T$, Weyl's theorem holds, that is, $\sigma(T) -\sigma_{w}(T) = \pi_{00}(T)$. Moreover, if $T$ is algebraically $(m,n)$-paranormal, then spectral mapping theorem and Weyl's theorem  hold.