Weyl's theorem for commuting tuple of paranormal and $\ast$-paranormal operators

Abstract

In this article, we show that a commuting pair $T=(T_1,T_2)$ of $\ast$-paranormal operators $T_1$ and $T_2$ with quasitriangular property satisfy the Weyl's theorem-I, that is $${\sigma }_T(T)\setminus {\sigma }_{T_W}(T)={\pi }_{00}(T)$$ and a commuting pair of paranormal operators satisfy Weyl's theorem-II, that is $${\sigma }_T(T)\setminus \omega (T)={\pi }_{00}(T),$$ where ${\sigma }_T(T),{\sigma }_{T_W}(T),\omega (T)$ and ${\pi }_{00}(T)$ are the Taylor spectrum, the Taylor Weyl spectrum, the joint Weyl spectrum and the set consisting of isolated eigenvalues of $T$ with finite multiplicity, respectively. Moreover, we prove that Weyl's theorem-II holds for $f(T)$, where $T$ is a commuting pair of paranormal operators and $f$ is an analytic function in a neighbourhood of ${\sigma }_T(T)$.

There is a change in the definition of function of an operator