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

We show that a commuting pair $T=(T_1,T_2)$ of $\ast $-paranormal operators $T_1$ and $T_2$ with quasitriangular property satisfies Weyl’s theorem-I, that is, $$ \sigma _{\rm T}(T)\setminus \sigma _{\rm T_W}(T)=\pi _{00}(T) $$ and a commuting pair of paranormal operators satisfies Weyl’s theorem-II, that is, $$ \sigma _{\rm T}(T)\setminus \omega (T)=\pi _{00}(T), $$ where $\sigma _{\rm T}(T),\, \sigma _{\rm T_W}(T),\,\omega (T)$ and $\pi _{00}(T)$ are the Taylor spectrum, the Taylor Weyl spectrum, the joint Weyl spectrum and the set 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 $\sigma _{\rm T}(T)$.