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

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 $σ_T(T)\setminusσ_{T_W}(T)=π_{00}(T)$ and a commuting pair of paranormal operators satisfy Weyl's theorem-II, that is $σ_T(T)\setminusω(T)=π_{00}(T),$ where $σ_T(T),\, σ_{T_W}(T),\,ω(T)$ and $π_{00}(T)$ 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 $σ_T(T)$.

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