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

TL;DR

This paper extends Weyl's theorem to commuting pairs of paranormal and $ ext{\ast}$-paranormal operators, establishing spectral equalities and invariance under analytic functional calculus.

Contribution

It proves Weyl's theorem for pairs of commuting paranormal and $ ext{\ast}$-paranormal operators, including invariance under analytic functions.

Findings

Weyl's theorem-I holds for $ ext{\ast}$-paranormal operators with quasitriangular property.

Weyl's theorem-II holds for paranormal operators.

Weyl's theorem-II is valid for $f(T)$ with analytic $f$ near the spectrum.

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 $\sigma_T(T)$.