Weyl's Theorem for pairs of commuting hyponormal operators

Abstract

Let $\mathbf{T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property $$ \textrm{dim} \; \textrm{ker} \; (\mathbf{T}-\boldsymbolλ) \ge \textrm{dim} \; \textrm{ker} \; (\mathbf{T} - {\boldsymbolλ})^*), $$ for every $\boldsymbolλ$ in the Taylor spectrum $σ(\mathbf{T})$ of $\mathbf{T}$. We prove that the Weyl spectrum of $\mathbf{T}$, $ω(\mathbf{T})$, satisfies the identity $$ ω(\mathbf{T})=σ(\mathbf{T}) \setminus π_{00}(\mathbf{T}), $$ where $π_{00}(\mathbf{T})$ denotes the set of isolated eigenvalues of finite multiplicity. Our method of proof relies on a (strictly $2$-variable) fact about the topological boundary of the Taylor spectrum; as a result, our proof does not hold for $d$-tuples of commuting hyponormal operators with $d>2$.