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 $$ \textrm{dim} \; \textrm{ker} \; (\mathbf{T}-\boldsymbol\lambda) \ge \textrm{dim} \; \textrm{ker} \; (\mathbf{T} - {\boldsymbol\lambda})^*), $$ for every $\boldsymbol\lambda$ in the Taylor spectrum $\sigma(\mathbf{T})$ of $\mathbf{T}$. We prove that the Weyl spectrum of $\mathbf{T}$, $\omega(\mathbf{T})$, satisfies the identity $$ \omega(\mathbf{T})=\sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T}), $$ where $\pi_{00}(\mathbf{T})$ 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 $d>2$.