Weyl's Theorem for pairs of commuting hyponormal operators

Let $\mathbf{T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property $$\text{dim}\text{ker}(\mathbf{T}-\mathit{\lambda })\ge \text{dim}\text{ker}(\mathbf{T}-\mathit{\lambda }{)}^{\ast }),$$ for every $\mathit{\lambda }$ 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 $d>2$.