Weyl's Theorem for pairs of commuting hyponormal operators

TL;DR

This paper extends Weyl's theorem to pairs of commuting hyponormal operators satisfying a quasitriangular property, characterizing their Weyl spectrum in terms of the Taylor spectrum and isolated eigenvalues.

Contribution

It proves a Weyl's theorem for pairs of commuting hyponormal operators with the quasitriangular property, linking the Weyl spectrum to the Taylor spectrum minus isolated eigenvalues.

Findings

Weyl spectrum equals Taylor spectrum minus isolated eigenvalues of finite multiplicity.

The proof uses properties of the topological boundary of the Taylor spectrum.

The result does not extend to higher-dimensional tuples of 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 $d>2$.