Weyl’s Theorem for pairs of commuting hyponormal operators

Chavan, Sameer; Curto, Raúl

doi:10.1090/proc/13479

Weyl’s Theorem for pairs of commuting hyponormal operators
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by Sameer Chavan and Raúl Curto PDF

Proc. Amer. Math. Soc. 145 (2017), 3369-3375 Request permission

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 $d>2$.

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