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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Additional Information
- Sameer Chavan
- Affiliation: Department of Mathematics & Statistics, Indian Institute of Technology, Kanpur, 208016, India
- MR Author ID: 784696
- Email: chavan@iitk.ac.in
- Raúl Curto
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
- MR Author ID: 53500
- Email: raul-curto@uiowa.edu
- Received by editor(s): August 27, 2016
- Published electronically: April 12, 2017
- Additional Notes: The second named author was partially supported by NSF Grant DMS-1302666.
- Communicated by: Stephan Ramon Garcia
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3369-3375
- MSC (2010): Primary 47A13; Secondary 47B20
- DOI: https://doi.org/10.1090/proc/13479
- MathSciNet review: 3652790