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Weyl's Theorem for pairs of commuting hyponormal operators

Authors: Sameer Chavan and Raúl Curto
Journal: Proc. Amer. Math. Soc. 145 (2017), 3369-3375
MSC (2010): Primary 47A13; Secondary 47B20
Published electronically: April 12, 2017
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Abstract: Let $ \mathbf {T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property

$\displaystyle \textrm {dim} \; \textrm {ker} \; (\mathbf {T}-\boldsymbol \lambd... \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

$\displaystyle \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

Raúl Curto
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242

Keywords: Hyponormality, Taylor spectrum, Weyl spectrum
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
Article copyright: © Copyright 2017 American Mathematical Society

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