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Geometric spectral theory for compact operators


Authors: Isaak Chagouel, Michael Stessin and Kehe Zhu
Journal: Trans. Amer. Math. Soc. 368 (2016), 1559-1582
MSC (2010): Primary 47A13, 47A10
DOI: https://doi.org/10.1090/tran/6588
Published electronically: June 15, 2015
MathSciNet review: 3449218
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Abstract: For an $ n$-tuple $ \mathbb{A}=(A_1,\cdots ,A_n)$ of compact operators we define the joint point spectrum of $ \mathbb{A}$ to be the set

$\displaystyle \sigma _p(\mathbb{A})=\{(z_1,\cdots ,z_n)\in \mathbb{C}^n:\ker (I+z_1A_1+\cdots +z_nA_n)\not =(0)\}.$

We prove in several situations that the operators in $ \mathbb{A}$ pairwise commute if and only if $ \sigma _p(\mathbb{A})$ consists of countably many, locally finite, hyperplanes in $ \mathbb{C}^n$. In particular, we show that if $ \mathbb{A}$ is an $ n$-tuple of $ N\times N$ normal matrices, then these matrices pairwise commute if and only if the polynomial

$\displaystyle p_{\mathbb{A}}(z_1,\cdots ,z_n)=\det (I+z_1A_1+\cdots +z_nA_n)$

is completely reducible, namely,

$\displaystyle p_{\mathbb{A}}(z_1,\cdots ,z_n)=\prod _{k=1}^N(1+a_{k1}z_1+\cdots +a_{kn}z_n)$

can be factored into the product of linear polynomials.

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Additional Information

Isaak Chagouel
Affiliation: Department of Mathematics and Statistics, State University of New York, Albany, New York 12222
Email: ichagouel@albany.edu

Michael Stessin
Affiliation: Department of Mathematics and Statistics, State University of New York, Albany, New York 12222
Email: mstessin@albany.edu

Kehe Zhu
Affiliation: Department of Mathematics and Statistics, State University of New York, Albany, New York 12222
Email: kzhu@albany.edu

DOI: https://doi.org/10.1090/tran/6588
Keywords: Normal operator, compact operator, projective spectrum, joint point spectrum, characteristic polynomial, completely reducible polynomial, complete commutativity
Received by editor(s): December 18, 2013
Published electronically: June 15, 2015
Article copyright: © Copyright 2015 American Mathematical Society