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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Geometric spectral theory for compact operators
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by Isaak Chagouel, Michael Stessin and Kehe Zhu PDF
Trans. Amer. Math. Soc. 368 (2016), 1559-1582 Request permission

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 \[ \sigma _p(\mathbb {A})=\{(z_1,\cdots ,z_n)\in \mathbb {C}^n:\ker (I+z_1A_1+\cdots +z_nA_n)\not =(0)\}.\] 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 \[ p_{\mathbb {A}}(z_1,\cdots ,z_n)=\det (I+z_1A_1+\cdots +z_nA_n)\] is completely reducible, namely, \[ p_{\mathbb {A}}(z_1,\cdots ,z_n)=\prod _{k=1}^N(1+a_{k1}z_1+\cdots +a_{kn}z_n)\] 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
  • MR Author ID: 187055
  • Email: kzhu@albany.edu
  • Received by editor(s): December 18, 2013
  • Published electronically: June 15, 2015
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 1559-1582
  • MSC (2010): Primary 47A13, 47A10
  • DOI: https://doi.org/10.1090/tran/6588
  • MathSciNet review: 3449218