Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47A13, 47A10

Retrieve articles in all journals with MSC (2010): 47A13, 47A10


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

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