## Geometric spectral theory for compact operators

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- by Isaak Chagouel, Michael Stessin and Kehe Zhu PDF
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**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.## References

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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