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Funktsional. Anal. i Prilozhen., 2017, Volume 51, Issue 2, Pages 72–82 (Mi faa3438)  

Spectra of $3\times 3$ upper triangular operator matrices

Xiufeng Wua, Junjie Huanga, Alatancang Chenab

a School of Mathematical Sciences, Inner Mongolia University, Hohhot, P. R. China
b Department of Mathematics, Hohhot University for Nationalities, Hohhot, P. R. China

Abstract: Let ${H}_1$, ${H}_2$, and ${H}_3$ be complex separable Hilbert spaces. Given $A\in {B}({H}_1)$, $B\in{B}({H}_2)$, and $C\in{B} ({H}_3)$, write $M_{D,E,F}=(\begin{smallmatrix} A & D&E
0 & B&F
0&0&C \end{smallmatrix})$, where $D\in {B}({H}_2,{H}_1)$, $E\in{B}({H}_3,{H}_1)$, and $F\in{B}({H}_3,{H}_2)$ are unknown operators. This paper gives a complete description of the intersection $\bigcap_{D,E,F} \sigma(M_{D,E,F})$, where $D$, $E$, and $F$ range over the respective sets of bounded linear operators. Further, we show that $\sigma(A)\cup\sigma(B)\cup\sigma(C)=\sigma(M_{D,E,F})\cup W$, where $W$ is the union of certain gaps in $\sigma(M_{D,E,F})$, which are subsets of $(\sigma(A)\cap\sigma(B))\cup(\sigma(B)\cap\sigma(C))\cup(\sigma(A) \cap\sigma(C))$. Finally, we obtain a necessary and sufficient condition for the relation $\sigma(M_{D,E,F})=\sigma(A)\cup\sigma(B)\cup\sigma(C)$ to hold for any $D$, $E$, and $F$.

Keywords: spectrum, perturbation, $3\times 3$ upper triangular operator matrix.

Funding Agency Grant Number
National Natural Science Foundation of China 11461049
11371185
National Science Fund for Distinguished Young Scholars 2013JQ01
2013ZD01
This work is supported by the NNSF of China (projects nos. 11461049 and 11371185) and the NSF of Inner Mongolia (projects nos. 2013JQ01 and 2013ZD01).


DOI: https://doi.org/10.4213/faa3438

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English version:
Functional Analysis and Its Applications, 2017, 51:2, 135–143

Bibliographic databases:

UDC: 517.983+517.984
Received: 10.09.2015
Revised: 05.05.2016
Accepted:06.05.2016

Citation: Xiufeng Wu, Junjie Huang, Alatancang Chen, “Spectra of $3\times 3$ upper triangular operator matrices”, Funktsional. Anal. i Prilozhen., 51:2 (2017), 72–82; Funct. Anal. Appl., 51:2 (2017), 135–143

Citation in format AMSBIB
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\by Xiufeng Wu, Junjie Huang, Alatancang Chen
\paper Spectra of $3\times 3$ upper triangular operator matrices
\jour Funktsional. Anal. i Prilozhen.
\yr 2017
\vol 51
\issue 2
\pages 72--82
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\crossref{https://doi.org/10.4213/faa3438}
\elib{http://elibrary.ru/item.asp?id=29106592}
\transl
\jour Funct. Anal. Appl.
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\vol 51
\issue 2
\pages 135--143
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