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This article is cited in 1 scientific paper (total in 1 paper)
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 $M_{D,E,F}=\left(\begin{smallmatrix} A & D&E\\
0 & B&F\\
0&0&C
\end{smallmatrix}\right)$, 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.
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
Linking options:
https://www.mathnet.ru/eng/faa3438https://doi.org/10.4213/faa3438 https://www.mathnet.ru/eng/faa/v51/i2/p72
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