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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.
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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\paper Spectra of $3\times 3$ upper triangular operator matrices
\jour Funktsional. Anal. i Prilozhen.
\yr 2017
\vol 51
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\pages 72--82
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\vol 51
\issue 2
\pages 135--143
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