RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 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 1146104911371185 National Science Fund for Distinguished Young Scholars 2013JQ012013ZD01 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

Full text: PDF file (199 kB)
First page: PDF file
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.983+517.984
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
\Bibitem{WuHuaChe17} \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 \mathnet{http://mi.mathnet.ru/faa3438} \crossref{https://doi.org/10.4213/faa3438} \elib{http://elibrary.ru/item.asp?id=29106592} \transl \jour Funct. Anal. Appl. \yr 2017 \vol 51 \issue 2 \pages 135--143 \crossref{https://doi.org/10.1007/s10688-017-0175-2} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000403405500005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85020760733}