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A note on the spectrum of an upper triangular operator matrix

Author(s): Mohamed Barraa; Mohamed Boumazgour
Journal: Proc. Amer. Math. Soc. 131 (2003), 3083-3088.
MSC (1991): Primary 47A10, 47A55, 47B47
Posted: January 28, 2003
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Abstract: Let $M_C= [\begin{smallmatrix} A& C 0 & B \end{smallmatrix}]$ be a $2\times 2$ upper triangular operator matrix acting on the Banach space $E\oplus F$ . We investigate the set of the operators for which $\sigma(M_C)=\sigma(A)\cup\sigma(B)$ , where $\sigma(.)$ denotes the spectrum.

References:

1.: H.K. Du and J. Pan, Perturbation of spectrums of $2\times 2$ operator matrices, Proc. Amer. Math. Soc. 121(1994), 761-776. MR 94i:47004
2.: L. Fialkow, A note on the range of the operator $X\longrightarrow AX-XB$ , Illinois J. Math. 25(1981), 112-124. MR 84b:47021
3.: P.R. Halmos, A Hilbert space problem book, Springer Verlag, New York, 1973. MR 84e:47001
4.: J.K. Han, H.Y. Lee and W.Y. Lee, Invertible completions of $2\times 2$ upper triangular operator matrices, Proc. Amer. Math. Soc. 129(2000), 119-123. MR 2000c:47003

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Additional Information:

Mohamed Barraa
Affiliation: Département de Mathématiques, Faculté des Sciences Semlalia, B.P 2390, Marrakech, Maroc
Email: barraa@hotmail.com

Mohamed Boumazgour
Affiliation: Département de Mathématiques, Faculté des Sciences Semlalia, B.P 2390, Marrakech, Maroc
Address at time of publication: Département de Mathématiques et Statistique, Pavillon Alexandre Vachon, Université Laval, Québec, Canada G1K 7P4
Email: boumazgour@ucam.ac.ma, boumazgo@mat.ulaval.ca

DOI: 10.1090/S0002-9939-03-06862-X
PII: S 0002-9939(03)06862-X
Keywords: Spectrum, $2\times 2$ upper triangular operator matrix, generalized derivation
Received by editor(s): March 1, 2002
Received by editor(s) in revised form: April 23, 2002
Posted: January 28, 2003
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2003, American Mathematical Society


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