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A note on Weyl's theorem for operator matrices

Authors: Slavisa V. Djordjevic and Young Min Han
Journal: Proc. Amer. Math. Soc. 131 (2003), 2543-2547
MSC (2000): Primary 47A10, 47A55
Published electronically: November 27, 2002
MathSciNet review: 1974653
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Abstract: When $A\in\mathcal B(X)$ and $B\in\mathcal B(Y)$ are given we denote by $M_C$ an operator acting on the Banach space $X\oplus Y$of the form

\begin{displaymath}M_{C}=\left (\begin{matrix}A&C\\ 0&B\end{matrix}\right), \ \text{where} \ C\in \mathcal B(Y,X). \end{displaymath}

In this note we examine the relation of Weyl's theorem for $A\oplus B$ and $M_C$ through local spectral theory.

References [Enhancements On Off] (What's this?)

  • 1. P. Aiena and O. Monsalve, Operators which do not have the single valued extension property, J. Math. Anal. Appl. 250 (2000), 435-448. MR 2001g:47005
  • 2. S.V. Djordjevic, B.P. Duggal and Y.M. Han, The single valued extension property and Weyl's theorem (preprint).
  • 3. H.K. Du and J. Pan, Perturbation of spectrums of $2\times2$ operator matrices, Proc. Amer. Math. Soc. 121 (1994), 761-766. MR 94i:47004
  • 4. J.K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), 61-69. MR 51:11181
  • 5. J.K. Han, H.Y. Lee and W.Y. Lee, Invertible completions of $2\times 2$ upper triangular operator matrices, Proc. Amer. Math. Soc. 128 (2000), 119-123. MR 2000c:47003
  • 6. R.E. Harte, Invertibility and Singularity for Bounded Linear Operators, Dekker, New York, 1988. MR 89d:47001
  • 7. R.E. Harte and W.Y. Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 349 (1997), 2115-2124. MR 98j:47024
  • 8. M. Houimdi and H. Zguitti, Propriétés spectrales locales d'une matrice carree des operateurs, Acta Math. Vietnamica 25 (2000), 137-144. MR 2001d:47011
  • 9. J.J. Koliha, Isolated spectral points, Proc. Amer. Math. Soc. 124 (1996), 3417-3424. MR 97a:46057
  • 10. K.B. Laursen and M.M. Neumann, An Intruduction to Local Spectra Theory, London Mathematical Society Monographs, New Series 20, Clarendon Press, Oxford 2000. MR 2001k:47002
  • 11. W.Y. Lee, Weyl's theorem for operator matrices, Integral Equations and Operator Theory 32 (1998), 319-331. MR 99g:47023
  • 12. W.Y. Lee, Weyl's spectra of operator matrices, Proc. Amer. Math. Soc. 129 (2001), 131-138. MR 2001f:47003
  • 13. K.K. Oberai, Spectral mapping theorem for essential spectra, Rev. Roumaine Math. Pures Appl. 25 (1980), 365-373. MR 81j:47007

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Additional Information

Slavisa V. Djordjevic
Affiliation: University of Niš, Faculty of Science, P.O. Box 91, 18000 Niš, Yugoslavia

Young Min Han
Affiliation: Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419

Keywords: Upper triangular operator matrix, Weyl's theorem, single valued extension property
Received by editor(s): January 21, 2002
Received by editor(s) in revised form: March 27, 2002
Published electronically: November 27, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society

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