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$a$-Weyl's theorem for operator matrices


Authors: Young Min Han and Slavisa V. Djordjevic
Journal: Proc. Amer. Math. Soc. 130 (2002), 715-722
MSC (2000): Primary 47A50, 47A53
DOI: https://doi.org/10.1090/S0002-9939-01-06110-X
Published electronically: July 31, 2001
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Abstract:

If $M_{C}=\left(\begin{smallmatrix}A&C\\ 0&B\end{smallmatrix}\right)$ is a $2\times 2$ upper triangular matrix on the Hilbert space $H\oplus K$, then $a$-Weyl's theorem for $A$ and $B$ need not imply $a$-Weyl's theorem for $M_{C}$, even when $C=0$. In this note we explore how $a$-Weyl's theorem and $a$-Browder's theorem survive for $2\times 2$ operator matrices on the Hilbert space.


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  • 1. S.K. Berberian, An extension of Weyl's theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273-279. MR 40:3335
  • 2. S.K. Berberian, The Weyl spectrum of an operator, Indiana Univ.Math. J. 20 (1970), 529-544. MR 43:5344
  • 3. B. Chevreau, On the spectral picture of an operator, J. Operator Theory 4 (1980), 119-132. MR 81k:47002
  • 4. L.A. Coburn, Weyl's theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285-288. MR 34:1846
  • 5. S.V. Djordjevic and D.S. Djordjevic, Weyl's theorems: continuity of the spectrum and quasihyponormal operators, Acta Sci. Math. (Szeged) 64 (1998), 259-269. MR 2000c:47009
  • 6. S.V. Djordjevic and B.P. Duggal, Weyl's theorems and continuity of spectra in the class of p-hyponormal operators, Studia Math 143 (2000), 23-32. CMP 2001:08
  • 7. S.V. Djordjevic and Y.M. Han, Browder's theorems and spectral continuity, Glasgow Math. J. 42 (2000), 479-486. CMP 2001:04
  • 8. I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators (vol I), Birkhäuser, Basel, 1990. MR 93d:47002
  • 9. 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
  • 10. R.E. Harte, Fredholm, Weyl and Browder theory, Proc. Royal Irish Acad. 85A (2) (1985), 151-176. MR 87h:47029
  • 11. R.E. Harte, Invertibility and Singularity for Bounded Linear Operators, Dekker, New York, 1988. MR 89d:47001
  • 12. R.E. Harte and W.Y. Lee, Another note on Weyl's theorem, Trans. Amer. Math. Soc. 349 (1997), 2115-2124. MR 98j:47024
  • 13. R.E. Harte, W.Y. Lee and L.L. Littlejohn, On generalized Riesz points, J. Operator Theory (to appear).
  • 14. W.Y. Lee and S.H. Lee, A spectral mapping theorem for the Weyl spectrum, Glasgow Math. J. 38(1) (1996), 61-64. MR 97c:47023
  • 15. W.Y. Lee, Weyl's Theorem For Operator Matrices, Integral Equations and Operator Theory 32 (1998), 319-331. MR 99g:47023
  • 16. W.Y. Lee, Weyl Spectra of Operator Matrices, Proc. Amer. Math. Soc. 129 (2001), 131-138. CMP 2001:01
  • 17. K.K. Oberai, On the Weyl spectrum (II), Illinois J. Math. 21 (1977), 84-90. MR 55:1102
  • [18] V. Rakocevic, On the essential approximate point spectrum II, Mat. Vesnik 36 (1984), 89-97. MR 88h:47019
  • 18. V. Rakocevic, Approximate point spectrum and commuting compact perturbations, Glasgow Math. J. 28 (1986), 193-198. MR 87k:47006
  • 19. H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373-392.

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Additional Information

Young Min Han
Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea
Address at time of publication: Department of Mathematics, 14 MacLean Hall, University of Iowa, Iowa City, Iowa 52242-1419
Email: ymhan@math.skku.ac.kr, yhan@math.uiowa.edu

Slavisa V. Djordjevic
Affiliation: Department of Mathematics, Faculty of Philosophy, University of Niš, Ćirila and Metodija 2, 18000 Niš, Yugoslavia
Email: slavdj@archimed.filfak.ni.ac.yu

DOI: https://doi.org/10.1090/S0002-9939-01-06110-X
Keywords: Weyl spectrum, essential approximate point spectrum, Browder essential approximate point spectrum, $a$-Weyl's theorem, Weyl's theorem, $a$-Browder's theorem, Browder's theorem
Received by editor(s): February 29, 2000
Received by editor(s) in revised form: August 25, 2000
Published electronically: July 31, 2001
Additional Notes: This work was supported by the Brain Korea 21 Project (through Seoul National University)
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society

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