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Block diagonalization in Banach algebras

Author: Robin Harte
Journal: Proc. Amer. Math. Soc. 129 (2001), 181-190
MSC (1991): Primary 47A13; Secondary 15A21, 15A18
Published electronically: August 17, 2000
MathSciNet review: 1784022
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``Reduction" of linear operators is effected by commuting projections; the spectrum of the operator is then the union of the spectra of its range and null space restrictions. Disjointness of these partial spectra implies that the projection ``double commutes" with the operator, which in turn can be recognised as a curious kind of ``exactness". Variants of this exactness correspond to various kinds of disjointness between the partial spectra.

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

  • 1. Hong-Ke Du and Jin Pan, Perturbation of spectrums of $2\times 2$operator matrices, Proc. Amer. Math. Soc. 121 (1994) 761-766. MR 94i:47004
  • 2. Jin Kyu Han, Hong Youl Lee and Woo Young Lee, Invertible completions of $2\times 2$ upper triangular operator matrices, Proc. Amer. Math. Soc. 128 (2000) 119-123. MR 2000c:47003
  • 3. R.E. Harte, Invertibility and singularity, Dekker New York 1988. MR 89d:47001
  • 4. R.E. Harte, Invertibility and singularity for operator matrices, Proc. Royal Irish Acad. 88A (1988) 103-118. MR 90a:15019
  • 5. R.E. Harte, Unspectral sets, Rendiconti del Circ. Mat. Palermo 56 (1998) 69-77. CMP 2000:01
  • 6. R.E. Harte and C. Hernandez, On the Taylor spectrum of left-right multipliers, Proc. Amer. Math. Soc. 126 (1998) 397-404. MR 98e:46057
  • 7. K. Hoffman and R. Kunze, Linear algebra, Prentice Hall New York 1971. MR 43:1998
  • 8. J.J. Koliha, Block diagonalization, Mathematica Bohemica (to appear).
  • 9. R.R. London and H.P. Rogosinski, Decomposition theory in the teaching of elementary linear algebra, Amer. Math. Monthly 97 (1990) 478-485. MR 91d:15022
  • 10. H. Radjavi and P. Rosenthal, Invariant subspaces, Springer New York 1973. MR 51:3924
  • 11. Ch. Schmoeger, Remarks on commuting exponentials on Banach algebras, Proc. Amer. Math. Soc. 127 (1999) 1337-1338. MR 99h:46090
  • 12. D. A. Suprunenko and R.I. Tyshkevich, Commutative matrices, Academic Press New York 1968.
  • 13. J. H. M. Wedderburn, Lectures on matrices, AMS Colloq. Publ. 17, Amer. Math. Soc. Providence 1934.
  • 14. E. M. E. Wermuth, A remark on commuting operator exponentials, Proc. Amer. Math. Soc. 125 (1997) 1685-1688. MR 97g:39011

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Additional Information

Robin Harte
Affiliation: School of Mathematics, Trinity College, Dublin 2, Ireland

Keywords: Commuting idempotent, double commutant, spectral disjointness, exactness conditions
Received by editor(s): December 15, 1997
Received by editor(s) in revised form: March 10, 1998, October 6, 1998, and March 31, 1999
Published electronically: August 17, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society