Integral Equations and Operator Theory

, Volume 28, Issue 2, pp 245–250 | Cite as

The courant-fischer theorem and the spectrum of selfadjoint block band Toeplitz operators

  • Peter Zizler
  • Keith F. Taylor
  • Shigeru Arimoto
Short Communications


We show that ifT(F) is a selfadjoint block Toeplitz operator generated by a trigonometric matrix polynomialF, then the spectrum ofT(F) as well as the limiting set Λ(F) of the eigenvalues of the truncationsT n (F) is the union of a finite collection of segments (the spectral range ofF) and at most a finite set of points for which we give an upper bound.

MSC 1991

Primary 47B35 Secondary 15A18 15A47 15A60 42A82 45E10 65F15 


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  1. [1]
    A. Böttcher, B. Silbermann:Analysis of Toeplitz Operators. Akademie-Verlag, Berlin 1989 and Springer-Verlag, Berlin, Heidelberg, New York 1990.Google Scholar
  2. [2]
    I. Gohberg, I.A. Feldman:Convolution Equations and Projection Methods for Their Solution. Amer. Math. Soc. Transl. of Math. Monographs, Vol. 41, Providence, R.I., 1974.Google Scholar
  3. [3]
    I. Gohberg, S. Goldberg, M.A. Kaashoek:Classes of Linear Operators II. Birkhäuser Verlag, Basel, Boston, Berlin 1993.Google Scholar
  4. [4]
    T. Kato:Perturbation Theory for Linear Operators. Springer-Verlag, Berlin 1976.Google Scholar
  5. [5]
    P. Lancaster, M. Tismenetsky:The Theory of Matrices. Academic Press, New York 1985.Google Scholar
  6. [6]
    E.E. Tyrtyshnikov: A unifying approach to some old and new theorems on distribution and clustering.Linear Algebra Appl. 232 (1996), 1–43.Google Scholar
  7. [7]
    H. Widom: Toeplitz matrices. In:Studies in Real and Complex Analysis (I.I. Hirshman, Jr., ed.), M.A.A. Studies in Mathematics, Vol. 3 (1965), 179–209.Google Scholar
  8. [8]
    H. Widom: Asymptotic behavior of block Toeplitz matrices and determinants. II.Advances in Math. 21 (1976), 1–29.Google Scholar

Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • Peter Zizler
    • 1
  • Keith F. Taylor
    • 2
  • Shigeru Arimoto
    • 3
  1. 1.Dept. of Math & StatsUniv. of SaskatchewanSaskatoonCANADA
  2. 2.Dept. of Math & StatsUniv. of SaskatchewanSaskatoonCANADA
  3. 3.Dept. of ChemistryUniv. of SaskatchewanSaskatoonCANADA

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