Integral Equations and Operator Theory

, Volume 40, Issue 3, pp 342–381

Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections

  • V. S. Rabinovich
  • S. Roch
  • B. Silbermann


The central theme of the present paper are band and band-dominated operators, i.e. norm limits of band operators. In the first part, we generalize the results from [24] and [25] concerning the Fredholm properties of band-dominated operators and the applicability of the finite section method to the case of operators with operator-valued coefficients. We characterize these properties in terms of the limit operators of the given band-dominated operator. The main objective of the second part is to apply these results to pseudodifferential operators on cones in ℝn which is possible after a suitable discretization.

MSC 2000

Primary 47 L 80 Secondary 35 S 05 47 A 53 47 G 30 65 R 20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Ben-Artzi, I. Gohberg, Fredholm properties of band matrices and dichotomy.-In: Topics in Operator Theory. The Constantin Apostol Memorial Issue (ed. I. Gohberg), Operator Theory: Advances and Appl. 32, Birkhäuser Verlag, Basel 1988, 37–52.Google Scholar
  2. [2]
    A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators.-Akademie-Verlag, Berlin 1989 and Springer-Verlag, Berlin, Heidelberg, New-York 1990.Google Scholar
  3. [3]
    A. Böttcher, Yu. Karlovich, V. Rabinovich, The method of limit operators for the theory of one-dimensional singular integral operators with slowly oscillating data.-To appear in Operator Theory, 2000.Google Scholar
  4. [4]
    J. Dixmier, LesC *-algébres et leurs représentations.-Paris, Gauthier-Villars Éditeur, 1969.Google Scholar
  5. [5]
    R. G. Douglas, On the invertibility of a class of Toeplitz operators on the quarter plane.-Ind. Univ. Math. J.21(1972), 1031–1036.Google Scholar
  6. [6]
    R. G. Douglas, R. Howe, On the invertibility of a class of Toeplitz operators on the quarter plane.-Trans. Am. Math. Soc.158(1971), 203–217.Google Scholar
  7. [7]
    J. Favard, Sur les equations differentiales a coefficients presque periodicues.-Acta Math.51(1927), 31–81.Google Scholar
  8. [8]
    I. Gohberg, I. Feldman, Convolutions equations and projection methods for their solutions.-Nauka, Moskva 1971 (Russian, Engl. translation: Amer. Math. Soc. Transl. of Math. Monographs 41, Providence, R.I. 1974).Google Scholar
  9. [9]
    I. Gohberg, M.A. Kaashoek, Projection method for block Toeplitz operators with operator-valued symbols.-In: Toeplitz operators and related topics. Operator Theory: Advances and Applications 71, Birkhäuser Verlag, Basel-Boston-Berlin 1994, 79–104.Google Scholar
  10. [10]
    I. C. Gohberg, M. G. Krein, Systems of integral equations on the semi-axis with kernels depending on the difference of arguments.-Usp. Mat. Nauk13(1958), 5, 3–72 (Russian).Google Scholar
  11. [11]
    I. Gohberg, N. Krupnik, One-dimensional Linear Singular Integral Equations, I and II.-Birkhäuser Verlag, Basel, Boston, Berlin 1992.Google Scholar
  12. [12]
    M. G. Krein, Integral equations on the semi-axis with kernels depending on the difference of arguments.-Usp. Mat. Nauk13(1958), 2, 3–120 (Russian).Google Scholar
  13. [13]
    A. V. Kozak, A local principle in the theory of projection methods.-Dokl. Akad. Nauk SSSR212(1973), 6, 1287–1289 (Russian).Google Scholar
  14. [14]
    A. V. Kozak, I. B. Simonenko, Projection methods for solving discrete convolutions equations.-Sib. Mat. Zh.20(1979), 6, 1249–1260 (Russian).Google Scholar
  15. [15]
    V. G. Kurbatov, Linear differential-difference operators.-Vornej. Univ., Voronej 1990 (Russian).Google Scholar
  16. [16]
    B. V. Lange, V. S. Rabinovich, On the Noether property of multi-dimensional discrete convolutions.-Mat. Zam.37 (1985), 3, 407–421.Google Scholar
  17. [17]
    B. V. Lange, V. S. Rabinovich, On the Noether property of multi-dimensional operators of convolution type with measurable bounded coefficients.-Izv. Vyssh. Uchebn. Zaved., Mat.6(1985), 22–30 (Russian).Google Scholar
  18. [18]
    B. V. Lange, V. S. Rabinovich, Pseudo-differential operators on ℝn and limit operators.-Mat. Sb.129(1986), 2, 175–185 (Russian, Engl. transl.: Math. USSR Sb.577(1987), 1, 183–194).Google Scholar
  19. [19]
    E. M. Muhamadiev, On the invertibility of differential operators in partial derivatives of elliptic type.-Dokl. Akad. Nauk SSSR205(1972), 6, 1292–1295 (Russian).Google Scholar
  20. [20]
    E. M. Muhamadiev, On the normal solvability and the Noether property of elliptic operators on ℝn.-Zap. Nauchn. Sem. LOMI110(1981), 120–140 (Russian).Google Scholar
  21. [21]
    V. S. Rabinovich, Fredholmness of pseudo-differential operators on ℝn in the scale ofL p,q-spaces.-Sib. Mat. Zh.29(1988), 4, 635–646 (Russian. Engl. transl.: Math. J.29(1988),4,6635–646).Google Scholar
  22. [22]
    V. S. Rabinovich, Singular integral operators on complicated contours and pseudodifferential operator.-Mat. Zam.58(1995), 1, 67–85 (Russian, Engl. transl.: Math. Notes58(1995), 1, 722–734).Google Scholar
  23. [23]
    V. S. Rabinovich, Criterion for local invertibility of pseudodifferential operators with operator symbols and some applications.-In: Proceedings of the St. Petersburg Mathematical Society, Vol. V, AMS Translation Series193(1998), 2, 239–260.Google Scholar
  24. [24]
    V. S. Rabinovich, S. Roch, B. Silbermann, Fredholm theory and finite section method for band-dominated operators.-Integral Equations Oper. Theory30(1998), 452–495.Google Scholar
  25. [25]
    V. S. Rabinovich, S. Roch, B. Silbermann, Algebras of approximation sequences: Finite sections of band-dominated operators.-Submitted to Acta Appl. Math.Google Scholar
  26. [26]
    I. B. Simonenko, A new general method of investigating linear operator equations of the singular integral equations type, I.-Izv. Akad. Nauk SSSR, Ser. Mat.3(1965), 567–586 (Russian).Google Scholar
  27. [27]
    I. B. Simonenko, On multidimensional discrete convolutions.-Mat. Issled.3(1968), 1, 108–127 (Russian).Google Scholar
  28. [28]
    I. B. Simonenko, Operators of convolution type in cones.-Mat. Sb.74(116), 298–313 (1967) (Russian).Google Scholar
  29. [29]
    M. E. Taylor, Pseudodifferential operators.-Princeton University Press, Princeton, New Jersey, 1981.Google Scholar

Copyright information

© Birkhäuser Verlag 2001

Authors and Affiliations

  • V. S. Rabinovich
    • 1
    • 2
  • S. Roch
    • 3
  • B. Silbermann
    • 4
  1. 1.Department of Mechanics and MathematicsRostov State UniversityRostov-na-DonuRussia
  2. 2.Instituto Politecnico Nacional ESIME-Zacatenco SEPIMecicoMexico
  3. 3.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtFRG
  4. 4.Fakultät für MathematikTechnische Universität ChemnitzChemnitzFRG

Personalised recommendations