Advertisement

Rendiconti del Circolo Matematico di Palermo

, Volume 45, Issue 3, pp 377–396 | Cite as

Completely continuous and related multilinear operators

  • David K. Ruch
Article
  • 24 Downloads

Abstract

Completely continuous multilinear operators are defined and their properties investigated. This class of operators is shown to form a closed multi-ideal. Unlike the linear case, compact multilinear operators need not be completely continuous. The completely continuous maps are shown to be the closure of a subspace of the finite rank operators. Hilbert-Schmidt operators are also considered. An application to finding error bounds for solutions of multipower equations is presented.

Keywords

Banach Space Continuous Operator Finite Rank Reflexive Banach Space Multilinear Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Anselone P. M., Moore R. H.,An extension of the Newton-Kantorovič method for solving nonlinear equations with an application to elasticity, J. Math. Anal. Applic.,13 (1966), 476–501.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Antosik P., Swartz C.,Matrix Methods in Analysis, Lecture Notes in Mathematics 1113, Springer-Verlag, 1985.Google Scholar
  3. [3]
    Argyros I. K.,Quadratic equations and applications to Chandreshekhar’s and related equations, Bull. Austral. Math. Soc.,32 (1985), 275–292.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Argyros I. K.,On the approximation of solutions of compact operator equations in Banach space, Proyecciones,14 (1988), 29–46.Google Scholar
  5. [5]
    Balakrishnan A. V.,Applied Functional Analysis, Springer-Verlag, New York, 1976.MATHGoogle Scholar
  6. [6]
    Braunß A., Junek H.,Bilinear mappings and operator ideals, Rend. Circ. Mat. Palermo, Suppl.10 (1985), 25–35.Google Scholar
  7. [7]
    Chandrasekhar S.,Radiative Transfer, Dover, New York, 1960.Google Scholar
  8. [8]
    Geiss S.,Ein Faktorisierungssatz fur multilineare funktionale, Math. Nachr.,134 (1987), 149–159.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Kupsch J.,Estimates of the unitary integral, Commun. Math. Phys.,19 (1960), 65–82.CrossRefMathSciNetGoogle Scholar
  10. [10]
    Pietsch A.,Ideals of multilinear functionals, Proc. of the II Intern. Conf. on Operator Ideals and their Applic. in Theoretical Physics, Leipzig, 1983.Google Scholar
  11. [11]
    Pietsch A.,Operator Ideals, Deutscher Verlag der Wissenschaften, Berlin, 1979.Google Scholar
  12. [12]
    Porter D., Stirling D.,Integral Equations, Cambridge Univ. Press, Cambridge, 1990.MATHGoogle Scholar
  13. [13]
    Porter W.,Synthesis of polynomic systems, Siam J. Math. Anal.,11 (1980), 308–315.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Ramanujan M. S., Schock E.,Operator ideals and spaces of bilinear operators, Lin. Multilin. Alg.,18 (1985), 307–318.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Ruch D. K.,Characterizations of compact bilinear maps, Lin. Multilin. Alg.,25 (1989), 297–307.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Ruch D. K., Van Fleet P. J.,On multipower equations: Some iterative solutions and applications, Zeitschrift für Analysis und ihse Annehclungen,15 (1996), no. 1, 201–222.MATHGoogle Scholar
  17. [17]
    Singer I.,Bases in Banach Spaces I, Springer-Verlag, New York, 1970.MATHGoogle Scholar
  18. [18]
    Wang J.,A cubic spline wavelet basis of C[0,1], preprint, 1994.Google Scholar

Copyright information

© Springer 1996

Authors and Affiliations

  • David K. Ruch
    • 1
  1. 1.Department of MathematicsSam Houston State UniversityHuntsvilleUSA

Personalised recommendations