Skip to main content
SpringerLink
Log in

Matrix representations of polynomial operators

  • Published:
Rendiconti del Circolo Matematico di Palermo Aims and scope Submit manuscript

Summary

LetH be a separable Hilbert space. Every bounded,n-linear operatorL onH n toH(n=0,1,2,…) is shown to have a unique matrix representation with respect to each complete orthonormal sequence {ϕk} ∞1 . Conversely, every operator onH n toH possessing a matrix representation is proved to be a bounded,n-linear operator. The foregoing conclusions then apply to polynomial operatorsP onH toH wherePx=L 0+L1x+L2x2+…+Lnxn and eachL k is ak-linear operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akhiezer N. I. and Glazman I. M.,Theory of Linear Operators in Hilbert Space, Vol. I, Ungar, New York (1961).

    MATH  Google Scholar 

  2. Cesari L.,Functional Analysis and Galerkin’s Method, Michigan Math. Journal, 11 (1964), 336–384.

    MathSciNet  Google Scholar 

  3. Chandrasekhar S.,Radiative Transfer, Dover, New York, (1960).

    Google Scholar 

  4. Hopf E.,Uber die Anfangswertaufgabe fur hydrodynamischen Grundgleichungen, Math. Nachrichten, 4 (1951), 213–231.

    MATH  MathSciNet  Google Scholar 

  5. Kantorovich L. V. and Akilov G. P.,Functional Analysis in Normed Spaces, MacMillan (1964).

  6. Locker John,An existence analysis for nonlinear equations in Hilbert space, Transactions A.M.S., 128 (1967), 403–413.

    Article  Google Scholar 

  7. Marcus B.,Solutions of infinite polynomial systems by iteration, Rendiconti Circolo Matematico di Palermo, Serie II, vol. 11 (1962), 5–24.

    MATH  Google Scholar 

  8. Marcus B.,Error bounds for solutions of infinite polynomial systems by iteration, Rendiconti Circolo Matematico di Palermo, Serie II, vol. 13 (1964), 5–10.

    MATH  Google Scholar 

  9. McFarland J. E.,An iterative solution of the quadratic equation in Banach space, Proceedings of A.M.S., (1958), 824–830.

  10. Phillips John R.,Eigenfunction expansions for self-adjoint bilinear operators in Hilbert space, Technical report 27, Oregon State University, May (1966).

  11. Prenter P. M.,A Weierstrass Theorem for real, separable Hilbert spaces, MRC report 868, April (1968) and Journal of Approximation Theory (to appear).

  12. Prenter P. M.,Lagrange and Hermite interpolation in Banach spaces, MRC report 921, October (1968).

  13. Prenter P. M.,A Weierstrass Theorem for real normed linear spaces, MRC report 957, January (1969) and Bulletin of A.M.S., 75 (1969), 860–862.

    MATH  MathSciNet  Google Scholar 

  14. Prenter P. M.,On Polynomial Operators and Equations. Nonlinear Functional Analysis, Proceeding of an advanced symposium, MRC, edited by L. B. Rall, Wiley (1970) (to appear).

  15. Rall L. B.,Quadratic equations in Banach spaces, Rendiconti Circolo Matematico di Palermo, Serie II, vol. 10 (1961), 314–332.

    MathSciNet  Google Scholar 

  16. Rall L. B.,Solutions of abstract polynomial equations by iterative methods, MRC Technical Summary Report 892, August (1968), 1–35.

    Google Scholar 

  17. Taylor A. E.,An Introduction to Functional Analysis, Wiley, New York (1958).

    Google Scholar 

  18. Urabe Minoru,Galerkin’s procedure for nonlinear periodic systems, Arch. Rational Mech. Anal., 20 (1965), 120–152.

    MathSciNet  Google Scholar 

  19. Urabe Minoru,Galerkin’s procedure for nonlinear periodic systems and its extension to multipoint boundary value problems for general nonlinear systems. Numerical solutions of nonlinear differential equations, Proceedings of an advanced symposium, MRC, edited by Donald Greenspan, Wiley (1966).

Download references

Author information

Authors and Affiliations

  1. Fort Collins, U.S.A.

    P. M. Prenter

Authors
  1. P. M. Prenter
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Sponsored by Mathematics Research Center, United States Army, Madison, Wisconsin, under contract No. D.A.-31-124-ARO-D-462.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prenter, P.M. Matrix representations of polynomial operators. Rend. Circ. Mat. Palermo 21, 103–118 (1972). https://doi.org/10.1007/BF02844236

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02844236

Keywords

Search

Navigation