Ukrainian Mathematical Journal

, Volume 46, Issue 3, pp 304–317 | Cite as

Factorization of operators. Theory and applications

  • L. A. Sakhnovich


A survey of the development of Krein's factorization method and its applications is given.


Factorization Method 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. G. Krein, “On integral equations that generate second-order differential equations,”Dokl. Akad. Nauk SSSR,97, No. 1, 21–24 (1954).Google Scholar
  2. 2.
    M. G. Krein, “On a new method of the solution of linear integral equations of the first and second kinds,”Dokl. Akad. Nauk SSSR,100, No. 3, 413–416 (1955).Google Scholar
  3. 3.
    I. Ts. Gokhberg and M. G. Krein,Theory of the Volterra Operators and Its Applications [in Russian], Nauka, Moscow (1967).Google Scholar
  4. 4.
    L. A. Sakhnovich, “Factorization of operators inL 2(a, b)”Funkts. Anal. Prilozh.,13, No. 3, 40–45 (1979).Google Scholar
  5. 5.
    D. R. Larson, “Nest algebras and similarity transformations,”Ann. Math.,121, 409–427 (1985).Google Scholar
  6. 6.
    N. T. Andersen, “Compact perturbations of reflexive algebras,”J. Funct. Anal.,38, 366–400 (1980).Google Scholar
  7. 7.
    K. R. Davidson,Nest Algebras, Longman Sci. Techn., Pitman Res. Notes Math. (1988).Google Scholar
  8. 8.
    K. T. Andrews and I. D. Ward, “Factorization of diagonally dominant operators onL 1 (0, 1,X)”Trans. Am. Math. Soc.,291, No. 2, 789–800 (1985).Google Scholar
  9. 9.
    V. E. Zakharov and A. B. Shabat, “A scheme of integration of nonlinear equations. I,”Funkts. Anal. Prilozh.,8, No. 3, 43–53 (1974).Google Scholar
  10. 10.
    L. P. Nizhnik and M. D. Pochinaiko, “Nonlinear Schrödinger equation as an integrable Hamiltonian system in a two-dimensional space,”Usp. Mat. Nauk,37, No. 4, 111–112 (1982).Google Scholar
  11. 11.
    L. A. Sakhnovich, “Problems of factorization and operator identities,”Usp. Mat. Nauk,41, No. 1, 3–55 (1986).Google Scholar
  12. 12.
    L. A. Sakhnovich,Nonlinear Equations and Inverse Problems on a Semiaxis [in Russian], Preprint No. 87.48, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987).Google Scholar
  13. 13.
    M. A. Barkar' and I. Ts. Gokhberg, “On the factorization of operators in Banach spaces,”Mat. Issled.,1, No. 2, 90–123 (1966).Google Scholar
  14. 14.
    L. A. Sakhnovich, “Equations with difference kernels on a finite segment,”Usp. Mat. Nauk,34, No. 4, 69–129 (1980).Google Scholar
  15. 15.
    M. G. Krein, “Integral equations on a half line,”Usp. Mat. Nauk,13, No. 5, 3–120 (1958).Google Scholar
  16. 16.
    E. C. Titchmarsh,Introduction to the Theory of Fourier Integrals, Oxford University Press, Oxford (1937).Google Scholar
  17. 17.
    A. C. Dixon, “On the solving nuclei of certain integral equation,”Proc. London Math. Soc.,27, 233–272 (1926).Google Scholar
  18. 18.
    V. P. Khavin, S. V. Khrushchev, and N. K. Nikol'skii (editors),Linear and Complex Analysis Problem Book, Berlin (1984).Google Scholar
  19. 19.
    R. Kadison and I. Singer, “Triangular operator algebras,”Am. J. Math.,82, 227–259 (1960).Google Scholar
  20. 20.
    G. E. Kisilevskii, “On the generalization of the Jordan theory for some class of linear operators in a Hilbert śpace,”Dokl. Akad. Nauk SSSR,176, No. 4, 768–770 (1967).Google Scholar
  21. 21.
    T. Cárleman, “Über die Abelsche Integralgleichung mit Konstanten Integrationsgrenzen,”Math. Z.,15, 111–120 (1921).Google Scholar
  22. 22.
    A. G. Buslaev, “Factorization of a special integral operator,”Ukr. Mat. Zh.,40, No. 6, 780–784 (1988).Google Scholar
  23. 23.
    M. G. Krein, “On the logarithm of an infinitely decomposable Hermite-positive function,”Dokl. Akad. Nauk SSSR,45, 99–102 (1944).Google Scholar
  24. 24.
    M. G. Krein, “Continual analogs of the propositions concerning polynomials orthogonal on a unit circle,”Dokl. Akad. Nauk SSSR,105, No. 4, 637–640 (1955).Google Scholar
  25. 25.
    M. G. Krein, “On a continual analog of one Christoffel formula in the theory of orthogonal polynomials,”Dokl. Akad. Nauk SSSR,113, No. 5, 970–973 (1957).Google Scholar
  26. 26.
    V. A. Marchenko,Nonlinear Equations and Operator Algebras [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
  27. 27.
    S. P. Novikov (editor),Theory of Solitons. A Method of Inverse Problem [in Russian], Nauka, Moscow (1980).Google Scholar
  28. 28.
    L. A. Takhtadzhyan and L. D. Faddeev,Hamilton Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986).Google Scholar
  29. 29.
    V. P. Potapov, “Multiplicative structure ofJ-nonexpanding matrix functions,”Tr. Mosk. Mat. Obshch.,4, 125–236 (1955).Google Scholar
  30. 30.
    Z. L. Leibenzon, “A relationship between the inverse problems and completeness of eigenfunctions,”Dokl. Akad. Nauk SSSR,145, No. 3, 519–522 (1962).Google Scholar
  31. 31.
    M. S. Brodskii and G. É. Kisilevskii, “A criterion for attributing Volterra operators with imaginary kernel components to the class of one-cellular operators,”Izv. Akad. Nauk SSSR, Ser. Mat.,30, No. 6, 1213–1228 (1966).Google Scholar
  32. 32.
    L. A. Sakhnovich, “On dissipative Volterra operators,”Mat. Sb.,76, No. 3, 323–343 (1968).Google Scholar
  33. 33.
    L. A. Sakhnovich, “Spectral functions of a 2nth-order canonical system,”Mat. Sb.,181, No. 11, 1510–1524 (1990).Google Scholar
  34. 34.
    M. S. Livshits,Operators, Oscillations, and Waves. Open Systems [in Russian], Nauka, Moscow (1966).Google Scholar
  35. 35.
    M. S. Brodskii,Triangular and Jordan Representations of Linear Operators [in Russian], Nauka, Moscow (1969).Google Scholar
  36. 36.
    L. A. Sakhnovich, “On the reduction of nonself-adjoint operator to the triangular form,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 1, 180–186 (1959).Google Scholar
  37. 37.
    L. A. Sakhnovich, “Investigation of a triangular model of nonself-adjoint operators,”Izv. Vyssh. Uchebn. Zaved., Mat., No. 4, 141–149 (1959).Google Scholar
  38. 38.
    M. S. Brodskii, I. Ts. Gokhberg, and M. G. Krein, “General theorems on triangular representations of linear operators and on multiplicative representations of their characteristic functions,”Funkts. Anal. Prilozh.,3, No. 4, 1–27 (1969).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • L. A. Sakhnovich
    • 1
  1. 1.Odessa Electrotechnical Institute of CommunicationsOdessa

Personalised recommendations