Advertisement

On the Representation of Doubly Stochastic Integral Operators by Means of Disjoint Isomorphisms

  • V. N. Sudakov
Chapter
Part of the Seminars in Mathematics book series (SM)

Abstract

A doubly stochastic integral operator is an integral operator of the form
$$\left( {Bf} \right)\left( y \right) = \int\limits_X {k\left( {x,y} \right)f\left( x \right)d\mu \left( x \right)} ,$$
pwhere k (x,y) is a negative function given on the product M =X × Y of spaces with measures (X,μ) and (Y,v), where \(\int\limits_X {k\left( {x,y} \right)d\mu \left( x \right)} = 1\) for almost all y ∈ Y and \(\int\limits_Y {k\left( {x,y} \right)dv\left( y \right)} = 1\) for almost all x∈X.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literature Cited

  1. 1.
    Halmos, P. R., Measure Theory, Van Nostrand, New York (1950).CrossRefGoogle Scholar
  2. 2.
    Rokhlin, V. A., On the fundamental notions of measure theory, Matem. Sborn., 15(67) (1): 107–150 (1949).Google Scholar
  3. 3.
    Bourbaki, N., Integration [Russian translation], Nauka, Moscow (1967).Google Scholar
  4. 4.
    Birkhoff, G., Three observations on linear algebra, Univ. Nac. Tucuman Revista, (A) 5:147–151 (1946)Google Scholar
  5. 5.
    Ryser, J. H., Combinatorial Mathematics (Carus Mathematical Monographs, No. 14 ), Mathematical Association of America, Wiley, New York (1963).Google Scholar
  6. 6.
    Birkhoff, G. D., Lattice Theory, American Mathematical Society, New York (1967).Google Scholar
  7. 7.
    Isbell, J. R., Birkhoff’s Problem III, Proc. Amer, Math. Soc., 6: 217–218 (1955).Google Scholar
  8. 8.
    Isbell, J. R., Infinite doubly stochastic matrices, Canad. Math. Bull., 5: 1–4 (1962).CrossRefGoogle Scholar
  9. 9.
    Rattray, B. A. and Peck, J. E., Infinite stochastic matrices, Trans. Roy. Soc. Canada, Sec. III, 49: 55–57 (1955).Google Scholar
  10. 10.
    Kendall, D. G., On infinite doubly stochastic matrices and Birkhoff’s Problem III, J. London Math, Soc., 35: 81–84 (1960).Google Scholar
  11. 11.
    Pevesz, P., A probabilistic solution of Problem III of G. Birkhoff, Acta Math. Acad. Sci. Hungar., 13: 187–198 (1962).CrossRefGoogle Scholar
  12. 12.
    Peck, J. E. S., Doubly stochastic measures, Fund. Math., 40: 113–124 (1953).Google Scholar
  13. 13.
    Brown, J. R., Approximation theorems for Markov operators, Pacific J. Math., 16 (1) (1966).Google Scholar
  14. 14.
    Choo-whan Kim, Uniform approximation of doubly stochastic operators, Pacific J. Math., 26 (3) (1968).Google Scholar
  15. 15.
    Sudakov, V. N., On the existence of random variables independent of several given variables, Trans. Fourth Prague Conf. Information Theory, Statistics, Decision Functions, and Random Processes, Prague (1965).Google Scholar
  16. 16.
    Sudakov, V. N., On the independent complement to two partitions in the case of the existence of bounded density, Trudy Matem. Inst. Steklov. (submitted 1968 ).Google Scholar
  17. 17.
    Romanovskii, L V., and Sudakov, V. N., On the existence of independent partitions, Trudy Matem. Inst. Steklov., 79: 5–10 (1965).Google Scholar

Copyright information

© Springer Science+Business Media New York 1972

Authors and Affiliations

  • V. N. Sudakov

There are no affiliations available

Personalised recommendations