Advertisement

Journal of Soviet Mathematics

, Volume 49, Issue 4, pp 1058–1063 | Cite as

The Perron-Frobenius theory for almost periodic representations of semigroups in the spaces Lp

  • Vu. Kuok Fong
Article

Abstract

The classical Perron-Frobenius theory of nonnegative matrices is generalized to nonnegative almost periodic representations of topological semigroups in the spaces Lp(Ω, σ, Μ), where (Ω, σ, Μ) is a space with a σ-finite measure, 1≤p<. With each such representation one connects the associated action of its Sushkevich kernel onto some naturally arising space with measure; this allows that the investigation of the spectral properties of the representation be reduced to the investigation of the ergodic properties of the corresponding action. In particular, it is established, that the boundary spectrum of an indecomposable representation is a subgroup of the dual group of the Sushkevich kernel (coincides with it if the considered semigroup is Abelian). In the general case the boundary spectrum is cyclic (i.e., the union of the subgroups of the dual group of the Sushkevich kernel). The results of the paper are new even at the consideration of the semigroups of degree one of an operator (if other words, the representations of the semigroup Z+); this yields the generalized Perron-Frobenius theory for nonnegative a. p. operators.

Keywords

Spectral Property Dual Group Ergodic Property Topological Semigroup Nonnegative Matrice 
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.

Literature cited

  1. 1.
    M. Yu. Lyubich and Yu. I. Lyubich, “The spectral theory of the almost periodic representations of semigroups,” Ukr. Mat. Zh.,36, No. 5, 632–636 (1984).Google Scholar
  2. 2.
    M. Yu. Lyubich and Yu. I. Lyubich, “Splitting-off of the boundary spectrum for almost periodic operators and representations of semigroups,” Teor. Funktsii Funktsional. Anal. i Prilozhen. (Kharkov), No. 45, 69–84 (1986).Google Scholar
  3. 3.
    Yu. I. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birkhauser, Basel (1988).Google Scholar
  4. 4.
    M. Yu. Lyubich and Yu. I. Lyubich, “The Perron-Frobenius theory for almost periodic operators and representations of semigroups,” Teor. Funktsii Funktsional. Anal. i Prilozhen. (Kharkov), No. 46, 54–72 (1986).Google Scholar
  5. 5.
    H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York (1974).Google Scholar
  6. 6.
    J. Lamperti, “On the isometries of certain function-spaces,” Pac. J. Math.,8. No. 3, 459–466 (1958).Google Scholar
  7. 7.
    A. Ionescu Tulcea, “Ergodic properties of isometries in Lp spaces, 1<p<∞,” Bull. Am. Math. Soc.,70, No. 3, 366–371 (1964).Google Scholar
  8. 8.
    V. A. Rokhlin, “On the fundamental concepts of measure theory,” Mat. Sb.,25 (67), 107–150 (1949).Google Scholar
  9. 9.
    I. P. Kornfel'd, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow (1980).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • Vu. Kuok Fong

There are no affiliations available

Personalised recommendations