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Mat. Sb., 2006, Volume 197, Number 5, Pages 3–50 (Mi msb1135)  

This article is cited in 5 scientific papers (total in 5 papers)

Variational principles for the spectral radius

A. B. Antonevicha, K. Zajkowski

a Belarusian State University

Abstract: The spectral radius of a functional operator with positive coefficients generated by a set of maps (a dynamical system) is shown to be a logarithmically convex functional of the logarithms of the coefficients. This yields the following variational principle: the logarithm of the spectral radius is the Legendre transform of a convex functional $T$ defined on a set of vector-valued probability measures and depending only on the original dynamical system. A combinatorial construction of the functional $T$ by means of the random walk process corresponding to the dynamical system is presented in the subexponential case. Examples of the explicit calculation of the functional $T$ and the spectral radius are presented.
Bibliography: 28 titles.
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DOI: https://doi.org/10.4213/sm1135

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English version:
Sbornik: Mathematics, 2006, 197:5, 633–680

Bibliographic databases:

UDC: 517.983.23+517.984.5
MSC: Primary 47B38, 47A10; Secondary 47B33
Received: 18.08.2005

Citation: A. B. Antonevich, K. Zajkowski, “Variational principles for the spectral radius”, Mat. Sb., 197:5 (2006), 3–50; Sb. Math., 197:5 (2006), 633–680

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Ostaszewska U., Zajkowski K., “Spectral exponent of finite sums of weighted positive operators in $L^p$-spaces”, Positivity, 11:4 (2007), 549–562  crossref  mathscinet  zmath  isi
    2. Ostaszewska U., Zajkowski K., “Variational principle for the spectral exponent of polynomials of weighted composition operators”, J. Math. Anal. Appl., 361:1 (2010), 246–251  crossref  mathscinet  zmath  isi
    3. Antonevich A.B., Bakhtin V.I., Lebedev A.V., “On $t$-entropy and variational principle for the spectral radii of transfer and weighted shift operators”, Ergod. Th. Dynam. Sys., 2010  crossref  mathscinet  zmath  isi
    4. Tarasenko A., Karelin A., Pérez Lechuga G., González-Hernández M., “Modelling systems with renewable resources based on functional operators with shift”, Appl. Math. Comput., 216:7 (2010), 1938–1944  crossref  mathscinet  zmath  isi
    5. A. B. Antonevich, E. U. Leonova, “The Extended Legendre Transform and Related Variational Principles”, Math. Notes, 107:3 (2020), 369–382  mathnet  crossref  crossref  isi  elib  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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