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Doklady Rossijskoj Akademii Nauk. Mathematika, Informatika, Processy Upravlenia, 2021, Volume 499, Pages 20–25
DOI: https://doi.org/10.31857/S268695432104010X (Mi danma18) 		 
MATHEMATICS

Symplectic geometry of the Koopman operator

V. V. Kozlovab

a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, 119991 Russia
b Demidov Yaroslavl State University, Yaroslavl, 150000 Russia
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DOI: https://doi.org/10.31857/S268695432104010X
Abstract: We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.
Keywords: Koopman operator, ergodicity, symplectic structure, quadratic invariants, discrete spectrum, Lebesgue spectrum.
Funding agency Grant number
Russian Science Foundation 21-71-30011
This work was supported by the Russian Science Foundation, project no. 21-71-30011.
Received: 12.05.2021
Revised: 12.05.2021
Accepted: 19.05.2021
English version:
Doklady Mathematics, 2021, Volume 104, Issue 1, Pages 175–179
DOI: https://doi.org/10.1134/S1064562421040104
Bibliographic databases:
Document Type: Article
UDC: 519.21+514.154
Language: Russian
Citation: V. V. Kozlov, “Symplectic geometry of the Koopman operator”, Dokl. RAN. Math. Inf. Proc. Upr., 499 (2021), 20–25; Dokl. Math., 104:1 (2021), 175–179
Citation in format AMSBIB
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