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On one-parameter Koopman groups

Published online by Cambridge University Press:  28 December 2015

A. F. M. ter ELST  and
M. LEMAŃCZYK
A. F. M. ter ELST
Affiliation:
Department of Mathematics, University of Auckland, Private bag 92019, Auckland 1142, New Zealand email terelst@math.auckland.ac.nz
M. LEMAŃCZYK
Affiliation:
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, 12/18 Chopin street, 87-100 Toruń, Poland email mlem@mat.umk.pl
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Abstract

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We characterize Koopman one-parameter $C_{0}$ -groups, in the class of all unitary one-parameter $C_{0}$ -groups on $L_{2}(X)$ , as those that preserve $L_{\infty }(X)$ and for which the infinitesimal generator is a derivation on the bounded functions in its domain.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 5 , August 2017 , pp. 1635 - 1656
Copyright
© Cambridge University Press, 2015 

References

Arendt, W., Batty, C. J. K., Hieber, M. and Neubrander, F.. Vector-valued Laplace Transforms and Cauchy Problems (Monographs in Mathematics, 96) . Birkhäuser, Basel, 2001.CrossRefGoogle Scholar
Cornfeld, I. P., Fomin, S. and Sinai, Y. G.. Ergodic Theory (Grundlehren der mathematischen Wissenschaften, 245) . Springer, New York, 1982.CrossRefGoogle Scholar
Choksi, J. R.. Unitary operators induced by measurable transformations. J. Math. Mech. 17 (1967), 785–801.Google Scholar
Cotlar, M. and Ricabarra, R. A.. Sobre transformaciones de conjuntos y operadores de Koopman. Rev. Un. Mat. Argentina 14 (1950), 232–254.Google Scholar
Denker, M.. On unitary operators inducing measure-preserving transformations. Math. Z. 160 (1978), 163–172.CrossRefGoogle Scholar
Derndinger, R. and Nagel, R.. Der Generator stark stetiger Verbandshalbgruppen auf C (X) und dessen Spektrum. Math. Ann. 245 (1979), 159–177.CrossRefGoogle Scholar
Engel, K.-J. and Nagel, R.. One-parameter Semigroups for Linear Evolution Equations (Graduate Texts in Mathematics, 194) . Springer, New York, 2000.Google Scholar
Goodrich, K., Gustafson, K. and Misra, B.. On converse to Koopman’s lemma. Phys. A 102 (1980), 379–388.CrossRefGoogle Scholar
Gallavotti, G. and Pulvirenti, M.. Classical KMS condition and Tomita–Takesaki theory. Comm. Math. Phys. 46 (1976), 1–9.CrossRefGoogle Scholar
Glasner, E., Tsirelson, B. and Weiss, B.. The automorphism group of the Gaussian measure cannot act pointwise. Israel J. Math. 148 (2005), 305–329.CrossRefGoogle Scholar
Halmos, P. R.. Lectures on Ergodic Theory. Chelsea Publishing Co., New York, 1956.Google Scholar
Kechris, A. S.. Classical Descriptive Set Theory (Graduate Texts in Mathematics, 156) . Springer, New York, 1995.CrossRefGoogle Scholar
Katok, A. and Lemańczyk, M.. Some new cases of realization of spectral multiplicity function for ergodic transformations. Fund. Math. 206 (2009), 185–215.CrossRefGoogle Scholar
Katok, A. and Thouvenot, J.-P.. Spectral properties and combinatorial constructions in ergodic theory. Handbook of Dynamical Systems. Vol. 1B. Elsevier, Amsterdam, 2006, pp. 649–743.CrossRefGoogle Scholar
Lamperti, J.. On the isometries of certain function-spaces. Pacific J. Math. 8 (1958), 459–466.CrossRefGoogle Scholar
Lemańczyk, M.. Spectral theory of dynamical systems. Encyclopedia of Complex and Systems Science. Ed. Meyers, R. A.. Springer, New York, 2009, pp. 8554–8575.CrossRefGoogle Scholar
Ridge, W. C.. Characterization of abstract composition operators. Proc. Amer. Math. Soc. 45 (1974), 393–396.CrossRefGoogle Scholar
Stone, M. H.. On one-parameter unitary groups in Hilbert space. Ann. of Math. (2) 33 (1932), 643–648.CrossRefGoogle Scholar
Voigt, J.. One-parameter semigroups acting simultaneously on different L p -spaces. Bull. Soc. Roy. Sci. Liège 61 (1992), 465–470.Google Scholar