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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Transformations conjugate to their
inverses have even essential values


Authors: Geoffrey Goodson and Mariusz Lemanczyk
Journal: Proc. Amer. Math. Soc. 124 (1996), 2703-2710
MSC (1991): Primary 28D05; Secondary 47A35
MathSciNet review: 1327016
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Abstract: Let $T$ be an ergodic automorphism defined on a standard Borel probability space for which $T$ and $T^{-1}$ are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of $T$. It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97--124) that if $T$ is ergodic having simple spectrum and isomorphic to its inverse, and if $S$ is a conjugation between $T$ and $T^{-1}$ (i.e. $S$ satisfies $TS=ST^{-1}$), then $S^{2}=I$, the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation $S$, the unitary operator induced by $T$ on $L^{2}(X,\mu )$ must have a multiplicity function whose essential values on the ortho-complement of the subspace $\{ f\in L^{2}(X,\mu ): f(S^{2})=f \} $ are always even. In particular, we see that $S$ can be weakly mixing, so the corresponding $T$ must have even maximal spectral multiplicity (regarding $\infty $ as an even number).


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Additional Information

Geoffrey Goodson
Affiliation: Department of Mathematics, Towson State University, Towson, Maryland 21204-7097
Email: e7m2grg@toe.towson.edu

Mariusz Lemanczyk
Affiliation: Institute of Mathematics, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland
Email: mlem@mat.uni.torun.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03344-8
PII: S 0002-9939(96)03344-8
Received by editor(s): November 1, 1994
Received by editor(s) in revised form: February 27, 1995
Additional Notes: The second author was partially supported by a KBN grant.
Communicated by: Mary Rees
Article copyright: © Copyright 1996 American Mathematical Society