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

DOI:
https://doi.org/10.1090/S0002-9939-96-03344-8

MathSciNet review:
1327016

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an ergodic automorphism defined on a standard Borel probability space for which and are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of . 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 is ergodic having simple spectrum and isomorphic to its inverse, and if is a conjugation between and (i.e. satisfies ), then , the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation , the unitary operator induced by on must have a multiplicity function whose essential values on the ortho-complement of the subspace are always even. In particular, we see that can be weakly mixing, so the corresponding must have even maximal spectral multiplicity (regarding 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:
https://doi.org/10.1090/S0002-9939-96-03344-8

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