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The converse of the Inverse-Conjugacy Theorem for unitary operators
and ergodic dynamical systems

Author: Geoffrey R. Goodson
Journal: Proc. Amer. Math. Soc. 128 (2000), 1381-1388
MSC (1991): Primary 28D05; Secondary 47A35
Published electronically: August 5, 1999
MathSciNet review: 1641701
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Abstract: We show that the converse to the main theorem of Ergodic transformations conjugate to their inverses by involutions, by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97-124), holds in the unitary category. Specifically it is shown that if $U$ is a unitary operator defined on an $L^{2}$ space which preserves real valued functions, and if $U^{-1}S=SU$ implies $S^{2}=I$ whenever $S$ is another such operator, then $U$ has simple spectrum. The corresponding result for measure preserving transformations is shown to be false. The counter-example we have involves Gaussian automorphisms. We show that a Gaussian automorphism is always conjugate to its inverse, so that the Inverse-Conjugacy Theorem is applicable to such maps having simple spectrum. Furthermore, there are Gaussian automorphisms having non-simple spectrum for which every conjugation of $T$ with $T^{-1}$ is an involution.

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Geoffrey R. Goodson
Affiliation: Department of Mathematics, Towson University, Towson, Maryland 21252

Received by editor(s): October 28, 1997
Received by editor(s) in revised form: June 25, 1998
Published electronically: August 5, 1999
Communicated by: Mary Rees
Article copyright: © Copyright 2000 American Mathematical Society