The converse of the Inverse-Conjugacy Theorem for unitary operators and ergodic dynamical systems
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- by Geoffrey R. Goodson PDF
- Proc. Amer. Math. Soc. 128 (2000), 1381-1388 Request permission
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.References
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Additional Information
- Geoffrey R. Goodson
- Affiliation: Department of Mathematics, Towson University, Towson, Maryland 21252
- Email: ggoodson@towson.edu
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1381-1388
- MSC (1991): Primary 28D05; Secondary 47A35
- DOI: https://doi.org/10.1090/S0002-9939-99-05143-6
- MathSciNet review: 1641701