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 | References | Similar Articles | Additional Information

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 is a unitary operator defined on an space which preserves real valued functions, and if implies whenever is another such operator, then 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 with is an involution.

**[1]**I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ,*Ergodic theory*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR**832433****[2]**Geoffrey R. Goodson, Andrés del Junco, Mariusz Lemańczyk, and Daniel J. Rudolph,*Ergodic transformations conjugate to their inverses by involutions*, Ergodic Theory Dynam. Systems**16**(1996), no. 1, 97–124. MR**1375129**, 10.1017/S0143385700008737**[3]**Geoffrey Goodson and Mariusz Lemańczyk,*Transformations conjugate to their inverses have even essential values*, Proc. Amer. Math. Soc.**124**(1996), no. 9, 2703–2710. MR**1327016**, 10.1090/S0002-9939-96-03344-8**[4]**G. R. Goodson and V. V. Ryzhikov,*Conjugations, joinings, and direct products of locally rank one dynamical systems*, J. Dynam. Control Systems**3**(1997), no. 3, 321–341. MR**1472356**, 10.1007/BF02463255**[5]**M. Lema\'{n}czyk, J. de Sam Lazaro,*Spectral analysis of certain factors for Gaussian dynamical systems*, Israel J. Math.**98**(1997), 307-328. CMP**97:15****[6]**Paul R. Halmos,*Introduction to Hilbert Space and the theory of Spectral Multiplicity*, Chelsea Publishing Company, New York, N. Y., 1951. MR**0045309**

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Additional Information

**Geoffrey R. Goodson**

Affiliation:
Department of Mathematics, Towson University, Towson, Maryland 21252

Email:
ggoodson@towson.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05143-6

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