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

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.