Transformations conjugate to their inverses have even essential values

Let $T$ be an ergodic automorphism defined on a standard Borel probability space for which $T$ and $T^{-1}$ are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of $T$. 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 $T$ is ergodic having simple spectrum and isomorphic to its inverse, and if $S$ is a conjugation between $T$ and $T^{-1}$ (i.e. $S$ satisfies $TS=ST^{-1}$), then $S^{2}=I$, the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation $S$, the unitary operator induced by $T$ on $L^{2}(X,\mu )$ must have a multiplicity function whose essential values on the ortho-complement of the subspace $\{ f\in L^{2}(X,\mu ): f(S^{2})=f \}$ are always even. In particular, we see that $S$ can be weakly mixing, so the corresponding $T$ must have even maximal spectral multiplicity (regarding $\infty$ as an even number).