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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 L2 space which preserves real valued functions, and if U-S = SU implies S2 = 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. ?0. INTRODUCTION Let U: H -* H be a unitary operator defined on a separable Hilbert space H. U is said to have simple spectrum if there exists an h E H for which Z(h) = H, where Z(h) is the closed linear span of the set {UTh: n E Z}. We are mainly interested in the case where the Hilbert space is a function space. Let (X, F, ,u) denote a standard Borel probability space, and let T: X -+ X be an invertible measure preserving transformation (automorphism). To say that the automorphism T has simple spectrum means that the unitary operator UT: L2(X, 1t) -* L2(X, ,u), UTf(x) = f(Tx), induced by T, has simple spectrum. It is known that if T has simple spectrum, then T is ergodic, and that unitary operators preserving real valued functions are conjugate to their inverses. The following theorem and corollary were given in [2] (see also [3] and [4] for related results). We use I to denote both the identity operator, and the identity automorphism. Inverse Conjugacy Theorem. Let U: L2(X, F, 1t) -+ L2(X, F, p) be a unitary operator having simple spectrum and preserving real valued functions. If S is another unitary operator preserving real valued functions and U1S SU, then S2 = J. Received by the editors October 28, 1997 and, in revised form, June 25, 1998. 1991 Mathematics Subject Classification. Primary 28D05; Secondary 47A35. (?2000 American Mathematical Society

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