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

Abstract

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. §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 ∈ H for which Z(h) = H, where Z(h) is the closed linear span of the set {U n h : n ∈ Z}. We are mainly interested in the case where the Hilbert space is a function space. Let (X, F , µ) 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, 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.