On unitary operators inducing measure-preserving transformations

Abstract

We give a necessary and sufficient condition that a unitary operator U on an abstract separable Hilbert space is inducible by a measure preserving transformation T on a probability space (f2, ~, m). Several authors have studied this question, also in connection with nonsingular transformations ([1, 2, 3, 4, 6]). In [1] and [2] Choksi characterizes these unitary operators in terms of spectral measures and certain subsets of H, which he called cg-families. In this paper we define @=families-closely related to Choksi's f am i ly and show that such a family exists iff U is inducible. There are two advantages in defining this family. First the definition involves fewer and easier conditions and secondly, a ~ family bears a separable structure, which a cg-family does not; essentially we reduce the condition of norm-elosedness of a cg-family. Our conditions give some hints in which way one should derive a measure preserving transformation inducing a given unitary operator for concrete examples. H always denotes a separable (real) Hilbert space with inner product ( . , . ) and norm Jr.lr. The letter U is used to denote a unitary operator acting on H with the following property: U has an eigenvalue 1 and an eigenvector I (with HIJI = 1) with respect to the eigenvalue 1. Obviously this is no restriction to our problem, but simplifies the statements in this paper.