Eigen operators of ergodic transformations

Abstract

Introduction.In a recent paper by A. Beck and J. T. Schwartz [2], mention is made of the following problem: Let 3E be a 5-space and T a bounded linear operator on with ||r|| =1.Let (S, 2, m) be a measure space, with an ergodic measure-preserving transformation h defined on it.Let X be a strongly measurable function, not identically 0, from (S, 2, m) into 26 and suppose that (1) X(h(s)) = T(X(s)) a.e. in S.Then we shall call the operator T an eigen operator of h.The problem is to determine when these conditions can be satisfied.One is struck by the resemblance of equation ( 1) to the eigenvalue equation i.e.X is defined to be an eigenvalue of the m.p.t.h if there is a nonzero complex-valued measurable function f on S such that(2) /(*(*)) = \f(s) a.e. in S.The importance of eigenvalues as a tool in the study of m.p.t.'s suggests that eigen operators, which include eigenvalues as a special case, may prove a yet finer tool in this work.In this paper, we will investigate the nature of eigen operators and their relation to their corresponding ergodic transformations.It will be shown that the solutions of this equation (1) depend on the theory of ergodic translations in a monothetic measure group.Definitions.We take (S, 2, m) to be a <r-finite measure space in the sense of Halmos [3].A measure-preserving transformation (m.p.t.) h is a oneto-one mapping of S onto itself so that if Ai = h(A2)= {h(s)\s (EAi}, then Ai is 2-measurable if and only if ^42 is and m(Ai) =m(A2).If h(A)=A, .4G2,necessarily implies that either w(^4)=0 or m(S-A)=0, then h is said to be ergodic.Yet be a complex 5-space.Then the set of all open sets in 36 generates a cr-field of subsets of H known as the Borel field of %, and each set in the Borel field is called a Borel set.A mapping X from S into TL is called strongly measurable if