Invertible Composition Operators on L 2 (λ)

Abstract

Let C<, be a composition operator on L?(X), where X is a a-finite measure defined on the Borel subsets of a standard Borel space. In this paper a necessary and sufficient condition for the invertibility of C<, is given in terms of invertibility of 0. Also all invertible composition operators on L2(R) induced by monotone continuous functions are characterised. Introduction. Let (X, S, X) be a a-finite measure space, and let ( be a measurable transformation from X into itself. Let L3(X) denote the Hilbert space of all square-integrable functions on X. Define a composition transformation C. on L3(X) as C-f = f o p for every f E L3(X). In case C, is a bounded operator with the range in L3(A), we call it a composition operator induced by 4. The Banach space of all bounded linear operators will be denoted by S. The purpose of this paper is to study the invertible composition operators. The invertible composition operators on HP (of the unit disk) are studied by Schwartz [5], where he proves that the invertibility of the inducing function on the unit disk into itself is a necessary and sufficient condition for the invertibility of the corresponding composition operator. This is true because the inducing functions are analytic functions and hence they are nicely behaved. In case of composition operators on L2(X) the above statement is not completely true as is shown later by an example. However, a little more addition to the hypothesis makes it go through both ways in case of L2(X). 2. An invertibility theorem. DEFINITION. Let (X,S,X) be a measure space. Let 0 be a measurable transformation on X into itself. Then 0 is said to be one-to-one if there exists a measurable transformation 4' on X into itself such that (4p o +)(x) = x a.e. 0 is said to be onto if there exists a measurable transformation X such that (po w) (x) = x a.e. 0 is said to be invertible if there is a measurable transformation 4, such that (0 o A)(x) = (4o +)(x) = x a.e. Such 4 is called the inverse of 0 and is denoted as 0DEFINITION. A standard Borel space X is a Borel subset of a complete separable metric space T. The class S will consist of all the sets of the form X n B, where B is a Borel subset of T. Received by the editors March 21, 1975. AMS (MOS) subject classifications (1970). Primary 47B99; Secondary 47B99.