Unitary equivalence to integral operators

Abstract

A bounded operator A on L²(X) is called an integral operator if there exists a measurable function k on X x X such that, for each f) in L ²(X), ∫\k(x,y)ƒ(y)\d μ (y) < ∞ a.e. and Aƒ(x)= ∫ k(x,y)ƒ(y)d μ (y) a.e. (Throughout this paper, (X, μ ) will denote a separable, sigma -finite measure space which is not purely atomic.) An integral operator is called a Carleman operator if the inducing kernel k satisfies the stronger requirement: ∫\k(x,y)\ ²d μ (y) < ∞ for almost every x in X.