Two characterization theorems for integral operators

Abstract

Let (X,μ) be a separable σ-finite measure space. A bounded operator A on L²(X) is called an integral operator if it is induced by an equation: Af(x)=∫k(x,y)f(y)dμ(y), where k is a measurable function on X×X such that ∫|k(x,y)f(y)|dμ(y) < > ∞ a.e. for every f in L²(X). In this paper, some results on Carleman operators, due to von Neumann, Tarjonski and Weidmann, are extended to the case of the general integral operator.