Two characterization theorems for integral operators

Let (X,) be a separable -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)d(y), where k is a measurable function on X × X such that |k(x,y)f(y)|d(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.