Ideals generated by integral operators (Q1100711)

Let (X,\(\mu)\) be a space with \(\sigma\)-finite positive measure \(\mu\) so that \(L_ 2(X,\mu)\) is a separable space. The linear continuous operator \(T:L_ 2(X,\mu)\to L_ 2(X,\mu)\), is defined as integral operator if there exists a (\(\mu\times \mu)\) measurable function K:X\(\times X\to C\), so that for every \(f\in L_ 2(X,\mu)\), \[ Tf(s)=\int_{X}K(s,t)f(t)d\mu (t), \] for almost every \(d\in X.\) In this paper the author proves the following theorem: If \(T:L_ 2(X,\mu)\to L_ 2(X,\mu)\) is a linear integral operator with kernel K(s,t), then the operator AT is an integral operator for every linear continuous operator \(A:L_ 2(X,\mu)\to L_ 2(X,\mu)\) if and only if \(\int_{X}\int_{X}| K(s,t)|^ 2d\mu (t)d\mu (s)<\infty\).