Schatten class Hankel operators on the Bergman space (Q806008)

Let \(H_ f,H_{\bar f}\) be Hankel operators on the Bergman spaces of a bounded symmetric domain D. The author gives the condition on f where both operators \(H_ f\) and \(H_{\bar f}\) are in Schatten p-class \({\mathfrak S}_ p\) for \(2\leq p<\infty\). Let \(k_ z\) be the normalized Bergman kernel, dA(w) the a Euclidean measure on \(C^ n\) and G denote the connected component of the biholomorphic automorphism group of D. Let also \(\tilde f(z)=(fk_ z,k_ z)\) denote the Berezin transform of f in \(L^ 2\) and \[ \tilde f(z)=\int_{D}f(w)| k_ z(w)|^ 2dA(w) \] for f in \(L^ 1\) and put \[ MQ(f)(z)=[| \tilde f|^ 2(z)- | \tilde f(z)|^ 2]^{1/2}. \] The main result contains the following Theorem. Suppose \(2\leq p<\infty\). For f in \(L^ 2\), both \(H_ f\) and \(H_{\bar f}\) are in \({\mathfrak S}_ p\) if and only if \[ \int_{D}MQ(f)^ p(z)d\mu (z)<+\infty \] where \(d\mu\) (z) is the G- invariant measure K(z,z)dA(z) of D. The paper of \textit{K. Zhu} ``Schatten class Hankel operators on the Bergman space of the unit ball'', preprint, 1989, is also relevant.