Hankel operators between weighted Bergman spaces in the ball
Let m be the Lebesgue measure on the unit ball \(B\subset {\mathbb{C}}^ n\); \(\mu_{\alpha}=c_{\alpha}(1-| z|^ 2)^{\alpha} dm(z)\), where \(-1<\alpha <\infty\) and \(c_{\alpha}\) is chosen such that \(\mu_{\alpha}(B)=1\). Introduce the weighted Bergman space \(A^{\alpha}\) as the closed subspace of all holomorphic functions in \(L^ 2(d\mu_{\alpha})\). And let \(P_{\alpha}\) be the selfadjoint projection of \(L^ 2(d\mu_{\alpha})\) onto \(A^{\alpha}.\) The author studies the Hankel operator \(H_ f: A^{\beta}\to L^ 2(d\mu_{\alpha})\), where \((H_ f)(g)=(I-P_{\alpha})\bar fg\); the symbol f is assumed to be holomorphic. The necessary and sufficient conditions to be a compact operator and to belong to the Schatten-von Neumann class \(S_ p\) for the operator \(H_ f\) are given.
scientific article; zbMATH DE number 4157302