Products of Hankel and Toeplitz operators on the Bergman space
Let \(L^2_\alpha\) denote the Bergman space consisting of the analytic functions in \(L^2(D, dA)\), where \(dA\) is the normalized Lebesgue area measure on the unit disk \(D\). Let \(P\) be the orthogonal projection from \(L^2(D, dA)\) onto \(L^2_\alpha\). For \(f\in L^2(D, dA)\), the Toeplitz operator \(T_f\) and the Hankel operator \(H_f\) with symbol \(f\) are defined densely on \(L^2_\alpha\) by \(T_f(h)= P(fh)\) and \(H_f(h)= (1- P)(fh)\) for all polynomials \(h\). The authors consider the question for which \(f\) and \(g\) in \(L^2_\alpha\) the densely defined products \(T_fT_g\) are bounded on the Bergman space. Hankel products \(H_f H^*_g\) and the mixed Haplitz products \(H_fT_g\) and \(T_gH^*_f\) are also investigated. In particular, conditions for compactness of the various Haplitz products are established.
scientific article; zbMATH DE number 1398382