A new kind of Hankel-Toeplitz type operator connected with the complementary series (Q5930100)

Denote by \({\mathcal A}^{\nu}\) the weighted Bergman spaces either over the unit disk \({\mathbf D}\) or over the unit ball \(\mathbf B\) having dimension \(d >1\). The authors define the Hankel-Toeplitz operator \(H_{\phi}\) with symbol \(\phi\) which acts on \({\mathcal A}^{\nu}\), and prove that the operator \(H_{\phi}\) is Hilbert-Schmidt if and only if its symbol \(\phi\) belongs to the Hilbert space \(H^{\lambda}\) of the complementary series of representations of \(SU(1,1)\), where \(\lambda=2-2\nu\). In the case of the unit ball \(\mathbf B\) the authors define covariantly \(k\) different Hankel-Toeplitz type operators \(H_\psi^{(k)}\) acting on \({\mathcal A}^{\nu}\), where \(0\leq k <{1\over 2}(d-2\nu)\), and show that the operators \(H_\psi^{(k)}\) can also be obtained with the help of certain sesquilinear differential operators.