Sums of products of Toeplitz and Hankel operators on the Dirichlet space (Q649059)

Let \(T_u\) and \(H_v\) denote the Toeplitz and the Hankel operators on the Dirichlet space \({\mathcal D}\) of the unit disk. The authors find necessary and sufficient conditions guaranteeing that each of the sums of products \(\sum_{i=1}^N\prod_{j=1}^M T_{u_{ij}}\), \(\sum_{i=1}^N H_{u_{i}} H_{v_i}\), and \(\sum_{i=1}^N T_{u_{i}} H_{v_i}\) is equal to zero. Here, all symbols \(u_{ij}, u_i, v_i\) are supposed to belong to the Sobolev space \({\mathcal L}^{1,\infty}\).