Defect operators associated with submodules of the Hardy module (Q2909228)

Let \(H^2(S)\) be the Hardy space on the unit sphere \(S\) in \(\mathbb C^n\), \(n\geq 2\). Then \(H^2(S)\) is a natural Hilbert module over the ball algebra \(A(\mathbb B)\). Let \(M_{z_1}, \dots, M_{z_n}\) be the module operators corresponding to the multiplication by the coordinate functions. Each submodule \(\mathcal M\subset H^2(S)\) gives rise to the module operators \(Z_{\mathcal M,j}={M_{z_j}}{|\mathcal M}\), \(j=1,\dots, n\), on \(\mathcal M\). Denote by \(D_{\mathcal M}\) the operator \(\sum_{i=1}^n [ Z_{\mathcal M,i}^*,Z_{\mathcal M,i}]\). The main result of the paper is an estimate for the distribution of the \(s\)-numbers of that operator which implies that, whenever \(\mathcal M\neq \{ 0\}\), it does not belong to the Schatten class \(\mathcal C_n\). A similar result is presented in relation with the submodules of the Drury-Arveson module \(H_n^2\). The history of the corresponding problems is given in detail.