Defect Operators Associated with Submodules of the Hardy Module

Let H(S) be the Hardy space on the unit sphere S in C, n ≥ 2. Then H(S) is a natural Hilbert module over the ball algebra A(B). Let Mz1 , ..., Mzn be the module operators corresponding to the multiplication by the coordinated functions. Each submodule M ⊂ H(S) gives rise to the module operators ZM,j = Mzj |M, j = 1, ..., n, on M. In this paper we establish the following commonly believed, but never previously proven result: whenever M 6 = {0}, the sum of the commutators [Z∗ M,1, ZM,1] + ...+ [Z ∗ M,n, ZM,n] does not belong to the Schatten class Cn.