Defect operators associated with submodules of the Hardy module

Let H²(S) be the Hardy space on the unit sphere S in ℂⁿ, n ≥ 2. Then H²(S) is a natural Hilbert module over the ball algebra A(B). Let M_z1,...,M_zn be the module operators corresponding to the multiplication by the coordinated functions. Each submodule M ⊂ H²(S) gives rise to the module operators Z _M,j = M_zj |M, j = 1,...,n, onM. In this paper we establish the following commonly believed, but never previously proven, result: whenever M ≠ {0}, the sum of the commutators [Z_M,1*, ZM,1] + ⋯ + [Z_M,n*, ZM,n] does not belong to the Schatten class ℂⁿ.