Essential Normality of Homogeneous Submodules

Let M ì H(mathbbB){M subset H(mathbb{B})} be a homogeneous submodule of the n-shift Hilbert module on the unit ball in mathbbCⁿ{mathbb{C}^{n}}. We show that a modification of an operator inequality used by Guo and Wang in the case of principal submodules is equivalent to the existence of factorizations of the form [M_zj^*,P_M] = (N+1)^-1A_j{[M_{z_j}^*,P_M] = (N+1)^{-1}A_j}, where N is the number operator on H(mathbbB){H(mathbb{B})}. Thus a proof of the inequality would yield positive answers to conjectures of Arveson and Douglas concerning the essential normality of homogeneous submodules of H(mathbbB){H(mathbb{B})}. We show that in all cases in which the conjectures have been established the inequality holds and leads to a unified proof of stronger results.