article

Title: Defect operators associated with submodules of the Hardy module

Authors: Quanlei Fang and Jingbo Xia

Issue: Volume 60 (2011), Issue 3, 729-750

Abstract:

$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 coordinated 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 $\mathcal{M}$. In this paper we establish the following commonly believed, but never previously proven, result: whenever $\mathcal{M} \neq \{0\}$, the sum of the commutators \[ [Z_{\mathcal{M},1}^{*},Z_{\mathcal{M},1}] + \dots + [Z_{\mathcal{M},n}^{*},Z_{\mathcal{M},n}] \] does not belong to the Schatten class $\mathcal{C}_n$.$