Decomposition of the tensor product of two Hilbert modules

Abstract

Given a pair of positive real numbers $\alpha ,\beta$ and a sesqui-analytic function $K$ on a bounded domain $\Omega \subset {\mathbb{C}}^m$, in this paper, we investigate the properties of the sesqui-analytic function ${\mathbb{K}}^{(\alpha ,\beta )}:=K^{\alpha +\beta }({\partial }_i{\bar{\partial }}_j\log K{)}_{i,j=1}^m,$ taking values in $m\times m$ matrices. One of the key findings is that ${\mathbb{K}}^{(\alpha ,\beta )}$ is non-negative definite whenever $K^{\alpha }$ and $K^{\beta }$ are non-negative definite. In this case, a realization of the Hilbert module determined by the kernel ${\mathbb{K}}^{(\alpha ,\beta )}$ is obtained. Let ${\mathcal{M}}_i$, $i=1,2,$ be two Hilbert modules over the polynomial ring $\mathbb{C}[z_1,\dots ,z_m]$. Then $\mathbb{C}[z_1,\dots ,z_{2m}]$ acts naturally on the tensor product ${\mathcal{M}}_1\otimes {\mathcal{M}}_2$. The restriction of this action to the polynomial ring $\mathbb{C}[z_1,\dots ,z_m]$ obtained using the restriction map $p\mapsto p_{|\Delta }$ leads to a natural decomposition of the tensor product ${\mathcal{M}}_1\otimes {\mathcal{M}}_2$, which is investigated. Two of the initial pieces in this decomposition are identified.