# Decomposition of the tensor product of two Hilbert modules

## TL;DR

This paper studies the properties of a sesqui-analytic function derived from a kernel on a domain, establishing conditions for non-negativity and providing a decomposition of tensor products of Hilbert modules over polynomial rings.

## Contribution

It introduces a new sesqui-analytic function related to kernels, proves its non-negativity under certain conditions, and decomposes tensor products of Hilbert modules over polynomial rings.

## Key findings

- $oldsymbol{ ext{$oldsymbol{ extbf{K}^{(oldsymbol{eta})}$ is non-negative definite when $K^eta$ is non-negative definite}}$
- $oldsymbol{ ext{A realization of the Hilbert module from $oldsymbol{ extbf{K}^{(oldsymbol{eta})}}$ is constructed}}$
- $oldsymbol{ ext{Initial pieces of the tensor product decomposition are identified}}$

## 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 $\mathbb K^{(\alpha, \beta)}:= K^{\alpha+\beta}\big(\partial_i\bar{\partial}_j\log K\big )_{i,j=1}^ m,$ 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 $\mathbb C[z_1, \ldots, z_m]$. Then $\mathbb C[z_1, \ldots, z_{2m}]$ acts naturally on the tensor product $\mathcal M_1\otimes \mathcal M_2$. The restriction of this action to the polynomial ring $\mathbb C[z_1, \ldots, z_m]$ 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.