  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.
