  In this note, we show that a quasi-free Hilbert module R defined over the
polydisk algebra with kernel function k(z, w) admits a unique minimal dilation
(actually an isometric co-extension) to the Hardy module over the polydisk if
and only if S^{-1}(z, w) k(z, w) is a positive kernel function, where S(z, w)
is the Szeg\"{o} kernel for the polydisk. Moreover, we establish the
equivalence of such a factorization of the kernel function and a positivity
condition, defined using the hereditary functional calculus, which was
introduced earlier by Athavale \cite{Ath} and Ambrozie, Englis and M\"{u}ller.
An explicit realization of the dilation space is given along with the isometric
embedding of the module R in it. The proof works for a wider class of Hilbert
modules in which the Hardy module is replaced by more general quasi-free
Hilbert modules such as the classical spaces on the polydisk or the unit ball
in {C}^m. Some consequences of this more general result are then explored in
the case of several natural function algebras.
