  For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space
$\mathcal H$ with $T=\prod_{i=1}^n T_i$, we show that $(T_1, \dots, T_n)$
dilates to commuting isometries $(V_1, \dots , V_n)$ on the minimal isometric
dilation space of $T$ with $V=\prod_{i=1}^n V_i$ being the minimal isometric
dilation of $T$ if and only if $(T_1^*, \dots , T_n^*)$ dilates to commuting
isometries $(Y_1, \dots , Y_n)$ on the minimal isometric dilation space of
$T^*$ with $Y=\prod_{i=1}^n Y_i$ being the minimal isometric dilation of $T^*$.
Then, we prove an analogue of this result for unitary dilations of $(T_1, \dots
, T_n)$ and its adjoint. We find a necessary and sufficient condition such that
$(T_1, \dots , T_n)$ possesses a unitary dilation $(W_1, \dots , W_n)$ on the
minimal unitary dilation space of $T$ with $W=\prod_{i=1}^n W_i$ being the
minimal unitary dilation of $T$. We show an explicit construction of such a
unitary dilation on both Sch$\ddot{a}$ffer and Sz. Nagy-Foias minimal unitary
dilation spaces of $T$. Also, we show that a relatively weaker hypothesis is
necessary and sufficient for the existence of such a unitary dilation when $T$
is a $C._0$ contraction, i.e. when ${T^*}^n \rightarrow 0$ strongly as $n
\rightarrow \infty $. We construct a different unitary dilation for $(T_1,
\dots , T_n)$ when $T$ is a $C._0$ contraction.
