  One of the most important results in operator theory is And\^o's \cite{ando}
generalization of dilation theory for a single contraction to a pair of
commuting contractions acting on a Hilbert space. While there are two explicit
constructions (Sch\"affer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the
minimal isometric dilation of a single contraction, there was no such explicit
construction of an And\^o dilation for a commuting pair $(T_1,T_2)$ of
contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In
this paper, we give two new proofs of And\^o's dilation theorem by giving both
Sch\"affer-type and Douglas-type explicit constructions of an And\^o dilation
with function-theoretic interpretation, for the general case. The results, in
particular, give a complete description of all possible factorizations of a
given contraction $T$ into the product of two commuting contractions. Unlike
the one-variable case, two minimal And\^o dilations need not be unitarily
equivalent. However, we show that the compressions of the two And\^o dilations
constructed in this paper to the minimal dilation spaces of the contraction
$T_1T_2$, are unitarily equivalent.
  In the special case when the product $T=T_1T_2$ is pure, i.e., if $T^{* n}\to
0$ strongly, an And\^o dilation was constructed recently in \cite{D-S-S},
which, as this paper will show, is a corollary to the Douglas-type
construction.
  We define a notion of characteristic triple for a pair of commuting
contractions and a notion of coincidence for such triples. We prove that two
pairs of commuting contractions with their products being pure contractions are
unitarily equivalent if and only if their characteristic triples coincide. We
also characterize triples which qualify as the characteristic triple for some
pair $(T_1,T_2)$ of commuting contractions such that $T_1T_2$ is a pure
contraction.
