  The goal of the present paper is to introduce and study noncommutative Hardy
spaces associated with the regular $\Lambda$-polyball, to develop a functional
calculus on noncommutative Hardy spaces for the completely non-coisometric
(c.n.c.) $k$-tuples in ${\bf B}_\Lambda(H)$, and to study the characteristic
functions and the associated multi-analytic models for the c.n.c. elements in
the regular $\Lambda$-polyball. In addition, we show that the characteristic
function is a complete unitary invariant for the class of c.n.c. $k$-tuples in
${\bf B}_\Lambda(H)$. These results extend the corresponding classical results
of Sz.-Nagy--Foia\c s for contractions and the noncommutative versions for row
contractions. In the particular case when $n_1=\cdots=n_k=1$ and
$\Lambda_{ij}=1$, we obtain a functional calculus and operator model theory in
terms of characteristic functions for $k$-tuples of contractions satisfying
Brehmer condition.
