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The goal of the present paper is to introduce and study noncommutative Hardyspaces associated with the regular $\Lambda$-polyball, to develop a functionalcalculus on noncommutative Hardy spaces for the completely non-coisometric(c.n.c.) $k$-tuples in ${\bf B}_\Lambda(H)$, and to study the characteristicfunctions and the associated multi-analytic models for the c.n.c. elements inthe regular $\Lambda$-polyball. In addition, we show that the characteristicfunction 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 resultsof Sz.-Nagy--Foia\c s for contractions and the noncommutative versions for rowcontractions. In the particular case when $n_1=\cdots=n_k=1$ and$\Lambda_{ij}=1$, we obtain a functional calculus and operator model theory interms of characteristic functions for $k$-tuples of contractions satisfyingBrehmer condition.

Functional calculus and multi-analytic models on regular $\Lambda$-polyballs