$\clw$-hypercontractions and their model

Abstract

We revisit the study of $\omega$-hypercontractions corresponding to a single weight sequence $\omega=\{\omega_k\}_{k\geq0}$ introduced by Olofsson in \cite{O} and find an analogue of Nagy-Foias characteristic function in this setting. Explicit construction of characteristic functions is obtained and it is shown to be a complete unitary invariant. By considering a multi-weight sequence $\clw$ and $\clw$-hypercontractions we extend Olofsson's work \cite{O} in the multi-variable setting. Model for $\clw$-hypercontractions is obtained by finding their dilations on certain weighted Bergman spaces over the polydisc corresponding to the multi-weight sequence $\clw$. This recovers and provides a different proof of the earlier work of Curto and Vasilescu \cite{CVPoly, CV} for $\gamma$-contractive multi-operators through a particular choice of multi-weight sequence.