$\clw$-hypercontractions and their model

TL;DR

This paper studies $ ext{w}$-hypercontractions, extending classical results to multi-variable settings, and constructs explicit characteristic functions that serve as complete invariants for these operators.

Contribution

It introduces a new model for $ ext{w}$-hypercontractions using dilations on weighted Bergman spaces, generalizing previous work to multi-variable cases.

Findings

Explicit characteristic functions are constructed and shown to be complete invariants.

Extension of Olofsson's work to multi-weight sequences and multi-variable operators.

Provides new proofs for earlier results on $ ext{γ}$-contractive multi-operators.

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.