Beurling quotient subspaces for covariant representations of product systems

TL;DR

This paper characterizes Beurling quotient subspaces for certain covariant representations of product systems, providing dilation theorems and conditions for unitary equivalence, and exploring factorization and invariant subspaces.

Contribution

It introduces a concrete characterization of Beurling quotient subspaces for pure doubly commuting isometric representations of product systems, advancing dilation theory.

Findings

Derived a regular dilation theorem for covariant representations satisfying Brehmer-Solel condition.

Provided necessary and sufficient conditions for unitary equivalence to induced representations.

Explored the relationship between Sz.Nagy-Foias factorization and joint invariant subspaces.

Abstract

We characterize Beurling quotient subspaces for pure doubly commuting isometric representations of product systems. As a consequence, we derive a concrete regular dilation theorem for a pure completely contractive covariant representation which satisfies Brehmer-Solel condition and using it and the above characterization, we provide a necessary and sufficient condition that when a completely contractive covariant representation is unitarily equivalent to the compression of the induced representation on the Beurling quotient subspace. Further, we study the relation between Sz.Nagy-Foias type factorization of isometric multi-analytic operators and joint invariant subspaces.