# Doubly commuting invariant subspaces for representations of product systems of $C^*$-correspondences

## TL;DR

This paper extends Wold-type decompositions to doubly commuting covariant representations of product systems of $C^*$-correspondences, providing new insights into invariant subspaces and their structure using Fock space methods.

## Contribution

It introduces a Shimorin-Wold-type decomposition for doubly commuting representations, extending previous results to a broader class of product systems.

## Key findings

- Extended Wold-type decomposition for doubly commuting covariant representations.
- Characterized wandering subspaces in the context of these representations.
- Established a Beurling-type theorem for invariant subspaces using Fock space techniques.

## Abstract

We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariant representation of a product system of $C^*$-correspondences. This extends a recent Wold-type decomposition by Jeu and Pinto for a $q$-doubly commuting isometries. Application to the wandering subspaces of doubly commuting induced representations is explored, and a version of Mandrekar's Beurling type theorem is obtained to study doubly commuting invariant subspaces using Fock space approach due to Popescu.