# Generating wandering subspaces for doubly commuting covariant representations

## TL;DR

This paper extends the theory of wandering subspaces to covariant representations of C*-correspondences, introducing new decompositions and characterizations for doubly commuting cases.

## Contribution

It develops a Halmos-Richter-type wandering subspace theorem and explores Cauchy duals and Shimorin's Wold-type decomposition for these representations.

## Key findings

- Established a wandering subspace theorem for doubly commuting covariant representations.
- Characterized generating wandering subspaces for product system representations.
- Connected doubly commutativity with the structure of wandering subspaces.

## Abstract

We obtain a Halmos-Richter-type wandering subspace theorem for covariant representations of C*-correspondences. Further the notion of Cauchy dual and a version of Shimorin's Wold-type decomposition for covariant representations of C*-correspondences is explored and as an application a wandering subspace theorem for doubly commuting covariant representations is derived. Using this wandering subspace theorem generating wandering subspaces are characterized for covariant representations of product systems in terms of the doubly commutativity condition.