A characterization of invariant subspaces for isometric representations of product system over $\mathbb{N}_0^{k}$

TL;DR

This paper characterizes invariant subspaces and commutants of doubly commuting pure isometric representations of product systems over ^k, providing explicit invariants and classification results.

Contribution

It offers an explicit description of the commutant and a complete characterization of invariant subspaces for these representations, advancing understanding in operator algebra theory.

Findings

Explicit representation of the commutant for doubly commuting pure isometric representations.

Complete characterization of invariant subspaces with isomorphic invariants.

Classification of a large class of commuting isometric representations.

Abstract

Using the Wold-von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over $\mathbb{N}_0^{k}.$ As an application, we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of commuting isometric representations of the product system.