Berger-Coburn-Lebow representation for pure isometric representations of product system over $\mathbb N^2_0$

TL;DR

This paper develops a Berger-Coburn-Lebow representation for pure isometric covariant representations of product systems over ^2, analyzing invariants, subspaces, and classification methods.

Contribution

It introduces a BCL-representation for these representations, compares it with other descriptions, and establishes classification and index concepts.

Findings

Established BCL-representation for pure isometric covariant representations.

Characterized invariant subspaces and connected defect and Fringe operators.

Proposed a congruence relation for classification of representations.

Abstract

We obtain Berger-Coburn-Lebow (BCL)-representation for pure isometric covariant representation of product system over $\mathbb{N}_0^2$. Then the corresponding complete set of (joint) unitary invariants is studied, and the BCL- representations are compared with other canonical multi-analytic descriptions of the pure isometric covariant representation. We characterize the invariant subspaces for the pure isometric covariant representation. Also, we study the connection between the joint defect operators and Fringe operators, and the Fredholm index is introduced in this case. Finally, we introduce the notion of congruence relation to classify the isometric covariant representations of the product system over $\mathbb{N}_0^2$.