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

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 ${\mathbb{N}}_0^2$.