Classify commuting isometric pairs with compact cross-commutator

Classify, up to unitary equivalence, all pairs of commuting isometries (V1,V2) acting on complex separable Hilbert spaces such that the cross-commutator [V2,V1] is a compact operator. Develop concrete representations and complete unitary invariants for this class, extending beyond the compact+normal case analyzed in the paper.

Background

The paper provides a complete representation and classification for compact normal pairs—commuting isometric pairs (V1,V2) whose cross-commutator [V2*,V1] is compact + normal—via building blocks (irreducible 1-, 2-, and 3-finite pairs and shift-unitary components) and explicit unitary invariants. A natural next step proposed by the authors is to remove the normality assumption and consider pairs with merely compact cross-commutator.

They point out that approaches based solely on defect operators may be insufficient, giving examples on H^2(D) where pairs share the same defect operator but are not unitarily equivalent, indicating the need for new techniques to address the purely compact case.