Classify n-tuples of commuting isometries with pairwise compact (or compact+normal) cross-commutators

Classify, up to unitary equivalence, all n-tuples (V1, ..., Vn) of commuting isometries acting on complex separable Hilbert spaces such that for every i ≠ j either [Vi,Vj] is a compact operator or [Vi*,Vj] is compact + normal. Provide structural models and complete unitary invariants for these higher-dimensional analogues.

Background

The authors note that while their paper addresses the two-variable case with compact+normal cross-commutators, extending such classifications to n>2 variables is substantially harder; operator and function theory often diverge as the dimension increases. They highlight connections to spherical isometries and essential normality problems on function Hilbert spaces over the unit ball, underscoring the broader context and difficulty of the multi-variable setting.

This question serves as the multi-variable analogue of the preceding two-variable classification problem and seeks representations and invariants in higher dimensions.