The core operator and congruent submodules

The Hardy spaces H² (D²) can be conveniently viewed as a module over the polynomial ring C [z₁, z₂]. Submodules of H² (D²) have connections with many areas of study in operator theory. A large amount of research has been carried out striving to understand the structure of submodules under certain equivalence relations. Unitary equivalence is a well-known equivalence relation in set of submodules. However, the rigidity phenomenon discovered in [Douglas et al., Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math. 117 (1) (1995) 75-92] and some other related papers suggests that unitary equivalence, being extremely sensitive to perturbations of zero sets, lacks the flexibility one might need for a classification of submodules. In this paper, we suggest an alternative equivalence relation, namely congruence. The idea is motivated by a symmetry and stability property that the core operator possesses. The congruence relation effectively classifies the submodules with a finite rank core operator. Near the end of the paper, we point out an essential connection of the core operator with operator model theory.