The core operator and congruent submodules

By Rongwei Yang. 2005

55 citations recorded by the source corpus.

Abstract The Hardy spaces H 2 ( D 2 ) can be conveniently viewed as a module over the polynomial ring C [ z 1 , z 2 ] . Submodules of H 2 ( D 2 ) 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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/2298df8d3d166f83de2795ebb23dab1e2c7b75f9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Kunyu Guo
2. Dechao Zheng
3. Ronald G. Douglas
4. Kai Wang
5. W. Arveson

In Applied Mathematics

1. C. Zhong
2. Shunhua Sun
3. Kunyu Guo
4. Dechao Zheng
5. Ronald G. Douglas
6. Kai Wang
7. W. Arveson

Citations in the corpus

• Multiplication operators on the Bergman space via the Hardy space of the bidisk (2009, background): “Yang [13], [37], [38] and [39]; the higher-order Hankel forms by S.”