Unitary invariants for Hilbert modules of finite rank

TL;DR

This paper introduces a refined curvature concept for classifying certain analytic Hilbert modules of finite rank, connecting operator theory with complex geometry through invariants and localization techniques.

Contribution

It develops a new notion of curvature for Hilbert modules with finite-dimensional localizations, extending the classification framework beyond constant rank cases.

Findings

Unitary classification of a broad class of Hilbert modules achieved.

Explicit computations inspired by linear group representation theory.

Connection established between Hilbert modules and holomorphic Hermitian bundles.

Abstract

A refined notion of curvature for a linear system of Hermitian vector spaces, in the sense of Grothendieck, leads to the unitary classification of a large class of analytic Hilbert modules. Specifically, we study Hilbert sub-modules, for which the localizations are of finite (but not constant) dimension, of an analytic function space with a reproducing kernel. The correspondence between analytic Hilbert modules of constant rank and holomorphic Hermitian bundles on domains of $\mathbb C^n$ due to Cowen and Douglas, as well as a natural analytic localization technique derived from the Hochschild cohomology of topological algebras play a major role in the proofs. A series of concrete computations, inspired by representation theory of linear groups, illustrate the abstract concepts of the paper.