Similarity of Quotient Hilbert modules in the Cowen-Douglas Class

TL;DR

This paper investigates the conditions under which certain Cowen-Douglas Hilbert modules are similar or quasi-affine to simpler modules, using complex geometric invariants like curvature and vector bundles.

Contribution

It provides necessary and sufficient conditions for similarity and quasi-affinity of Cowen-Douglas modules based on geometric and positivity criteria.

Findings

Characterization of similarity to tensor products of modules

Conditions for quasi-affinity to submodules of free modules

Use of curvature and vector bundle invariants in module analysis

Abstract

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and sufficient conditions for a class of Cowen-Douglas Hilbert modules satisfying some positivity conditions to be similar to $\mathcal{M} \otimes \mathbb{C}^m$. We also show that under certain uniform bound condition on the anti-holomorphic frame, a Cowen-Douglas Hilbert module is quasi-affinity to a submodule of the free module $\mathcal{M} \otimes \mathbb{C}^m$.