A classification of homogeneous operators in the Cowen-Douglas class

TL;DR

This paper classifies all homogeneous operators in the Cowen-Douglas class by explicitly constructing associated vector bundles, identifying irreducible components, and analyzing their similarity classes.

Contribution

It provides a complete classification of homogeneous operators in the Cowen-Douglas class through explicit constructions and analysis of their vector bundle structures.

Findings

Explicit construction of all homogeneous holomorphic Hermitian vector bundles over the unit disc.

Identification of irreducible components within these bundles.

Complete classification of homogeneous operators in the Cowen-Douglas class and their similarity classes.

Abstract

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc $\mathbb D$ is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over $\mathbb D$, the ones corresponding to operators in the Cowen-Douglas class ${\mathrm B}_n(\mathbb D)$ are identified. The classification of homogeneous operators in ${\mathrm B}_n(\mathbb D)$ is completed using an explicit realization of these operators. We also show how the homogeneous operators in ${\mathrm B}_n(\mathbb D)$ split into similarity classes.