Multiplicity-free homogeneous operators in the Cowen-Douglas class

TL;DR

This paper classifies all multiplicity-free homogeneous operators in the Cowen-Douglas class on the unit disk, providing an alternative construction and confirming their equivalence to previously known examples.

Contribution

It offers a new independent construction of all such operators and verifies their correspondence with earlier identified examples.

Findings

All multiplicity-free homogeneous Cowen-Douglas operators are characterized.

The new construction matches previously known examples.

The classification confirms the uniqueness of these operators.

Abstract

In a recent paper, the authors have constructed a large class of operators in the Cowen-Douglas class Cowen-Douglas class of the unit disc $\mathbb D$ which are {\em homogeneous} with respect to the action of the group M\"{o}b -- the M\"{o}bius group consisting of bi-holomorphic automorphisms of the unit disc $\mathbb{D}$. The {\em associated representation} for each of these operators is {\em multiplicity free}. Here we give a different independent construction of all homogeneous operators in the Cowen-Douglas class with multiplicity free associated representation and verify that they are exactly the examples constructed previously.