Rigidity of the flag structure for a class of Cowen-Douglas operators

TL;DR

This paper studies a special class of Cowen-Douglas operators with a flag structure, proving their irreducibility and rigidity, and providing a complete set of more manageable unitary invariants.

Contribution

It introduces a class of Cowen-Douglas operators with a flag structure, proves their rigidity, and derives a complete set of tractable unitary invariants.

Findings

Operators are irreducible.

Flag structure is rigid and determines the operator.

Complete set of unitary invariants obtained.

Abstract

The explicit description of irreducible homogeneous operators in the Cowen-Douglas class and the localization of Hilbert modules naturally leads to the definition of a smaller class of Cowen-Douglas operators possessing a flag structure. These operators are shown to be irreducible. It is also shown that the flag structure is rigid, that is, the unitary equivalence class of the operator and the flag structure determine each other. A complete set of unitary invariants, which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class, are obtained.