Curvature invariant and generalized canonical Operator models - I

TL;DR

This paper extends the canonical model framework for contraction operators to include Bergman spaces, classifies these operators in the Cowen-Douglas class using complex geometric invariants, and provides a complete unitary classification.

Contribution

It introduces a generalized model incorporating Bergman spaces and achieves a full classification of certain operators via their geometric invariants.

Findings

Operators classified by associated vector bundles and curvatures

Complete unitary classification achieved for Cowen-Douglas class operators

Framework generalizes classical Hardy space models

Abstract

One can view contraction operators given by a canonical model of Sz.-Nagy and Foias as being defined by a quotient module where the basic building blocks are Hardy spaces. In this note we generalize this framework to allow the Bergman and weighted Bergman spaces as building blocks, but restricting attention to the case in which the operator obtained is in the Cowen-Douglas class and requiring the multiplicity to be one. We view the classification of such operators in the context of complex geometry and obtain a complete classification up to unitary equivalence of them in terms of their associated vector bundles and their curvatures.