Similarity Invariants of Essentially normal Cowen-Douglas Operators and Chern Polynomials

TL;DR

This paper explores the geometric properties of essentially normal Cowen-Douglas operators, establishing that Chern polynomials and second fundamental forms serve as their similarity invariants, extending the Brown-Douglas-Fillmore theory.

Contribution

It introduces a new geometric approach to classify essentially normal Cowen-Douglas operators using Chern polynomials and second fundamental forms as invariants.

Findings

Chern polynomials are similarity invariants for the class of operators studied.

The second fundamental forms also serve as invariants in this context.

A version of the Brown-Douglas-Fillmore theorem is proved within Cowen-Douglas theory.

Abstract

In this paper, we systematically study a class of essentially normal operators by using the geometry method from the Cowen-Douglas theory and prove a Brown-Douglas-Fillmore theorem in the Cowen-Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.