# Geometric Similarity invariants of Cowen-Douglas Operators

## TL;DR

This paper explores how geometric invariants like curvature and the second fundamental form can fully characterize the similarity invariants of Cowen-Douglas operators, advancing the understanding of their geometric classification.

## Contribution

It demonstrates that curvature and the second fundamental form completely determine similarity invariants for a dense class of Cowen-Douglas operators, partially answering a longstanding question.

## Key findings

- Curvature and second fundamental form characterize similarity invariants.
- Results apply to a norm dense class of Cowen-Douglas operators.
- Advances geometric understanding of operator similarity.

## Abstract

In 1978, M. J. Cowen and R.G. Douglas introduce a class of operators (known as Cowen-Douglas class of operators) and associates a Hermitian holomorphic vector bundle to such an operator in a very influential paper. They give a complete set of unitary invariants in terms of involving the curvature and its covariant partial derivatives. At the same time they ask: can one use geometric ideas to characterize completely the similarity invariants of Cowen-Douglas operators? We give a partial answer to this question. In this paper, we show that the curvature and the second fundamental form completely characterize the similarity invariants for a norm dense class of Cowen-Douglas operators.