Geometry of holomorphic vector bundles and similarity of commuting operator tuples

TL;DR

This paper introduces a geometric criterion for the similarity of commuting operator tuples, providing new invariants and subclasses within the Cowen-Douglas class, advancing understanding in operator theory.

Contribution

It develops a novel geometric similarity invariant for operator tuples and identifies a new subclass within the Cowen-Douglas class, addressing open questions in the field.

Findings

Established a new similarity criterion for commuting operator tuples.

Derived a geometric invariant for tuples in the Cowen-Douglas class.

Identified a new subclass of commuting tuples within the Cowen-Douglas class.

Abstract

In this paper, a new criterion for the similarity of commuting tuples of operators on Hilbert spaces is introduced. As an application, we obtain a geometric similarity invariant of tuples in the Cowen-Douglas class which gives a partial answer to a question raised by R.G. Douglas about the similarity of quasi-free Hilbert modules. Moreover, a new subclass of commuting tuples of Cowen-Douglas class is obtained.