The Cowen-Douglas Theory for Operator Tuples and Similarity

TL;DR

This paper investigates the similarity problem for Cowen-Douglas operator tuples, establishing new results that connect complex geometry and operator theory, extending single-operator findings to multivariable cases without relying on corona theorems.

Contribution

It proves that similarity results for single Cowen-Douglas operators extend to commuting tuples without using corona theorems, advancing understanding in multivariable operator theory.

Findings

Similarity results for single operators extend to tuples.

Established connections between complex geometry and operator similarity.

Proved similarity results without relying on corona theorems in multivariable setting.

Abstract

We are concerned with the similarity problem for Cowen-Douglas operator tuples. The unitary equivalence counterpart was already investigated in the 1970's and geometric concepts including vector bundles and curvature appeared in the description. As the Cowen-Douglas conjecture show, the study of the similarity problem has not been so successful until quite recently. The latest results reveal the close correlation between complex geometry, the corona problem, and the similarity problem for single Cowen-Douglas operators. Without making use of the corona theorems that no longer hold in the multi-variable setting, we prove that the single operator results for similarity remain true for commuting Cowen-Douglas operator tuples as well.