On the metric of the jet bundle and similarity of Cowen-Douglas operators

Abstract

The study of Cowen-Douglas operators not only involves traditional operator-theoretic tools but also concepts and results from complex geometry on holomorphic vector bundles. We make use of the ratio of the metric matrices first considered by Clark and Misra and a model theorem by Agler to describe the similarity of backward shift operators on analytic function spaces whose multiplier algebras are the space of bounded analytic functions. It is well-known that, in general, it becomes much more complicated to formulate a sufficient condition for similarity than a necessary one. We also give a sufficient condition for a Cowen-Douglas operator to be similar to the backward shift operator on the Dirichlet space with weights by introducing a condition on the jet bundle of a holomorphic vector bundle. Note that the multiplier algebras of these spaces do not coincide with the space of bounded, analytic functions as in other analytic functions spaces, requiring a different approach. The results by M\"{u}ller on operator models related to Dirichlet shifts and by Kidane and Trent on the corona problem for the multiplier algebras of weighted Dirichlet spaces are indispensable tools in attaining this similarity result.