A subclass of the Cowen-Douglas class and similarity

We consider a subclass of the Cowen-Douglas class in which the problem of deciding whether two operators are similar becomes more manageable. A similarity criterion for Cowen-Douglas operators is known to be dependent on the trace of the curvatures of the corresponding eigenvector bundles. Unless the given eignvector bundle is a line bundle, the computation of the curvatures, in general, is not so simple as one might hope. By using a structure theorem given in \cite{JW}, we reduce the problem of finding the trace of the curvatures to looking at the curvatures of the associated line bundles. Moreover, several questions related to the similarity problem are also taken into account.