Similarity of Quotient Hilbert modules in the Cowen-Douglas Class

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a “simple” rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and sufficient conditions for a class of Cowen-Douglas Hilbert modules satisfying some positivity conditions to be similar to $\mathcal{M}\otimes {\mathbb{C}}^m$. We also show that under certain uniform bound condition on the anti-holomorphic frame, a Cowen-Douglas Hilbert module is quasi-affinity to a submodule of the free module $$\mathcal{M}$ \otimes $\mathbb{C}$^m$.