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 $\mathcal{ M} \otimes \mathbb{ C}^m$.
Categories: math.FA math.DG math.OA