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We consider 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 \(\mathscr {M}\), we find necessary and sufficient conditions for a class of Cowen–Douglas Hilbert modules satisfying some positivity conditions to be similar to  Open image in new window  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  Open image in new window

Similarity of quotient Hilbert modules in the Cowen–Douglas class