  We investigate the interaction between the existence of reproducing kernels
on infinite-dimensional Hermitian vector bundles and the positivity properties
of the corresponding bundles. The positivity refers to the curvature form of
certain covariant derivatives associated to reproducing kernels on the vector
bundles under consideration. The values of the curvature form are Hilbert space
operators, and its positivity is thus understood in the usual sense from
operator theory.
