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In their pioneering work, Cowen and Douglas establish a natural correspondence between analytic Hilbert modules and holomorphic Hermitian
vector bundles. They show this correspondence descends to the respective equivalence classes and what is more, they write down a complete set of invariants for these equivalence classes. However, there are natural
examples of submodules of analytic Hilbert modules which don't necessarily
come from any holomorphic Hermitian vector bundle. For such class of
submodules of analytic Hilbert modules, we construct a sheaf model and show
the associated sheaf of modules is coherent analytic. We use this model to
to give a decomposition of the reproducing kernel and extract several
unitary invariants for the submodules in our class. In the end, we discuss
a number of questions which we intend to explore in the future. |