Let Ω ⊂ C m be an open, connected and bounded set and A(Ω) be a function algebra of holomorphic functions on Ω. In this article, we study quotient Hilbert modules obtained from submodules, consisting of functions in M vanishing to order k along a smooth irreducible complex analytic set Z ⊂ Ω of codimension at least 2, of a quasi-free Hilbert module, M . Our motive is to investigate unitary invariants of such quotient modules. We completely determine unitary equivalence of aforementioned quotient modules and relate it to geometric invariants of a Hermitian holomorphic vector bundles. Then, as an application, we characterize unitary equivalence classes of weighted Bergman modules over A(D m ) in terms of those of quotient modules arising from the submodules of functions vanishing to order 2 along the diagonal in D m .
We construct a large family of positive definite kernels K : D n × D n → M(r, C), holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb × • • •×Möb (n times) of the bi-holomorphic automorphism group of D n . The adjoint of the n-tuple of the multiplication operators by the co-ordinate functions is then homogeneous with respect to this subgroup on the Hilbert space HK determined by K. We show that these n-tuples are irreducible, are in the Cowen-Douglas class Br(D n ) and are mutually pairwise unitarily inequivalent.
We construct a large family of positive-definite kernels K : D n × D n → M(r, C), holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup Möb × • • • × Möb (n times) of the bi-holomorphic automorphism group of D n . The adjoint of the n -tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces HK determined by K is then homogeneous with respect to this subgroup. We show that these n -tuples are irreducible, are in the Cowen-Douglas class Br(D n ) and that they are mutually pairwise unitarily inequivalent.
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