An explicit construction characterizing the operator-valued Bergman inner functions is given for a class of vector-valued standard weighted Bergman spaces in the unit disk. These operator-valued Bergman inner functions act as contractive multipliers from the Hardy space into the associated Bergman space, and they have a natural interpretation as transfer functions for a related class of discret...

The reproducing kernel function of a weighted Bergman space over domains in C is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kern...

We prove two sharp estimates for the subspace of a standard weighted Bergman space that consists functions vanishing at given point (with prescribed multiplicity).

For α > −1, let Aα denote the corresponding weighted Bergman space of the unit ball. For any self-adjoint subset G ⊂ L∞, let T(G) denote the C∗−subalgebra of B(Aα) generated by {Tf : f ∈ G}. Let CT(G) denote the commutator ideal of T(G). It was showed by D. Suárez (in 2004 for n = 1) and by the author (in 2006 for all n ≥ 1) that CT(L∞) = T(L∞) in the case α = 0. In this paper we show that in t...

We classify self-adjoint first-order differential operators on weighted Bergman spaces the unit disc and answer questions related to uncertainty principles for such operators. Our main tools are discrete series representations of $$\textrm{SU}(1,1)$$ . This approach has promise generalize other bounded symmetric domains.

We describe a notion of ampleness for line bundles on orbifolds with cyclic quotient singularities that is related to embeddings in weighted projective space, and prove a global asymptotic expansion for a weighted Bergman kernel associated to such a line bundle.

Let D be the open unit disk in the complex plane. For ε > 0 we consider the sector Σε = {z ∈ C : | arg z| < ε}. We prove that for every α ≥ 0 and for each ε > 0 there is a constant K > 0 depending only on α and ε such that for any function f in the weighted Bergman space Aα univalent on D, and f(0) = 0, then ∫ f−1(Σε) |f(z)|dAα(z) > K‖f‖1,α. This result extends a theorem of Marshall and Smith i...

In this article, we mainly study the weighted composition-differentiation operator on Bergman space A?2 and derivative Hardy S12, which characterize complex symmetric Du,?. Moreover, normality self-adjointness of Du,? are also discussed.