Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Abstract

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk D and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space H(k) shares certain properties with the classical Bergman space L2α of the unit disk. For example, the weighted Bergman kernels kβw(z)=(1−wz)−β, 1⩽β⩽2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H2⊆H(k)⊆L2a, where the inclusion maps are contractive, and Mζ, the operator of multiplication with the identity function ζ, defines a contraction operator on H(k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace M of H(k) is a Bergman-type kernel as well. For the weighted Bergman kernels kβ this result even holds for all Mζ-invariant subspace M of index 1, i.e., whenever the dimension of M/ζM is one. Second, if M is any multiplier invariant subspace of H(k), and if we set C∗=M⊖zM, then Mζ∣M is unitarily equivalent to Mζ acting on a space of C∗-valued analytic functions with an operator-valued reproducing kernel of the typekw(z)=(IC∗−zwV(z)V(w)*)kw(z), where V is a contractive analytic function V:D→L(E, C∗), for some auxiliary Hilbert space E. Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of H(kβ), 1⩽β⩽2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.