Bergman-type reproducing kernels, contractive divisors, and dilations

Let 92 be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Omega that we call B-kernels. When Omega 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 L; of the unit disk. For example, the weighted Bergman kernels k(w)(beta)(z) = (1-(w) over barz)(-beta), 1less than or equal to beta less than or equal to2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H-2 g YE(k) g La, where the inclusion maps are contractive, and M-zeta, the operator of multiplication with the identity function zeta, 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 k(beta) this result even holds for all M-zeta-invariant subspace M of index 1, i.e., whenever the dimension of M/zetaM is one. Second, if M is any multiplier invariant subspace of H(k), and if we set C-* = M circle minus zM, then Mzeta\M is unitarily equivalent to M-zeta acting on a space of C-*-valued analytic functions with an operator-valued reproducing kernel of the type I-w(z) = (I-c* - z (w) over barV(z) V(w)*) k(w)(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(k(beta)), 1less than or equal to beta less than or equal to 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. (C) 2002 Elsevier Science (USA).