Bergman-type reproducing kernels, contractive divisors, and dilations
Define a \(B\)-kernel \(k\) on \(\Omega\) as a special nonzero positive kernel being sesquianalytic on \(\Omega\times \Omega\) (where \(\Omega\) is a region in the complex plane), and let \({\mathcal H}(k)\) denote the unique Hilbert space of analytic functions on \(\Omega\), \(k\) as reproducing kernel. If \(\Omega\) is the unit disk and some further conditions on \(k\) are imposed, these kernels are called Berman-type reproducing kernels, and in this case the corresponding spaces \({\mathcal H}(k)\) lie between the Hardy space \(H^2\) and the classical Bergman space \(L_\alpha^2\), and the inclusion maps are contractive. The present paper is devoted to three types of reproducing kernels, namely, the \(B\)-kernels, (suitably defined) regular \(B\)-kernels, and, especially, Bergman-type kernels (and their specializations). The topics here considered concern: characterizations and general properties of those kernels; multiplier invariant subspaces of \({\mathcal H}(k)\) (with special stress on the case of index 1), factorization theorems for the reproducing kernels of such subspaces; strong contractive divisor, wandering subspace, dilation, and inner-outer factorization theorems for Bergman-type kernels.
scientific article; zbMATH DE number 1764703
scientific article