Bergman-type reproducing kernels, contractive divisors, and dilations
type: Item
instance of: scholarly article
title: Bergman-type reproducing kernels, contractive divisors, and dilations
zbMATH Open document ID: 1038.46021
published in: Journal of Functional Analysis
publication date: 2002-07-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47B32
Mathematics Subject Classification ID: 32A36
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 1764703
zbMATH Keywords: sesquianalytic reproducing kernel
zbMATH Keywords: Bergman-type kernel
zbMATH Keywords: Hilbert space
zbMATH Keywords: multiplier invariant subspace
zbMATH Keywords: factorization theorems
author: Scott McCullough
author: Stefan Richter
reviewed by: M. Kracht
MaRDI profile type: Publication
full work available at URL: jfan.2001.3874
OpenAlex ID: W2069102260
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DOI: 10.1006/JFAN.2001.3874
description: scientific article; zbMATH DE number 1764703