Hankel forms and the Fock space
type: Item
instance of: scholarly article
title: Hankel forms and the Fock space
zbMATH Open document ID: 0704.47022
DOI: 10.4171/RMI/46
author: Svante Janson
author: Jaak Peetre
author: Richard Rochberg
published in: Revista Matemática Iberoamericana
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/39342
review text: The paper develops in a systematic way a theory of Hankel forms over quite general domains in \({\mathbb{C}}^ n\). Namely, a domain \(\Omega \subset {\mathbb{C}}^ n\) and an absolutely continuous measure \(\mu\) on \(\Omega\) with continuous, strictly positive Radon-Nikodym derivative \(d\mu\) /dm (m is the Lebesgue measure) ar given. The basic Hilbert space \(A^ 2(\mu)={\mathcal H}ol(\Omega)\cap L^ 2(\mu)\) of square integrable holomorphic functions has the reproducing kernel K(z,w). The associated measure \(\nu\) is defined as \(d\nu (z)=d\mu /K(z,z).\) We put L for the reproducing kernel in the Hilbert space \(A^ 2(\nu)\) and Q for the orthogonal projection Q: \(L^ 2(\nu)\to A^ 2(\nu)\). The whole theory of Hankel forms on \(A^ 2(\mu)\) is developed under the following basic assumptions: V1. \(K(z,z)>0\), \(\forall z\in \Omega,\) V2. \(L(z,w)=\kappa (K(z,w))^ 2\) for some constant \(\kappa\geq 0,\) V3. If \(f\in {\mathcal H}ol(\Omega)\) and \(\int | L(z,w)f(w)| d\nu (w)<\infty\) for every z, then \[ \int L(z,w)f(w)d\nu (w)=f(z),\quad z\in \Omega. \] For a function b on \(\Omega\) the Hankel forms \(H_ b\) and \(\Gamma_ b\) on \(A^ 2(\mu)\) (resp. on \(A^ 2(\nu))\) are given by \[ H_ b(f,g)=\int_{\Omega}\bar bfg d\mu. \] Before stating the main result of the first part of the paper ({\S}{\S} 1-6) recall one more concept: the natural scale of weighted \(A^ p\)-spaces. Set \(\omega (z)=K(z,z)^{-1}\). An analytic function f is in \(A^ p_{\omega}\) iff \[ \int_{\Omega}| f|^ p \omega^{p-2}d\nu <\infty \] (f is in \(A^{\infty}_{\omega}\) iff \(\omega\) f is essentially bounded on \(\Omega\)). Let \(\alpha^{\infty}_{\omega}\) be the closure of \(A^ 2_{\omega}\) in the \(A^{\infty}_{\omega}\)-metric. The main result reads as follows. Theorem A. Under the assumptions V1-V3 we have (a) \(\Gamma_ b\) is bounded (in \(A^ 2(\mu))\) if and only if \(Qb\in A^{\infty}_{\omega},\) (b) \(\Gamma_ b\) is compact if and only if \(Qb\in a^{\infty}_{\omega},\) (c) \(\Gamma_ b\) is in \(…
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 4155253
zbMATH Keywords: Hankel forms
zbMATH Keywords: reproducing kernel
zbMATH Keywords: scale of weighted \(A^ p\)-spaces
zbMATH Keywords: Fock spaces
zbMATH Keywords: gauge transformations
reviewed by: Jan Janas
MaRDI profile type: Publication
OpenAlex ID: W2004171696
description: scientific article; zbMATH DE number 4155253