Analytic Hilbert spaces over the complex plane
type: Item
instance of: scholarly article
title: Analytic Hilbert spaces over the complex plane
zbMATH Open document ID: 1012.46031
author: Xiaoman Chen
author: Kunyu Guo
author: Shengzhao Hou
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-04T00:00:00Z
review text: Let \(X\) denote a Hilbert space of entire functions on the complex plane \(\mathbb{C}\). We say that \(X\) is an analytic Hilbert space in case (i) the polynomials are dense in \(X\) and (ii) the evaluation mapping \(E_\lambda (f)= f(\lambda)\) for all \(f\in X\) is continuous on \(X\) for each \(\lambda\in \mathbb{C}\). The point of departure for the present paper is the result that if \(M\neq \{0\}\) is a closed subspace of \(X\) and \(fM\subset M\) for some entire function \(f\), then \(f\) is a constant. Following the work of \textit{K. Guo} and \textit{D. Zheng} [J. Funct. Anal. 187, 308-342 (2001; Zbl 1045.47022)] for the Fock space \(L^2_a (\mathbb{C})\), the authors put \(X_0=\{f\in X:z f\in X\}\) and call any \(M\) such that \(X_0\cap M\) is dense in \(M\) a quasi-invariant subspace (QIS) in case \(zM\cap X\subset M\). For \(f\in X\) let \([f]\) be the intersection of all QIS's containing \(f\). The authors show that if the QIS \(M_1\) has finite codimension, then \(M_1=[p]\) for some polynomial \(p\). They define a QIS \(M_2\) to be similar to \(M_1\) in case some bounded invertible operator \(A\) maps \(M_1\) onto \(M_2\) in such a way that \(A(zf)= zA(f)\) and \(A^{-1} (zg)=zA^{-1} (g)\) for all \(f\in M_1\) and \(g\in M_2\). They show that if \(M_2\) is similar to \(M_1= [p]\), then \(M_2= [q]\) for some polynomial \(q\) with the same degree as \(p\). These results are extended to the case where \(A\) and \(A^{-1}\) are unitary. A natural order between entire functions \(f\) and \(g\) is defined, and \(X\) is shown to be ordered if and only if for any polynomial \(p\), \(pf\in X\) implies \(f\in X\). They show for an ordered analytic Hilbert space \(X\) that the QIS's \([p]\) and \([q]\) are similar if and only if \(p\) and \(q\) haye the same degree.
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 1763482
zbMATH Keywords: quasi-invariant subspace
zbMATH Keywords: analytic Hilbert space
reviewed by: James V. Whittaker
MaRDI profile type: Publication
full work available at URL: jmaa.2001.7839
OpenAlex ID: W2143987683
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DOI: 10.1006/JMAA.2001.7839
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description: scientific article; zbMATH DE number 1763482