Differentiation in star-invariant subspaces. I: Boundedness and compactness
type: Item
instance of: scholarly article
title: Differentiation in star-invariant subspaces. I: Boundedness and compactness
zbMATH Open document ID: 1011.47005
author: Konstantin M. Dyakonov
published in: Journal of Functional Analysis
publication date: 2002-11-28T00:00:00Z
review text: Given two inner functions \(\theta_1,\theta_2\) on the upper half-plane \(\mathbb{C}_+\), let \(K^p(\theta_1, \theta_2)= \overline\theta_1 H^p\cap \theta_2 \overline{H^p}\), where \(H^p=H^p (\mathbb{C}_+)\) is the Hardy space, \(p\geq 1\). It is shown that the operator \({d\over dx}: K^p(\theta_1, \theta_2)\to L^p\) is bounded iff \(\theta_1', \theta_2'\in L^\infty (\mathbb{R})\); moreover, the norm of the operator is equivalent to \(\|\theta_1' \|_\infty+ \|\theta_2' \|_\infty\). In addition, the operator is compact iff \(\theta_1', \theta_2'\in C_0 (\mathbb{R})\). This implies the following result. Let \({\mathcal R}_\Lambda^p\) be the closed subspace of \(L^p(\mathbb{R})\) generated by the rational functions \(\{(x-\lambda)^{-j}: 1\leq j\leq m(\lambda)\), \(\lambda\in \Lambda\}\), where \(\Lambda\) is a discrete subset of \(\mathbb{C}\setminus\mathbb{R}\). Then the operator \({d\over dx}: {\mathcal R}^p_\Lambda \to L^p\) is bounded iff \({\mathcal F}_\Lambda\in L^\infty (\mathbb{R}) \), and compact iff \(F_\Lambda\in C_0(\mathbb{R})\); here \[ {\mathcal F}_\Lambda (x)= \sum_{\lambda\in \Lambda}m (\lambda) {|\text{Im} \lambda|\over |x-\lambda |^2},\quad x\in\mathbb{R}. \] For part II, cf. ibid. 387--409 (2002; Zbl 1011.47006).
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 1836660
zbMATH Keywords: star-invariant subspaces
zbMATH Keywords: differentiation operator
zbMATH Keywords: Bernstein's inequality
zbMATH Keywords: inner functions
MaRDI profile type: Publication
full work available at URL: jfan.2001.3920
OpenAlex ID: W2027465856
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DOI: 10.1006/JFAN.2001.3920
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description: scientific article; zbMATH DE number 1836660