Toeplitz operators and arguments of analytic functions
type: Item
instance of: scholarly article
title: Toeplitz operators and arguments of analytic functions
zbMATH Open document ID: 1167.47023
author: Konstantin M. Dyakonov
published in: Mathematische Annalen
publication date: 2009-04-27T00:00:00Z
review text: For a Toeplitz operator \(T_\varphi\) with symbol \(\varphi\in L^\infty({\mathbb T})\), let \(\text{Ker}_pT_\varphi\) denote its kernel on the Hardy space \(H^p\), \(1\leq p\leq\infty\), of the unit circle \({\mathbb T}\). Relations between \(\text{Ker}_pT_\varphi\) and the smoothness of \(\varphi\) are studied. Let \(\Lambda^\alpha\) (resp., \(A^\alpha\)), \(0<\alpha<\infty\), denote the Lipschitz (Hölder-Zygmund) space (resp., its subspace of analytic functions). Suppose that \(\beta=\alpha-1/p\) and \(N\) is the integral part of \(1/(\alpha p)\). Let us formulate two sample results for unimodular symbols \(\varphi\in\Lambda^\alpha\). If \(1/p<\alpha<\infty\), then \(\text{Ker}_pT_\varphi\subset A^\beta\) and \(\|f\|_{\Lambda^\beta}\leq\text{const}\,\|\varphi\|_{\Lambda^\alpha}\|f\|_p\) for each \(f\in\text{Ker}_pT_\varphi\). If \(0<\alpha<\infty\), then \(\text{Ker}_pT_\varphi\subset A^\alpha\) and \(\|f\|_{\Lambda^\alpha}\leq\text{const}\,\|\varphi\|_{\Lambda^\alpha}^{N+2} \|f\|_p\) for every \(f\in\text{Ker}_pT_\varphi\). The constants in the above inequalities are independent of \(f\) and \(\varphi\). Similar and more general results for various smoothness classes are obtained, and several approaches are discussed.
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 5547452
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: symbol
zbMATH Keywords: kernel
zbMATH Keywords: Lipschitz space
zbMATH Keywords: unimodular function
zbMATH Keywords: BMO
zbMATH Keywords: Gevrey class
MaRDI profile type: Publication
full work available at URL: s00208-008-0310-8
OpenAlex ID: W2079565528
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DOI: 10.1007/S00208-008-0310-8
description: scientific article; zbMATH DE number 5547452