Duality of Bloch spaces and norm convergence of Taylor series
type: Item
instance of: scholarly article
title: Duality of Bloch spaces and norm convergence of Taylor series
zbMATH Open document ID: 0728.30026
published in: Michigan Mathematical Journal
publication date: 1991-01-01T00:00:00Z
review text: Let B be the Bloch space of the unit disc \({\mathbb{D}}\) with the norm \[ | f| =| f(0)| +\sup \{(1-| z|^ 2)| f'(z)|,\quad z\in {\mathbb{D}}\}. \] The Little Bloch space \(B_ 0\) consists of all holomorphic f(z) in \({\mathbb{D}}\) such that \((1-| z|^ 2)f(z)\) belongs to \(C_ 0({\mathbb{D}})\), the space of complex continuous functions on \({\bar {\mathbb{D}}}\) which vanish on \(\partial {\mathbb{D}}.\) For \(\alpha >-1\) the weighted Bergman space \(L_ a'(dA_{\alpha})\) is the subspace of \(L'({\mathbb{D}},dA_{\alpha})\) of all analytic functions, where \(dA_{\alpha}(z)=(1+\alpha)(1-| z|^ 2)^{\alpha}dA(z)\) and dA(z) is the area measure on \({\mathbb{D}}\). The author proves that under the \(pairing\) \(<f,g>_{\alpha}=\int_{{\mathbb{D}}}f(z)\overline{g(z)}dA(z)\) the following dualities hold (with equivalent norms) \[ B^*_ 0\cong L_ a'(dA_{\alpha})\text{ and } (L_ a'(dA_{\alpha}))^*\cong B. \] This is a generalization of the same equalities known for \(\alpha =0\). Moreover it is shown that there exist functions \(f\in B_ 0\) and \(g\in L_ a'(dA_{\alpha})\), whose Taylor series are not convergent in norm.
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 4202670
zbMATH Keywords: Bloch space
author: Kehe Zhu
reviewed by: Jan Janas
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004264
OpenAlex ID: W2068127963
DOI: 10.1307/MMJ/1029004264
description: scientific article; zbMATH DE number 4202670