Operator-valued Fourier-Haar multipliers on vector-valued \({L^1}\) spaces. II: A characterisation of finite dimensionality
type: Item
instance of: scholarly article
title: Operator-valued Fourier-Haar multipliers on vector-valued \({L^1}\) spaces. II: A characterisation of finite dimensionality
zbMATH Open document ID: 1478.42029
author: H. M. Wark
published in: Positivity
publication date: 2021-08-16T00:00:00Z
review text: The paper under review is concerned with the boundedness of operator-valued Fourier-Haar multipliers from \(L^1([0,1],X)\) to \(L^1([0,1],Y)\), where \(X\) and \(Y\) are Banach spaces. A necessary and sufficient condition for the boundedness is obtained when \(X\) is finite dimensional. It is also shown that the finite dimensionality of \(X\) is necessary for such a characterization to hold. Moreover, a lower bound for the operator norm is obtained for any Banach space \(X\) whose dual has cotype \(q\) for some \(2\leq q<\infty\). The results complement \(L^p([0,1],X)\) to \(L^q([0,1],Y)\) boundedness results obtained by \textit{M. Girardi} [J. Math. Anal. Appl. 325, No. 2, 1314--1326 (2007; Zbl 1105.42024)].
reviewed by: Jongchon Kim
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 7382081
zbMATH Keywords: Fourier-Haar multiplier
zbMATH Keywords: Bochner-Lebesgue spaces
zbMATH Keywords: finite-dimensional Banach spaces
MaRDI profile type: Publication
full work available at URL: s11117-020-00797-z
OpenAlex ID: W3108194708
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DOI: 10.1007/S11117-020-00797-Z
description: scientific article; zbMATH DE number 7382081