On operator valued sequences of multipliers and \(R\)-boundedness
type: Item
instance of: scholarly article
title: On operator valued sequences of multipliers and \(R\)-boundedness
zbMATH Open document ID: 1117.47015
author: I. M. Schoeman
author: Oscar Blasco
author: Jan Fourie
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-02-12T00:00:00Z
review text: Given Banach spaces \(X\) and \(Y\) as well as two spaces of sequences \(E(X)\) and \(F(Y)\) containing \(c_{00}(X)\) and \(c_{00}(Y)\), respectively, a sequence \((u_n) \subseteq L(X,Y)\) is called a \textit{multiplier sequence from \(E(X)\) to \(F(Y)\)} if the map \((x_i)\mapsto(u_ix_i)\) is bounded from \(E(X)\) into~\(F(Y)\). In the first part (Section~2) of the present paper, the authors discuss and summarize recent results on \((p,q)\)-summing multipliers, that is, multipliers for which \(E(X)=l_q^w(X)\) is the space of weakly \(q\)-summable sequences in \(X\) and \(F(Y)=l_p(Y)\) is the space of strongly \(p\)-summable sequences in \(Y\). In the second part (Section~3), the concept of Rademacher boundedness of a sequence \((u_i)\) of operators is considered, meaning that \((u_i)\) is a multiplier sequence from \(Rad(X)\) to \(Rad(Y)\). Several other related properties are considered and their relations among each other studied. In particular, conditions on the geometry of the spaces \(X\) and \(Y\) yield embedding results of the respective multiplier spaces. The setting of multiplier sequences proves also useful in simplifying proofs of several known results.
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 5124049
zbMATH Keywords: \((p
zbMATH Keywords: q)\)-summing multiplier
zbMATH Keywords: multiplier sequence
zbMATH Keywords: Rademacher bounded sequence
zbMATH Keywords: weakly Rademacher bounded
zbMATH Keywords: semi Rademacher bounded
zbMATH Keywords: almost summing
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.04.061
OpenAlex ID: W2028220968
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DOI: 10.1016/J.JMAA.2006.04.061
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description: scientific article; zbMATH DE number 5124049