Banach space sequences and projective tensor products.
type: Item
instance of: scholarly article
title: Banach space sequences and projective tensor products.
zbMATH Open document ID: 1043.46053
DOI: 10.1016/S0022-247X(02)00635-2
author: Jan Fourie
author: Ilse M. Röntgen
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-04-28T00:00:00Z
review text: Given \(1<p<\infty\) and a Banach space \(X\), the authors use \(L^p_{\text{ weak}}(X)\) to denote \[ \left\{(x_i)_i\in X^{\mathbb{N}}:\sum_ix^*(x_i)e_i\in L^p(0,1)\text{ for all } x^*\in X'\right\}, \] where \((e_i)_i\) is the Haar system renormalised to become a \(1\)-unconditional basis for \(L^p(0,1)\). They define \(L^p\langle X\rangle\) to be \[ \left\{(x_i)_i\in X^{\mathbb{N}}: \sum_i| x_i^*(x_i)| <\infty\text{ for all } (x_i)^*_i\in L^{p'}_{\text{weak}}(X')\right\}, \] (\(1/p+1/p'=1\)). Using ideas from Grothendieck's Résumé, they show that \(L^p\langle X\rangle\) is isometrically isomorphic to the space of nuclear operators from \(L^{p'}(0,1)\) into \(X\). From this, it follows that \(L^p\langle X\rangle\) is isometrically isomorphic to \(L^p_{ \text{new}}(0,1){\widehat\otimes_\pi}X\) where \(L^p_{\text{new}}(0,1)\) is \(L^p(0,1)\) renormed so that the unconditional basis constant of the Haar system is \(1\). Given Banach spaces \(X\) and \(Y\), \textit{J. L. Arregui} and \textit{O. Blasco} [J. Math. Anal. Appl. 274, 812--827 (2002; Zbl 1010.46014)] introduced a \((p,q)\)-summing multiplier as a sequence of bounded linear operators \((u_i)_i\), \(u_i\colon X\to Y\) such that there is \(c>0\) such that for any finite set \(x_1,x_2,\ldots, x_n\) in \(X\) \[ \left(\sum_{i=1}^n\| u_ix_i\| ^p \right )^{1/p}\leq c\sup_{\| x^*\| \leq 1} \left(\sum_{i=1}^n | x^*(x_i)| ^q\right)^{1/q}. \] They use \(\ell_{p,q}(X,Y)\) to denote the space of all \((p,q)\)-summing multipliers from \(X\) into \(Y\). The authors of the paper under review prove that if \(1\leq p<q\leq \infty\) then for all \((u_i)_i\) in \(\ell_{p,q} (X,Y)\) \((u_1,u_2,\ldots, u_n,0,0,\ldots)\) converges to \((u_i)\) in \(\ell_{p,q}(X,Y)\). The space of all sequences \((\alpha_i)\) such that \(u_i=\alpha_i\text{Id}_X\) belongs to \(\ell_{p,p}(X,X)\) is denoted by \(m_p(X)\). The authors show that \(m_p(X)=m_p(X^{**})\) and that \((\alpha_n)_n\) is in \(M_p(X)\) if and only if for all \((x^*_n)_n\) in \(\e…
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 1901859
zbMATH Keywords: vector-valued sequence spaces
zbMATH Keywords: \((p
zbMATH Keywords: q)\)-summing multipliers
MaRDI profile type: Publication
cites work: (\(p\),\(q\))-summing sequences
cites work: On summing multipliers and applications
cites work: OBSERVATIONS ABOUT THE PROJECTIVE TENSOR PRODUCT OF BANACH SPACES, I—
cites work: Observations About the Projective Tensor Product of Banach Spaces, II — Lp(0, 1) ⊗X, 1 <p< ∞
cites work: Absolutely \(p\)-summing, \(p\)-nuclear operators and their conjugates
cites work: Q4148368
cites work: Q3239535
cites work: Q4136031
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description: scientific article; zbMATH DE number 1901859