Banach space sequences and projective tensor products.
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 \(\ell^{p'}(X^*)\), \(\sum_ne_n^*\otimes \alpha_nx_n^*\colon \ell_p\to X^*\) is nuclear. Reviewer's remark: As noted by the authors, a number of results in this paper have also been obtained by \textit{Q.-Y. Bu} [Quaest. Math. 25, 209--227 (2002; Zbl 1038.46060) and ibid., 303--310 (2002; Zbl 1038.46062)].
scientific article; zbMATH DE number 1901859
scientific article