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Coordinatewise multiple summing operators in Banach spaces. (English) Zbl 1205.46026
The present paper is a continuation and improvement of [A. Defant and P. Sevilla-Peris, J. Funct. Anal. 256, No. 5, 1642–1664 (2009; Zbl 1171.46034)].
Let \(1\leq p\leq q<\infty\) and \(E_{1},\dots ,E_{m}\) and \(F\) be Banach spaces. A continuous \(m\)-linear mapping \(T:E_{1}\times\cdots\times E_{m}\rightarrow F\) is multiple \((q;p)\)-summing if there exists a constant \(L_{m}\geq0\) such that
\[ \left( \sum_{j_{1},\dots ,j_{m}=1}^{N}\| T(x_{j_{1}}^{(1)},\dots ,x_{j_{m} }^{(m)})\|^{q}\right) ^{1/q}\leq L_{m}\prod\limits_{k=1}^{m}\left( \sup_{\varphi\in B_{E_{k}^{\ast}}}\sum\limits_{j=1}^{N}\left| \varphi(x_{j}^{(k)})\right| ^{p}\right) ^{1/p}\text{ } \]
for every positive integer \(N\) and \(x_{j}^{(k)}\in E_{k}\), with \(k\in \{1,\dots ,m\}\) and \(j\in\{1,\dots ,N\}\).
This class of multilinear mappings was introduced, independently, by M. C. Matos [Collect. Math. 54, No. 2, 111–136 (2003; Zbl 1078.46031)] and by F. Bombal, D. Pérez-García and I. Villanueva [Q. J. Math. 55, No. 4, 441–450 (2004; Zbl 1078.46030)]. However, this idea of summability of multilinear maps appears, for particular bilinear maps, in the famous Littlewood \(4/3\) theorem and, for multilinear maps, in the generalization of Littlewood’s \(4/3\) theorem to multilinear mappings due to H. F. Bohnenblust and E. Hille [Ann. Math. (2) 32, No. 3, 600–622 (1931; Zbl 0001.26901 and JFM 57.0266.05)]. More precisely, a reformulation of the Bohnenblust-Hille inequality asserts that every continuous \(m\)-linear form \(T:E_{1}\times\dots\times E_{m}\rightarrow \mathbb{C}\) is multiple \((\frac{2m}{m+1};1)\)-summing and \(L_{m}\leq 2^{\frac{m-1}{2}}\). The result is also true for Banach spaces over the real scalar field.
The main result of the paper is a beautiful vector-valued extension of Bohnenblust-Hille inequality, involving the notion of cotype. Also, the results of the paper re-prove a recent result by Marcela Luciano Vilela de Souza (from her PhD thesis, 2003) and by F. Bombal, D. Pérez-García and I. Villanueva [loc. cit.] which asserts that every bounded \(m\)-linear mapping with values in a cotype \(q\) space is multiple \((q;1)\)-summing.
The nice proof of the main result combines the notion of coordinatewise multiple summing operators, introduced by the authors, and some lemmas (Khinchin’s inequality, a variant of an inequality of Blei, and an inequality involving cotype).
Moreover, the results of the paper are used, for example, for the generalization of a theorem due to S. Kwapien and in the convergence theory of products of vector valued Dirichlet series.

46G25 (Spaces of) multilinear mappings, polynomials
47L22 Ideals of polynomials and of multilinear mappings in operator theory
Full Text: DOI
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