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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.

##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 47L22 Ideals of polynomials and of multilinear mappings in operator theory
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