Coordinatewise multiple summing operators in Banach spaces
type: Item
instance of: scholarly article
title: Coordinatewise multiple summing operators in Banach spaces
zbMATH Open document ID: 1205.46026
author: Andreas Defant
author: Dumitrŭ Popa
author: Ursula Schwarting
published in: Journal of Functional Analysis
publication date: 2010-07-07T00:00:00Z
review text: The present paper is a continuation and improvement of [\textit{A.\,Defant} and \textit{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 \textit{M.\,C.\thinspace Matos} [Collect.\ Math.\ 54, No.\,2, 111--136 (2003; Zbl 1078.46031)] and by \textit{F.\,Bombal, D.\,Pérez-García} and \textit{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 \textit{H.\,F.\thinspace Bohnenblust} and \textit{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 rece…
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 47L22
zbMATH DE Number: 5731607
zbMATH Keywords: Bohnenblust-Hille inequality
zbMATH Keywords: multiple summing operators
zbMATH Keywords: Dirichlet series
zbMATH Keywords: polynomials
zbMATH Keywords: Littlewood's \(4/3\) theorem
zbMATH Keywords: multilinear maps
reviewed by: Daniel Pellegrino
MaRDI profile type: Publication
full work available at URL: j.jfa.2010.01.008
OpenAlex ID: W2081054398
cites work: Inclusion mappings between \(l^p\) spaces
cites work: Summability of multilinear mappings: Littlewood, Orlicz and beyond
cites work: Analysis in integer and fractional dimensions
cites work: On the absolute convergence of Dirichlet series
cites work: Multilinear extensions of Grothendieck's theorem
cites work: Inclusions and coincidences for multiple summing multilinear mappings
cites work: Complex interpolation and summability properties of multilinear operators
cites work: When every multilinear mapping is multiple summing
cites work: Sur l'ordre de grandeur des polynômes de Dirichlet
cites work: Absolut-(p , 1)-summierende identische Operatoren von l u in l v
cites work: Q4023934
cites work: A new multilinear insight on Littlewood's 4/3-inequality
cites work: Bohr's strip for vector valued Dirichlet series
cites work: The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive
cites work: Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables
cites work: Q4850055
cites work: Q4257163
cites work: Q4663509
cites work: Eigenvalue distribution of compact operators
cites work: The translation \(\frac 12\) in the theory of Dirichlet series
cites work: Some remarks on (p,q)-absolutely summing operators in $l_{p}$-spaces
cites work: Q4136031
cites work: Fully, absolutely summing and Hilbert-Schmidt multilinear mappings
cites work: The inclusion theorem for multiple summing operators
cites work: Multiple summing operators on Banach spaces.
cites work: Q4183861
cites work: Q4722695
cites work: Reverse inclusions for multiple summing operators
cites work: H. Bohr's vision of ordinary Dirichlet series; old and new results
DOI: 10.1016/J.JFA.2010.01.008
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description: scientific article; zbMATH DE number 5731607