Multiple summing operators on \(C(K)\) spaces
type: Item
instance of: scholarly article
title: Multiple summing operators on \(C(K)\) spaces
zbMATH Open document ID: 1063.46032
DOI: 10.1007/BF02432914
author: David Pérez-García
author: Ignacio Villanueva
published in: Arkiv för Matematik
publication date: 2004-09-28T00:00:00Z
review text: Let \(X\) be a Banach space and \(B_X\) be its unit ball. The weak \(p\)-norm of a finite family \((x_i)_{i=1}^m \subseteq X\) is denoted by \(w_p(x_i)\). The authors consider the Banach spaces \(\Pi_{(q;p_1,\ldots , p_n)}^n(X_1,\ldots,X_n;Y)\) of all multiple \((q;p_1,\ldots , p_n)\)-summing multilinear operators for Banach spaces \(X_1,\ldots,X_n,Y\) and \(1\leq q,p_1,\ldots , p_n < \infty \). A continuous multilinear operator \(T:X_1 \times \ldots \times X_n \rightarrow Y\) is multiple \((q;p_1,\ldots , p_n)\)-summing if where is a constant \(C\) such that \[ \left(\sum_{i_1=1}^{m_1} \cdots \sum_{i_n=1}^{m_n} \| T(x_{i_1},\ldots,x_{i_n})\| ^q\right)^{1/q} \leq C\cdot w_{p_1}(x_{i_1})\cdot \cdots \cdot w_{p_n}(x_{i_n}) \] for all finite families \((x_{i_1})_{i_1=1}^{m_1} \subseteq X_1,\ldots,(x_{i_n})_{i_n=1}^{m_n} \subseteq X_n\). The associated norm is defined by \(\pi_{(q;p_1,\ldots , p_n)}(T) = \inf\{C:C\) as above\(\}\). Write \((q;p)\) instead of \((q;p_1,\ldots , p_n)\) if \(p_1 = \cdots= p_n =p\). The operator \(T\) is integral if there is a regular \(Y''\)-valued Borel measure \(G\) of bounded variation on the product \(B=B_{X_1'} \times \cdots \times B_{X_n'}\) such that \[ T(x_1,\ldots, x_n) = \int_B x_1'(x_1)\cdots x_n'(x_n) dG(x_1',\ldots, x_n') \] for all \((x_1,\ldots, x_n) \in X_1 \times \ldots \times X_n\). A Banach space \(Y\) is a GT space if every bounded linear operator from \(Y\) into \(\ell_2\) is \((1;1)\)-summing. The main results are the following, inter alia: In [\textit{H. P. Rosenthal} and \textit{S. J. Szarek}, Lect. Notes Math. 1470, 108--132 (1991; Zbl 0758.47022)] it was shown that \(u_1\otimes u_2 \in L(X_1 \widehat{\otimes }_\varepsilon X_2;Y_1 \widehat{\otimes }_\pi Y_2)\) if \(u_j \in L(X_j;Y_j)\) for \({\mathcal L}_\infty \) spaces \(X_j\) and \({\mathcal L}_1 \) spaces \(Y_j\), \( j=1,2\). In the present paper, it is shown that this is also true if \(u_j \in L(X_j;Y_j)\) for \({\mathcal L}_\infty \) spaces \(X_j\) and GT spac…
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47B10
zbMATH DE Number: 2103843
zbMATH Keywords: multiple \((q
zbMATH Keywords: p)\)-summing mappings
zbMATH Keywords: integral multilinear mappings
zbMATH Keywords: polymeasures
zbMATH Keywords: \(p\)-summing operators
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
cites work: Norm Attaining Bilinear Forms on Spaces of Continuous Functions
cites work: Multilinear Mappings of Nuclear and Integral Type
cites work: A Hahn-Banach extension theorem for analytic mappings
cites work: On the absolute convergence of Dirichlet series
cites work: Multilinear extensions of Grothendieck's theorem
cites work: Multilinear operators on spaces of continuous functions
cites work: Integral operators on the product of \(C(K)\) spaces.
cites work: Polynomial characterization of \({\mathcal L}_{\infty}\)-spaces
cites work: Quotient Algebras of Uniform Algebras
cites work: Q4850055
cites work: Q4148368
cites work: Strongly \(p\)-summing multilinear operators.
cites work: The Dunford-Pettis property on tensor products
cites work: Tensor Powers of Operators and Nuclearity
cites work: Some results in the metric theory of tensor products
cites work: Q4296472
cites work: Fully, absolutely summing and Hilbert-Schmidt multilinear mappings
cites work: Q4521235
cites work: The inclusion theorem for multiple summing operators
cites work: Multiple summing operators on Banach spaces.
cites work: A composition theorem for multiple summing operators
cites work: Operator ideals and spaces of bilinear operators
cites work: Fully Nuclear and Completely Nuclear Operators with Applications to ℒ 1 - and ℒ ∞ -Spaces
cites work: Integral mappings between Banach spaces
full work available at URL: https://doi.org/10.1007/bf02432914
OpenAlex ID: W1987555529
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description: scientific article; zbMATH DE number 2103843