Cotype 2 estimates for spaces of polynomials on sequence spaces
type: Item
instance of: scholarly article
title: Cotype 2 estimates for spaces of polynomials on sequence spaces
zbMATH Open document ID: 1009.46032
DOI: 10.1007/BF02773168
author: Andreas Defant
author: Manuel Maestre
author: Pablo Sevilla-Peris
published in: Israel Journal of Mathematics
publication date: 2003-04-14T00:00:00Z
review text: For a Banach space \(E\) let \({\mathcal P}(^mE)\) be the Banach space of all scalar valued \(m\)-homogeneous continuous polynomials on \(E\) equipped with the topology of uniform convergence on bounded subsets of \(E\). If \(X\) is a Banach sequence space, then \(X_n\) is defined as the subspaces spanned by the sequences \(e_k:=(\delta_{kj})_{j \in {\mathbb{N}}}\), \(1\leq k \leq n\). In the main part of this paper, the authors show that the following conjecture is true for spaces \(X\) which are either 2-convex or 2-concave and have non-trivial convexity. Conjecture. Let \(X\) be a symmetric real Banach sequence space and \(m \geq 2\). Then \[ {\mathbb{C}}_2({\mathcal P}(^mX_n)) \asymp (n^{1/2})^{m-1}{\mathbb{C}}_2(X'_n), \] where \({\mathbb{C}}_2(E)\) denotes the cotype 2 constant of the Banach space \(E\). For this purpose, they prove that the space \({\mathcal P}(^mX_n) = \otimes^{m,s}_{\epsilon_s}X'_n\) (the \(m\)-fold symmetric tensor product of \(X'_n\) equipped with the symmetric injective tensor norm) can be replaced by \(\otimes^{m}_{\epsilon}X'_n\). Several steps are necessary to verify the result mentioned above (Sections 5 and 6 of this paper). In Section 7 it is shown that these results can be transfered to complex Banach spaces. In Section 8 the authors apply their results to special sequence spaces, that is, to \(\ell_p\) (\(1 \leq p \leq \infty\)), Orlicz spaces, and the Lorentz spaces \(d(w,p)\) and \(\ell_{p,q}\). In an Appendix a result is presented, which is not in the main scope of this paper, but it is based on techniques used in this paper and improves a result on cotype 2 estimates for injective tensor products of \(\ell_{p}^n\)'s given by \textit{B. Carl} and \textit{A. Defant} [J. Approximation Theory 88, No.~2, 228-256 (1997; Zbl 0892.46075)].
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1760720
zbMATH Keywords: polynomials on Banach spaces
zbMATH Keywords: spaces with cotype 2
zbMATH Keywords: Köthe function spaces
zbMATH Keywords: symmetric Banach sequence spaces
zbMATH Keywords: \(r\)-convex spaces
zbMATH Keywords: \(s\)-concave spaces
zbMATH Keywords: tensor products
zbMATH Keywords: \(\ell_p\) spaces
zbMATH Keywords: Orlicz spaces
zbMATH Keywords: Lorentz spaces
zbMATH Keywords: topology of uniform convergence on bounded subsets
reviewed by: Hans-Andreas Braunß
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773168
OpenAlex ID: W2064116667
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description: scientific article; zbMATH DE number 1760720