Little Grothendieck's theorem for sublinear operators
type: Item
instance of: scholarly article
title: Little Grothendieck's theorem for sublinear operators
zbMATH Open document ID: 1065.47018
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-08-19T00:00:00Z
review text: A sublinear operator \(T:X\rightarrow Y\) between a Banach space \(X\) and a Banach lattice \(Y\) is called \(2\)-summing, if the norms of the mappings \(\text{id} \otimes T: \ell_2^n \otimes_\varepsilon X \rightarrow \ell_2^n(Y)\) are uniformly bounded. Little Grothendieck's theorem states that every bounded linear operator from \(C(K)\) into a Hilbert space \(H\) is \(2\)-summing. In this article, the authors present some generalizations thereof within the framework of sublinear operators. In particular, they show that the theorem remains true for a bounded sublinear operator \(T:C(K) \rightarrow H\) if and only if it is \(2\)-regular, that is, \(\| (\sum_1^n | Tx_i| ^2)^{1/2}\| \leq C\), \(\| (\sum_1^n | x_i| ^2)^{1/2}\| \) for some \(C>0\) and all \(x_1, \ldots, x_n \in C(K)\). It is left open whether the assumption of \(2\)-regularity is necessary -- note that by a result of Krivine, every bounded linear operator between Banach lattices is \(2\)-regular.
reviewed by: Carsten Michels
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 2092944
zbMATH Keywords: Banach lattice
zbMATH Keywords: sublinear operator, \(p\)-summing operator
zbMATH Keywords: \(p\)-regular operator
author: Dahmane Achour
author: Lahcène Mezrag
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.04.018
OpenAlex ID: W1983448998
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DOI: 10.1016/J.JMAA.2004.04.018
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description: scientific article; zbMATH DE number 2092944