Some applications of Rademacher sequences in Banach lattices
type: Item
instance of: scholarly article
title: Some applications of Rademacher sequences in Banach lattices
zbMATH Open document ID: 0967.46019
DOI: 10.1023/A:1009767118180
author: K. Appert
published in: Positivity
publication date: 1998-11-18T00:00:00Z
review text: It has been already known for some time that (generalized) Rademacher sequences can be defined and successfully used in arbitrary Banach lattices. The paper under review contains several interesting applications of such sequences. It is worth mentioning that quite often the Rademacher sequences do not show up in the statements of results but they play a crucial role in the proofs. Two sample results follow. Theorem 3.1. For a Banach lattice \(E\) the following three assertions are equivalent. (1) The conjugate space \(E^*\) is atomic. (2) For each sequence \(\{f_n\}\) in \(E^*\) that converges to zero in the topology \(\sigma(E^*,E)\), it follows that \(\inf_n|f_n|= 0\). (3) There is no sequence \(\{f_n\}\) in \(E^*\) such that \(f_n\to 0\) in \(\sigma(E^*,E)\), and \(|f_n|= f\) for all \(n\) and some \(f\in E^*\). Theorem 4.7. Consider two infinite-dimensional spaces \(L_p(\mu)\) and \(L_q(\nu)\) with \(1\leq \min p\), \(q<\infty\). Then the following two statements are equivalent. (1) Each continuous operator \(T: L_p(\mu)\to L_q(\nu)\) is compact. (2) \(p>q\) and either (a) or (b) holds. (a) \(p>2\) and \((L_p(\mu))^*\) is atomic. (b) \(q< 2\) and \(L_q(\nu)\) is atomic. The second theorem is a nice and, to some extent, a final generalization of Pitt's theorem asserting that \({\mathcal L}(\ell_p,\ell_q)= {\mathcal K}(\ell_p,\ell_q)\) for \(p>q\) and of its generalization by H. Rosenthal to \(L_p\)-spaces.
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B28
zbMATH DE Number: 1203808
zbMATH Keywords: Rademacher sequences
zbMATH Keywords: Banach lattices
zbMATH Keywords: conjugate space
zbMATH Keywords: atomic
zbMATH Keywords: Pitt's theorem
zbMATH Keywords: \(L_p\)-spaces
reviewed by: Y. A. Abramovich
MaRDI profile type: Publication
full work available at URL: a:1009767118180
OpenAlex ID: W1495090929
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description: scientific article; zbMATH DE number 1203808