Strictly singular embeddings
type: Item
instance of: scholarly article
title: Strictly singular embeddings
zbMATH Open document ID: 1033.46026
DOI: 10.1023/A:1014438419455
author: S. Ya. Novikov
author: E. M. Semenov
author: Francisco L. Hernández
publication date: 2004-03-23T00:00:00Z
review text: Let \(E\) and \(F\) be rearrangement invariant (r.i.) Banach spaces on \((0,1)\) and \(E\subset F\). The present paper deals with conditions under which the natural embedding \(E\subset F\) is strictly singular. That means that for any infinite dimensional subspace \(B\subset E\), the norms of \(E\) and \(F\) are nonequivalent on \(B\). It is known that for every r.i. space \(E\neq L_{\infty}\) the embedding \(L_{\infty}\subset E\) is strictly singular, cf. \textit{S. Ya. Novikov} [Math. Notes 62, 457--468 (1997; Zbl 0914.46029)], and that this property characterizes \(L_{\infty}\), cf. \textit{A. Garcia del Amo, F. L. Hernandez, V. M. Sanchez} and \textit{E. M. Semenov} [J. Lond. Math. Soc. 62, 239--252 (2000; Zbl 0956.46021)]. In the paper under review, the embedding \(E\subset L_1\) is considered. Namely, this embedding is not strictly singular if and only if \(E\supset G\) and this property characterizes \(L_1\). Here \(G\) is the closure of \(L_{\infty}\) in the Orlicz space \(L_M\) with \(M(u)=e^{u^2}-1\). Another result of the paper is as follows. Let \(E\) be separable. The embedding \(E\subset F\) is strictly singular if and only if it is disjointly strictly singular and the norms of \(E\) and \(F\) are not equivalent on the subspace generated by the Rademacher system. (The embedding \(F\subset F\) is called disjointly strictly singular if its restriction to any subspace generated by a sequence of disjoint functions is not an isomorphism. The paper does not contain the proofs which have been published in [Positivity 7, 119--124 (2003; Zbl 1041.46022)].
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 1930640
zbMATH Keywords: rearrangement invariant Banach space
zbMATH Keywords: strictly singular embedding
zbMATH Keywords: disjointly strictly singular embedding
zbMATH Keywords: Rademacher system
zbMATH Keywords: exponential Orlicz space
reviewed by: A. M. Plichko
MaRDI profile type: Publication
full work available at URL: a:1014438419455
OpenAlex ID: W1597749455
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 1930640