Strictly singular and disjointly strictly singular inclusions
type: Item
instance of: scholarly article
title: Strictly singular and disjointly strictly singular inclusions
zbMATH Open document ID: 1161.46016
DOI: 10.1134/S1064562407020251
published in: Doklady Mathematics
publication date: 2009-01-19T00:00:00Z
review text: An operator between Banach spaces is said to be strictly singular \((SS)\) if it fails to be an isomorphism on any infinite-dimensional closed subspace. An operator from a Banach lattice \(E\) into a Banach space is said to be disjointly strictly singular \((DSS)\) if there is no sequence \((x_n)\) of disjointly supported non-null vectors in \(E\) such that the restriction of the operator to the subspace generated by the sequence is an isomorphism. The present note announces several results about strict singularity and disjointly strict singularity of inclusions between r.i. spaces. The presented theorems are of two different types: General results about inclusions maps and the study of strict and disjointly strict singularity of inclusion maps between concrete r.i. spaces. For instance, for general r.i. spaces \(E\) and \(F\), the fact that \(E\subset F\) is \(SS\) is equivalent to be \(DSS\) together with the fact that either the norms of \(E\) and \(F\) are not equivalent on the span of the Rademacher functions or \(E\) not containing the closure of \(L^\infty\) in the Orlicz space \(L_M\) for \(M(t)=e^{t^2}-1\). Several conditions for the inclusions between Orlicz or Marcinkiewicz spaces to be strictly singular are presented. Finally, the authors point out how their techniques allow them to solve in the negative a question by V. D. Milman.
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 5493401
zbMATH Keywords: stricty singular operators
zbMATH Keywords: Orlicz spaces
zbMATH Keywords: Marcinkiewicz spaces
author: E. M. Semenov
author: Francisco L. Hernández
MaRDI profile type: Publication
cites work: Q4136031
cites work: On \(l^ p\)-complemented copies in Orlicz spaces. II
cites work: Q4188284
cites work: Q5667036
cites work: Strictly singular embeddings between rearrangement invariant spaces
cites work: Disjointly Strictly-Singular Inclusions between Rearrangement Invariant Spaces
full work available at URL: s1064562407020251
OpenAlex ID: W1980090840
description: scientific article; zbMATH DE number 5493401