The strong property (B) for \(L_p\) spaces
type: Item
instance of: scholarly article
title: The strong property (B) for \(L_p\) spaces
zbMATH Open document ID: 1492.46012
published in: Mediterranean Journal of Mathematics
publication date: 2021-11-30T00:00:00Z
review text: Answering a question by \textit{R. W. Cross} et al. [Math. Nachr. 173, 91--114 (1995; Zbl 0834.47001)] (see also [\textit{A.~Martínez-Abejón} et al., Mat. Stud. 24, No.~2, 173--191 (2005; Zbl 1094.46502)]), the author proves that for every infinite dimensional subspace \(X\) of \(L_p(\nu)\) for a finite atomless \(\nu\) and \(1 \le p < \infty\) there exists a decomposition \(L_p(\nu) = Y \oplus Z\) into subspaces \(Y\) and \(Z\) both isomorphic to \(L_p(\nu)\) such that both subspaces \(X \cap Y\) and \(X \cap Z\) are infinite dimensional. The proof is pretty involved. As a consequence, he extends the following result by the aforementioned authors [\textit{R. W. Cross} et al., Math. Nachr. 173, 91--114 (1995; Zbl 0834.47001)] from \(1 < p < \infty\) to \(1 \le p < \infty\). Given a Banach space \(Y\) and a linear bounded operator \(T \colon L_p \to Y\) with a dense non-closed range \(T(L_p)\) in \(Y\), there exists a pair of linear bounded operators \(T_i \colon L_p \to Y\), \(i = 1,2\), with the following properties. (i) \(T_i(L_p)\) is dense and non-closed in \(Y\), \(i = 1,2\). (ii) \(T(L_p) = T_1(L_p) + T_2(L_p)\). (iii) \(T_1(L_p) \cap T_2(L_p) = \{0\}\).
reviewed by: M. M. Popov
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 7435776
zbMATH Keywords: property (B)
zbMATH Keywords: strong property (B)
zbMATH Keywords: \(L_p\) spaces
zbMATH Keywords: operator range
zbMATH Keywords: ultraproducts
author: Antonio Martínez-Abejón
MaRDI profile type: Publication
full work available at URL: s00009-021-01907-1
OpenAlex ID: W3217329123
cites work: Q5317928
cites work: Subspaces of L 1 , via Random Measures
cites work: \(L_p\) spaces
cites work: Q4606219
cites work: Almost exchangeable sequences of random variables
cites work: Q4836494
cites work: Ultrafilters on a countable set
cites work: New classes of \(L_ p-\)spaces
cites work: Q5544881
cites work: Operator Ranges in Banach Spaces, I
cites work: Sous-espaces de \(L^1\)
cites work: Q3329946
cites work: Q4148368
cites work: Étude sur les variétés et les opérateurs de Julia, avec quelques applications
cites work: Subspaces of $L^{1}$ containing $L^{1}$
cites work: On subprojectivity and superprojectivity of Banach spaces
cites work: Ultrapowers of \(L_1(\mu)\) and the subsequence splitting principle
cites work: The Unconditional Basic Sequence Problem
cites work: On subsequences of the Haar system in \(L_p [0,1]\), \((1<p<\infty)\)
cites work: Q4769223
cites work: Ultraproducts in Banach space theory.
cites work: Q3091370
cites work: Espaces de Banach stables
cites work: Q4772256
cites work: Q4136031
cites work: Q4188284
cites work: Some open problems on the classical function space \(L_1\).
cites work: Subprojective Banach spaces
cites work: On the impossibility of embedding of the space L in certain Banach spaces
cites work: Strictly Singular Operators and Their Conjugates
cites work: Q3995204
DOI: 10.1007/S00009-021-01907-1
description: scientific article; zbMATH DE number 7435776