On \({\mathcal L}_{p,\lambda}\) spaces for small \(\lambda\)
type: Item
instance of: scholarly article
title: On \({\mathcal L}_{p,\lambda}\) spaces for small \(\lambda\)
zbMATH Open document ID: 0578.46018
author: Dale Alspach
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: This paper presents some results related to the problem of whether a \({\mathcal L}_{p,\lambda}\) subspace of \(L_ p\) for \(1\leq p\neq 2<\infty\) is close to \(L_ p(\mu)\) for some measure \(\mu\) provided that \(\lambda\) is close enough to 1. We show that there is a strong relationship between the answer to this for any one p and all others. In particular it is shown that if for some p there is a positive solution to the problem and the distance to \(L_ p(\mu)\) goes to one as \(\lambda\) goes to one then there is a positive solution for all p with similar control on the distance to \(L_ p(\mu)\). The proof depends heavily on the results of L. Dor and G. Schechtman for the finite dimensional case. We also prove some representation results for some special classes of operators on \(L_ 1\) and a factorization theorem (through \(L_ 1)\) for certain projections on \(L_ p\).
Mathematics Subject Classification ID: 46B25
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 3925556
zbMATH Keywords: script \(L_ p\) space
zbMATH Keywords: conditional expectation
zbMATH Keywords: factorization theorem
MaRDI profile type: Publication
full work available at URL: pjm.1986.125.257
OpenAlex ID: W2326379666
DOI: 10.2140/PJM.1986.125.257
description: scientific article; zbMATH DE number 3925556