The range of a contractive projection in \(L_p(H)\).
type: Item
instance of: scholarly article
title: The range of a contractive projection in \(L_p(H)\).
zbMATH Open document ID: 1066.46025
author: Yves Raynaud
published in: Revista Matemática Complutense
publication date: 2004-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/44525
review text: The main purpose of the paper under review is to investigate structural aspects of ranges of (linear) contractive projections on Bochner spaces of Hilbert space valued functions. Since the results of \textit{R. G. Douglas} [Pac. J. Math. 15, 443--462 (1965; Zbl 0148.12203)] and \textit{T. Ando} [Pac. J. Math. 17, 391--405 (1966; Zbl 0192.23304)] on contractive projections on \(L_p\)-spaces, the vector-valued case has been a long standing problem. In this interesting paper, the author settles the case of Hilbert-space valued Bochner spaces. More precisely, it is shown that the range of a linear contractive projection on a Lebesgue-Bochner space of Hilbert valued functions is isometric to an \(l_p\)-direct sum of Hilbert space valued \(L_p\)-spaces. Also, the precise form of the contractive projections is given. The method of proof is very elegant and is a delicate generalization of the method that goes back to [\textit{S. J. Bernau} and \textit{H. E. Lacey}, Pac. J. Math. 53, 21-41 (1974; Zbl 0287.46038)]. One should note that the question of describing the range/form of linear contractive projections on \(L_p(X)\) remains open (even for the case of \(X=L_q\) with \(q\neq 2\)).
reviewed by: Narcisse Randrianantoanina
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46M07
zbMATH DE Number: 2109948
zbMATH Keywords: contractive projections
zbMATH Keywords: vector-valued \(L_p\)-space
MaRDI profile type: Publication
DOI: 10.5209/REV_REMA.2004.V17.N2.16763
description: scientific article; zbMATH DE number 2109948