Title :

L^p-projections sur des sous-espaces et des quotients d'espaces de Banach

Author(s) :

Agniel, Vidal [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Advances in Operator Theory

Pages :

38

Publisher :

Birkhäuser / Tusi Mathematical Research Group

Publication date :

2021

ISSN :

2662-2009

English keyword(s) :

Classes of operators
Projections
Orthogonality
p-orthogonality
Banach spaces
Subspaces
Quotients
L^p-projections
L^p
SQ^p

HAL domain(s) :

Mathématiques [math]/Analyse fonctionnelle [math.FA]
Mathématiques [math]
Mathématiques [math]/Algèbres d'opérateurs [math.OA]

English abstract : [en]

The aim of this paper is to study $L^p$-projections, a notion introduced by Cunningham in 1953, on subspaces and quotients of complex Banach spaces. An $L^p$-projection on a Banach space $X$, for $1\leq p \leq +\infty$, ...
Show more >The aim of this paper is to study $L^p$-projections, a notion introduced by Cunningham in 1953, on subspaces and quotients of complex Banach spaces. An $L^p$-projection on a Banach space $X$, for $1\leq p \leq +\infty$, is an idempotent operator $P$ satisfying $ \|f\|_X = \|( \|P(f)\|_X, \|(I-P)(f)\|_X) \|_{\ell_{p}}$ for all $f \in X$. This is an $L^p$ version of the equality $\|f\|^2=\|Q(f)\|^2 + \|(I-Q)(f)\|^2$, valid for orthogonal projections on Hilbert spaces. We study the relationships between $L^p$-projections on a Banach space $X$ and those on a subspace $F$, as well as relationships between $L^p$-projections on $X$ and those on the quotient space $X/F$.All the results in this paper are true for $1<p<+\infty$, $p\neq 2$. The cases $p=1,2$ or $+\infty$ can exhibit different behaviour. In this regard, we give a complete description of $L^{\infty}$-projections on spaces $L^{\infty}(\Omega)$. For this, we introduce a notion of $p$-orthogonality for two elements $x,y$ by requiring that $\textrm{Span}(x,y)$ admits an $L^p$-projection separating $x$ and $y$. We also introduce the notion of maximal $L^p$-projections for $X$, that is $L^p$-projections defined on a subspace $G$ of $X$ that cannot be extended to $L^p$-projections on larger subspaces. We prove results concerning $L^p$-projections and $p$-orthogonality of general Banach spaces or on Banach spaces with additional properties. Generalizations of some results to spaces $L^p(\Omega,X)$ as well as some results about $L^q$-projections on subspaces of $L^p(\Omega)$ are also discussed.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions
Frontières de la théorie des opérateurs

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL