L^p-projections on subspaces and quotients of Banach spaces
L^p-projections sur des sous-espaces et des quotients d'espaces de Banach
Abstract
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
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Submitted on : Monday, January 25, 2021-6:59:40 PM
Last modification on : Monday, September 4, 2023-12:04:05 PM
Long-term archiving on: Monday, April 26, 2021-7:29:16 PM
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- HAL Id : hal-03120770 , version 1
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Vidal Agniel. L^p-projections on subspaces and quotients of Banach spaces. Advances in Operator Theory, In press. ⟨hal-03120770⟩
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