L^p-projections on subspaces and quotients of Banach spaces ; L^p-projections sur des sous-espaces et des quotients d'espaces de Banach

International audience ; 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