$$L^p$$-projections on subspaces and quotients of Banach spaces

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 \(1\le p \le +\infty\), is an idempotent operator P satisfying \(\Vert f\Vert _X = \Vert ( \Vert P(f)\Vert _X, \Vert (I-P)(f)\Vert _X) \Vert _{\ell _{p}}\) for all \(f \in X\). This is an \(L^p\) version of the equality \(\Vert f\Vert ^2=\Vert Q(f)\Vert ^2 + \Vert (I-Q)(f)\Vert ^2\), 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 \(1<p<+\infty\), \(p\ne 2\). The cases \(p=1,2\) or \(+\infty\) can exhibit different behavior. 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 \(\text {Span}(x,y)\) 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.