Second derivatives of norms and contractive complementation in vector-valued spaces

Bas Lemmens¹ ; Beata Randrianantoanina² ; Onno van Gaans³

¹ Mathematics Institute University of Warwick CV4 7AL Coventry, United Kingdom
² Department of Mathematics and Statistics Miami University Oxford, OH 45056, U.S.A.
³ Mathematical Insitute Leiden University P.O. Box 9512 2300 RA Leiden, The Netherlands

Studia Mathematica, Tome 179 (2007) no. 2, pp. 149-166

Résumé

We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $\ell _p(X)$, where $X$ is a Banach space with a 1-unconditional basis and $p\in (1,2)\cup (2,\infty )$. If the norm of $X$ is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $\ell _p(X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection is then an averaging operator. We apply our results to the space $\ell _p(\ell _q)$ with $p,q\in (1,2)\cup (2,\infty )$ and obtain a complete characterization of its 1-complemented subspaces.

Détail

DOI : 10.4064/sm179-2-3

Keywords: consider complemented subspaces ranges contractive projections vector valued spaces ell where banach space unconditional basis cup infty norm twice continuously differentiable satisfies certain conditions connecting norm notion disjointness respect basis prove every complemented subspace ell admits basis mutually disjoint elements moreover every contractive projection averaging operator apply results space ell ell cup infty obtain complete characterization its complemented subspaces