We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces , where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). 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 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 with p,q ∈ (1,2) ∪ (2,∞) and obtain a complete characterization of its 1-complemented subspaces.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-2-3,
author = {Bas Lemmens and Beata Randrianantoanina and Onno van Gaans},
title = {Second derivatives of norms and contractive complementation in vector-valued spaces},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {149-166},
zbl = {1118.46023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-2-3}
}
Bas Lemmens; Beata Randrianantoanina; Onno van Gaans. Second derivatives of norms and contractive complementation in vector-valued spaces. Studia Mathematica, Tome 178 (2007) pp. 149-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm179-2-3/