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

Bas Lemmens, Beata Randrianantoanina, and Onno van Gaans. "Second derivatives of norms and contractive complementation in vector-valued spaces." Studia Mathematica 179.2 (2007): 149-166. <http://eudml.org/doc/284800>.

@article{BasLemmens2007,
abstract = {We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $ℓ_\{p\}(X)$, 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 $ℓ_\{p\}(X)$ 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 $ℓ_\{p\}(ℓ_\{q\})$ with p,q ∈ (1,2) ∪ (2,∞) and obtain a complete characterization of its 1-complemented subspaces.},
author = {Bas Lemmens, Beata Randrianantoanina, Onno van Gaans},
journal = {Studia Mathematica},
keywords = {unconditional basis; contractive projections},
language = {eng},
number = {2},
pages = {149-166},
title = {Second derivatives of norms and contractive complementation in vector-valued spaces},
url = {http://eudml.org/doc/284800},
volume = {179},
year = {2007},
}

TY - JOUR
AU - Bas Lemmens
AU - Beata Randrianantoanina
AU - Onno van Gaans
TI - Second derivatives of norms and contractive complementation in vector-valued spaces
JO - Studia Mathematica
PY - 2007
VL - 179
IS - 2
SP - 149
EP - 166
AB - We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $ℓ_{p}(X)$, 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 $ℓ_{p}(X)$ 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 $ℓ_{p}(ℓ_{q})$ with p,q ∈ (1,2) ∪ (2,∞) and obtain a complete characterization of its 1-complemented subspaces.
LA - eng
KW - unconditional basis; contractive projections
UR - http://eudml.org/doc/284800
ER -