# [Banach] Abstract of a paper by Bas Lemmens, Beata Randrianantoanina, and Onno van Gaans

Dale Alspach alspach at www.math.okstate.edu
Thu Nov 3 07:16:30 CST 2005

This is an announcement for the paper "Second derivatives of norms
and contractive complementation in vector-valued spaces" by Bas
Lemmens, Beata Randrianantoanina, and Onno van Gaans.

Abstract: 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.

Archive classification: Functional Analysis

Mathematics Subject Classification: 46B45, 46B04 (Primary) 47B37
(Secondary)

Remarks: 22 pages, LaTeX

The source file(s), lplqsub.tex: 52714 bytes, is(are) stored in
gzipped form as 0511044.gz with size 15kb. The corresponding postcript
file has gzipped size 80kb.

Submitted from: lemmens at maths.warwick.ac.uk

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http://arXiv.org/abs/math.FA/0511044

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