Abstract of a paper by Bas Lemmens, Beata Randrianantoanina, and Onno van Gaans
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@maths.warwick.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0511044 or http://arXiv.org/abs/math.FA/0511044 or by email in unzipped form by transmitting an empty message with subject line uget 0511044 or in gzipped form by using subject line get 0511044 to: math@arXiv.org.
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Dale Alspach