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