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

Lemmens, Bas, van Gaans, Onno, Randrianantoanina, Beata (2007) Second derivatives of norms and contractive complementation in vector valued spaces. Studia Mathematica, 179 (2). pp. 149-166. ISSN 0039-3223. (doi:10.4064/sm179-2-3) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:28443)

 The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided. Official URLhttp://dx.doi.org/10.4064/sm179-2-3

## 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 is an element of (1,2) boolean OR (2, infinity). 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 is an element of (1,2) boolean OR (2, infinity) and obtain a complete characterization of its 1-complemented subspaces.

Item Type: Article 10.4064/sm179-2-3 Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Central Services Bas Lemmens 17 Nov 2011 16:14 UTC 16 Nov 2021 10:06 UTC https://kar.kent.ac.uk/id/eprint/28443 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-6713-7683