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

Tom 179 / 2007

Bas Lemmens, Beata Randrianantoanina, Onno van Gaans Studia Mathematica 179 (2007), 149-166 MSC: Primary 46B45, 46B04; Secondary 47B37. DOI: 10.4064/sm179-2-3


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.


  • Bas LemmensMathematics Institute
    University of Warwick
    CV4 7AL Coventry, United Kingdom
  • Beata RandrianantoaninaDepartment of Mathematics and Statistics
    Miami University
    Oxford, OH 45056, U.S.A.
  • Onno van GaansMathematical Insitute
    Leiden University
    P.O. Box 9512
    2300 RA Leiden, The Netherlands

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