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

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Lemmens B., Randrianantoanina B., van Gaans O.

Studia Mathematica

2007

tom 179

nr 2

149-166

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 ∈ (1,2) ∪ (2,∞). 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 ∈ (1,2) ∪ (2,∞) and obtain a complete characterization of its 1-complemented subspaces.

Ograniczanie wyników

1 Studia Mathematica

1 Lemmens B.

1 Randrianantoanina B.

1 van Gaans O.

1 2007

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