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Calderón couples of rearrangement invariant spaces

N. Kalton — 1993

Studia Mathematica

We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with L . We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair ( L F , L ) forms a...

On vector-valued inequalities for Sidon sets and sets of interpolation

N. Kalton — 1993

Colloquium Mathematicae

Let E be a Sidon subset of the integers and suppose X is a Banach space. Then Pisier has shown that E-spectral polynomials with values in X behave like Rademacher sums with respect to L p -norms. We consider the situation when X is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if E is a set of interpolation ( I 0 -set). However, for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if X is restricted...

Uniqueness of unconditional bases in c 0 -products

P. CasazzaN. Kalton — 1999

Studia Mathematica

We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does c 0 ( X ) . We also give some positive results including a simpler proof that c 0 ( 1 ) has a unique unconditional basis and a proof that c 0 ( p n N n ) has a unique unconditional basis when p n 1 , N n + 1 2 N n and ( p n - p n + 1 ) l o g N n remains bounded.

Unconditional ideals in Banach spaces

G. GodefroyN. KaltonP. Saphar — 1993

Studia Mathematica

We show that a Banach space with separable dual can be renormed to satisfy hereditarily an “almost” optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition X * * * = X X * is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel X . We undertake a general study of h-ideals and u-ideals. For example we show that...

The complemented subspace problem revisited

N. J. Kalton — 2008

Studia Mathematica

We show that if X is an infinite-dimensional Banach space in which every finite-dimensional subspace is λ-complemented with λ ≤ 2 then X is (1 + C√(λ-1))-isomorphic to a Hilbert space, where C is an absolute constant; this estimate (up to the constant C) is best possible. This answers a question of Kadets and Mityagin from 1973. We also investigate the finite-dimensional versions of the theorem.

Lipschitz and uniform embeddings into

N. J. Kalton — 2011

Fundamenta Mathematicae

We show that there is no uniformly continuous selection of the quotient map Q : / c relative to the unit ball. We use this to construct an answer to a problem of Benyamini and Lindenstrauss; there is a Banach space X such that there is a no Lipschitz retraction of X** onto X; in fact there is no uniformly continuous retraction from B X * * onto B X .

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