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1
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Lipschitz and uniform embeddings into $ℓ_{∞}$

100%
Kalton N.
Fundamenta Mathematicae
|
2011
|
tom 212
|
nr 1
53-69
EN
We show that there is no uniformly continuous selection of the quotient map $Q: ℓ_{∞} → ℓ_{∞}/c₀$ 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 $B_X$.
2
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The complemented subspace problem revisited

100%
Kalton N.
Studia Mathematica
|
2008
|
tom 188
|
nr 3
223-257
EN
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.
3
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Rademacher series from Orlicz to the present day

100%
Kalton N.
Banach Center Publications
|
2004
|
tom 64
|
nr 1
105-116
EN
We survey some questions on Rademacher series in both Banach and quasi-Banach spaces which have been the subject of extensive research from the time of Orlicz to the present day.
4
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Best constants for Lipschitz embeddings of metric spaces into c₀

64%
Kalton N., Lancien G.
Fundamenta Mathematicae
|
2008
|
tom 199
|
nr 3
249-272
EN
We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c₀ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical $ℓ_p$-spaces into c₀ and give other applications. We prove that if a Banach space embeds almost isometrically into c₀, then it embeds linearly almost isometrically into c₀. We also study Lipschitz embeddings into c₀⁺.
5
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Lipschitz-free Banach spaces

64%
Godefroy G., Kalton N.
Studia Mathematica
|
2003
|
tom 159
|
nr 1
121-141
EN
We show that when a linear quotient map to a separable Banach space X has a Lipschitz right inverse, then it has a linear right inverse. If a separable space X embeds isometrically into a Banach space Y, then Y contains an isometric linear copy of X. This is false for every nonseparable weakly compactly generated Banach space X. Canonical examples of nonseparable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic are constructed. If a Banach space X has the bounded approximation property and Y is Lipschitz isomorphic to X, then Y has the bounded approximation property.
6
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Orbits in symmetric spaces, II

51%
Kalton N., Sukochev F., Zanin D.
Studia Mathematica
|
2010
|
tom 197
|
nr 3
257-274
EN
Suppose E is fully symmetric Banach function space on (0,1) or (0,∞) or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on f ∈ E so that its orbit Ω(f) is the closed convex hull of its extreme points. We also give an application to symmetrically normed ideals of compact operators on a Hilbert space.
7
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Best constants for some operators associated with the Fourier and Hilbert transforms

51%
Hollenbeck B., Kalton N., Verbitsky I.
Studia Mathematica
|
2003
|
tom 157
|
nr 3
237-278
EN
We determine the norm in $L^{p}(ℝ₊)$, 1 < p < ∞, of the operator $I - ℱ_{s}ℱ_{c}$, where $ℱ_{c}$ and $ℱ_{s}$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the $L^{p}$-norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best constants in other related inequalities are found. In a more general framework, we present an alternative proof of the important theorem of Cole relating best constant inequalities involving the Hilbert transform and the existence of subharmonic minorants, which extends to several variables and plurisubharmonic minorants.
8
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A new metric invariant for Banach spaces

51%
Baudier F., Kalton N., Lancien G.
Studia Mathematica
|
2010
|
tom 199
|
nr 1
73-94
EN
We show that if the Szlenk index of a Banach space X is larger than the first infinite ordinal ω or if the Szlenk index of its dual is larger than ω, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into X. We show that the converse is true when X is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.
9
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On the existence of almost greedy bases in Banach spaces

51%
Dilworth S., Kalton N., Kutzarova D.
Studia Mathematica
|
2003
|
tom 159
|
nr 1
67-101
EN
We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an almost greedy basis. We show that c₀ is the only $ℒ_{∞}$ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized.
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