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Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Eric Amar, Andreas Hartmann (2010)

Annales de l’institut Fourier

We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and of other scales of spaces), changing the size of the space seems necessary to deduce unconditionality...

Uniformly convex operators and martingale type.

Jörg Wenzel (2002)

Revista Matemática Iberoamericana

The concept of uniform convexity of a Banach space was gen- eralized to linear operators between Banach spaces and studied by Beauzamy [1]. Under this generalization, a Banach space X is uniformly convex if and only if its identity map Ix is. Pisier showe

Uniformly convex spaces, bead spaces, and equivalence conditions

Lech Pasicki (2011)

Czechoslovak Mathematical Journal

The notion of a metric bead space was introduced in the preceding paper (L. Pasicki: Bead spaces and fixed point theorems, Topology Appl., vol. 156 (2009), 1811–1816) and it was proved there that every bounded set in such a space (provided the space is complete) has a unique central point. The bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces. It appears that normed bead spaces are identical with uniformly convex spaces. On the...

Uniformly Gâteaux Differentiable Norms in Spaces with Unconditional Basis

Rychter, Jan (2000)

Serdica Mathematical Journal

*Supported in part by GAˇ CR 201-98-1449 and AV 101 9003. This paper is based on a part of the author’s MSc thesis written under the supervison of Professor V. Zizler.It is shown that a Banach space X admits an equivalent uniformly Gateaux differentiable norm if it has an unconditional basis and X* admits an equivalent norm which is uniformly rotund in every direction.

Uniqueness of measure extensions in Banach spaces

J. Rodríguez, G. Vera (2006)

Studia Mathematica

Let X be a Banach space, B ⊂ B X * a norming set and (X,B) the topology on X of pointwise convergence on B. We study the following question: given two (non-negative, countably additive and finite) measures μ₁ and μ₂ on Baire(X,w) which coincide on Baire(X,(X,B)), does it follow that μ₁ = μ₂? It turns out that this is not true in general, although the answer is affirmative provided that both μ₁ and μ₂ are convexly τ-additive (e.g. when X has the Pettis Integral Property). For a Banach space Y not containing...

Uniqueness of unconditional bases in c 0 -products

P. Casazza, N. 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 ( ℓ 1 ) has a unique unconditional basis and a proof that c 0 ( ℓ 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.

Uniqueness of unconditional bases of c 0 ( l p ) , 0 < p < 1

C. Leránoz (1992)

Studia Mathematica

We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of c 0 ( l p ) must be equivalent to a permutation of a subset of the canonical unit vector basis of c 0 ( l p ) . In particular, c 0 ( l p ) has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for c 0 ( l ₁ ) .

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