On Uniformly Finitely Extensible Banach spaces

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno and Plichko (2009) [39] and Castillo and Plichko (2010) [18]. We show that they have the Uniform Approximation Property of Pe?czyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal – do there exist automorphic spaces other than _c0(I)$c_0(I)$ and _?2(I)${?}_2(I)$? – showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI – among them, the super-reflexive HI space constructed by Ferenczi – and asymptotically _?2${?}_2$ spaces in the literature cannot be automorphic.     

Banach spaces without minimal subspaces

We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ₁ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability.     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Isometric embeddings of compact spaces into Banach spaces

We show the existence of a compact metric space K such that whenever K embeds isometrically into a Banach space Y, then any separable Banach space is linearly isometric to a subspace of Y. We also address the following related question: if a Banach space Y contains an isometric copy of the unit ball or of some special compact subset of a separable Banach space X, does it necessarily contain a subspace isometric to X? We answer positively this question when X is a polyhedral finite-dimensional space, c₀ or ?₁.     

On non-fragmentability of Banach spaces

In this paper, we use the game characterization of Kenderov and Moors [11] to construct an example of a non-fragmentable Banach space. More precisely, we will show that ifX is the tree-complete Banach algebra of Haydon and Zizler [3], (X/c ₀, weak) is not fragmentable by any metric. In particular, this shows thatX/c ₀ cannot be equivalently renormed to be rotund.     

On the central limit theorem in Banach spaces

It is shown, with the use of a concentration inequality of LeCam, that associated with an infinitely divisible random variable with values in a separable Banach space there is a Lévy-Khintchine formula. A partial converse of this fact is also proved. Relations between the continuity of the compound Poisson and the Gaussian variables associated with a Lévy measure are studied. A central limit theorem is obtained and examples are given.     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

On a problem of Rolewicz about Banach spaces that admit support sets

We construct an example of a nonseparable Banach space which does not admit a support set.² It is a consistent (and necessarily independent from the axioms of ZFC) example of a space C(K) of continuous functions on a compact Hausdorff K with the supremum norm. The construction depends on a construction of a Boolean algebra with some combinatorial properties. The space is also hereditarily Lindelöf in the weak topology but it doesn't have any nonseparable subspace nor any nonseparable quotient which is a C(K) space for K dispersed.     

On complemented subspaces of sums and products of Banach spaces

It is proved that there exist complemented subspaces of countable topological products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces.

     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

A universal property of reflexive hereditarily indecomposable Banach spaces

It is shown that every separable Banach space universal for the class of reflexive Hereditarily Indecomposable space contains isomorphically and hence it is universal for all separable spaces. This result shows the large variety of reflexive H.I. spaces.

     

Frame expansions in separable Banach spaces

Banach frames are defined by straightforward generalization of (Hilbert space) frames. We characterize Banach frames (and Xd-frames) in separable Banach spaces, and relate them to series expansions in Banach spaces. In particular, our results show that we can not expect Banach frames to share all the nice properties of frames in Hilbert spaces.     

Bases in spaces of homogeneous polynomials on Banach spaces

In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces.     

Kergin approximation in Banach spaces

We explore the convergence of Kergin interpolation polynomials of holomorphic functions in Banach spaces, which need not be of bounded type. We also investigate a case where the Kergin series diverges.     

The Banach envelope of Paley-Wiener type spaces

We give an explicit computation of the Banach envelope for the Paley-Wiener type spaces . This answers a question by Joel Shapiro.

     

A characterization of reflexive Banach spaces

A Banach space is not reflexive if and only if there exist a closed separable subspace of and a convex closed subset of with empty interior which contains translates of all compact sets in . If, moreover, is separable, then it is possible to put .

     

Complementarity problems for multivalued monotone operator in Banach spaces

We utilize S. Park's maximal element theorem in this paper to prove the existence theorems of solutions of the complementarity problems for multivalued monotone operator in Banach spaces.     

Coefficient quantization for frames in Banach spaces

Let (ei) be a fundamental system of a Banach space. We consider the problem of approximating linear combinations of elements of this system by linear combinations using quantized coefficients. We will concentrate on systems which are possibly redundant. Our model for this situation will be frames in Banach spaces.