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.     

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.     

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.     

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.     

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.     

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.     

On a new geometric property for Banach spaces

In this paper we study a geometric property for Banach spaces called condition (*), introduced by de Reynaet al in [3], A Banach space has this property if for any weakly null sequencex _n of unit vectors inX, ifx _* ⁿ is any sequence of unit vectors inX ^* that attain their norm at x_n’s, then . We show that a Banach space satisfies condition (*) for all equivalent norms iff the space has the Schur property. We also study two related geometric conditions, one of which is useful in calculating the essential norm of an operator.     

Geometric mean and triangles inscribed in a semicircle in Banach spaces

We consider the triangles with vertices x, −x and y where x,y are points on the unit sphere of a normed space. Using the geometric means of the variable lengths of the sides of these triangles, we define two geometric constants for Banach spaces. These constants are closely related to the modulus of convexity of the space under consideration, and they seem to represent a useful tool to estimate the exact values of the James and Jordan–von Neumann constants of some Banach spaces.     

A second-order Cauchy problem in a scale of Banach spaces and application to Kirchhoff equations

In a scale of Banach spaces we study the Cauchy problem for the equation u^′′=A(Bu(t),u), where A is a bilinear operator and B is a completely continuous operator. Obtained results are applied to prove existence of solutions in the Gevrey class for Kirchhoff equations.     

Even infinite-dimensional real Banach spaces

This article is a continuation of a paper of the first author [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about complex structures on real Banach spaces. We define a notion of even infinite-dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] and C(K) examples due to Plebanek [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We extend results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis [S. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159 (1) (2003) 1–32] provides examples of essentially incomparable complex structures which are not totally incomparable.     

Interpolation properties of scales of Banach spaces

It is shown that any interpolation scales joining weight spaces L _p or similar spaces have many remarkable properties. Not only are such scales intrinsically interpolation scales, but an analog of the Arazy-Cwikel theorem describing interpolation spaces between the spaces from the scale is valid.     

Colocality and twisted sums of Banach spaces

Using the relation between subspaces of Banach spaces and quotients of their duals, we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role (Theorem 1.7). An interesting point is that no restrictions are imposed on the quotients, only on the various subspaces. New examples of nontrivial twisted sums are given.