Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.     

Embeddings of Lorentz spaces of vector-valued martingales

We obtain new embedding theorems for Lorentz spaces of vector-valued martingales, thus generalizing the classical martingale inequalities. In contrast to earlier methods, we use martingale transformations defined by sequences of operators and identify the operator S ^(p)(f) for a martingale f ranging in a Banach space X with the maximal operator for some ℓ ^p (X)-valued martingale transform. The obtained inequalities are closely related to geometric properties of the Banach space in question.     

James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C¹ Lipschitz bump function if and only if there exists another C¹ smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

On Banach spaces whose norm-open sets are -sets in the weak topology

We investigate non-separable Banach spaces whose norm-open sets are countable unions of sets closed in the weak topology and a narrower class of Banach spaces with a network for the norm topology which is σ-discrete in the weak topology. In particular, we answer a question of Arhangel'skii exhibiting various examples of non-separable function spaces C(K) with a σ-discrete network for the pointwise topology and (consistently) we answer some questions of Edgar and Oncina concerning Borel structures and Kadec renormings in Banach spaces.     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

Stampacchia’s Property,Self-duality and Orthogonality Relations

We show that if the conclusion of the well known Stampacchia Theorem on variational inequalities holds on a real Banach space X, then X is isomorphic to a Hilbert space. Motivated by this, we obtain a relevant result concerning self-dual Banach spaces and investigate some connections between properties of orthogonality relations, self-duality and Hilbert space structure. Moreover, we revisit the notion of the cosine of a linear operator and show that it can be used to characterize real Banach spaces that are isomorphic to a Hilbert space. Finally, we present some consequences of our results to quadratic forms and to evolution triples.     

On a problem of Namioka on norm-attaining functionals

Isaac Namioka conjectured that every nonreflexive Banach space can be renormed is such a way that, in the new norm, the set of norm attaining functionals has an empty interior in the norm topology. We prove the rightness of this conjecture for spaces containing an isomorphic copy of ℓ₁. As a consequence, we prove also that the same result holds for a wide class of Banach spaces containing, for example, the weakly compactly generated ones.     

Almost Everywhere Convergent Fourier Series

We study some properties of the logconvex quasi-Banach space QA defined by Arias-de-Reyna and show several applications to convergence of Fourier series. In particular, we describe the Banach envelope of QA and prove that there exists a Lorentz space strictly bigger than the Antonov space in which the almost everywhere convergence of the Fourier series holds. We also give a necessary condition for a Banach rearrangement invariant space X to be contained in QA. As an application, we show that for some classes of Banach spaces there is no the largest Banach space in a given class which is contained in QA.     

Polynomial reflexivity in Banach spaces

We ask when the space ofN-homogeneous analytic polynomials on a Banach space is reflexive. This turns out to be related to whether polynomials are weakly sequentially continuous, and to the geometry of spreading models. For example, if these spaces are reflexive for allN, no quotient of the dual space may have a spreading model with an upperq-estimate, and every bounded holomorphic function on the unit ball has a Taylor series made up of weakly sequentially continuous polynomials (we assume the approximation property). Alencar, Aron and Dineen [AAD] gave the first example of some properties of a polynomially reflexive space (usingT*, the original Tsirelson space); we show that these properties and others are shared by all polynomially reflexive spaces. This paper forms a portion of the Ph. D. dissertation of the author, under the supervision of W. B. Johnson.     

Isomorphic Schauder decompositions in certain Banach spaces

We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ?_ψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.     

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X). BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332.     

Boundedly complete basic sequences,c 0-subspaces,and injections of Banach spaces

We study the connection between topological properties of subsets of a given Banach space and their images under linear, continuous one-to-one mappings on the one hand and the existence in a given Banach space of either a boundedly complete basic sequence (BCBS) or an isomorphic copy ofc _o (c _o -subspace) on the other hand. We present criteria for the existence of a BCBS. They are deduced from new characterisations ofG _δ-embeddings which we also present. We obtain a necessary and sufficient condition for separability of a dual Banach space in terms of saturation by BCBS. Criteria for the existence in a Banach space of ac _o -subspace are also presented. We describe the class of separable Banach spaces which contains either a BCBS or ac _o -subspace. This research was supported by the Rashi Foundation.     

Shearlet coorbit spaces and associated Banach frames

In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.     

Banach Spaces with the PC Property

A Banach space X possesses the PC (point of continuity) property if for any w-closed bounded subset A ? X the identity map (A,w)→(A, ∥ ? ∥) has a point of continuity (w is the weak topology in X). We deduce some criteria for Banach spaces to have the PC property and describe (for dual Banach spaces) relationships between spaces possessing the PC property and spaces possessing the RN or the WRN property.

     

Deformation theorems on non‐metrizable vector spaces and applications to critical point theory

Let E be a Banach space and Φ : E → ? a ??¹‐functional. Let ?? be a family of semi‐norms on E which separates points and generates a (possibly non‐metrizable) topology ??_?? on E weaker than the norm topology. This is a special case of a gage space, that is, a topological space where the topology is generated by a family of semi‐metrics. We develop some critical point theory for Φ : (E, ??) → ?. In particular, we prove deformation lemmas where the deformations are continuous with respect to ??_??. In applications this yields a gain in compactness when Φ does not satisfy the Palais–Smale condition because one can work with the weak topology. We also prove some foundational results on gage spaces. In particular, we introduce the concept of Lipschitz continuity in this setting and prove the existence of Lipschitz continuous partitions of unity. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Gelfand-Phillips Property with Respect to the Weak Topology

We consider a Gelfand-Phillips type property for the weak topology. The main results that we obtain are (1) for certain Banach spaces, E^?_? F inherits this property from E and F, and (2) the spaces L^p(μ, E) have this property when E does. A subset A of a Banach space E is a limited set if every (bounded linear) operator T:E → c₀ maps A onto a relatively compact subset of c₀. The Banach space E has the Gelfand-Phillips property if every limited set is relatively compact. In this note, we study the analogous notions set in the weak topology. Thus we say that A ? E is a Grothendieck set if every T: E → c₀ maps A onto a relatively weakly compact set; and E is said to have the weak type GP property if every Grothendieck set in E is relatively weakly compact. In the papers [3, 4 and 6], it is shown among other results that the ?-tensor product E and the spaces L^p(μ, E) inherit the Gelfand-Phillips property from E and F. In this paper, we study the same questions for the weak type GP property. It is easily verified that continuous linear images of Grothendieck sets are Grothendieck and that the weak type GP property is inherited by subspaces. Among the spaces with the weak type GP property one easily finds the separable spaces, and more generally, spaces with a weak* sequentially compact dual ball. Also, C(K) spaces where K is (DCSC) are weak type GP (see [3] and the discussion before Corollary 4 below). A Grothendieck space (a Banach space whose unit ball is a Grothendieck set) has the weak type GP if and only if it is reflexive.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

On a new generalization of metric spaces

In this paper, we introduce the \({\mathcal {F}}\)-metric space concept, which generalizes the metric space notion. We define a natural topology \(\tau _{{\mathcal {F}}}\) in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of \({\mathcal {F}}\)-metric spaces. Several examples are presented to illustrate our study.