Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

It is shown that every echelon space λ_∞(A), with A an arbitrary Köthe matrix, is a Grothendieck space with the Dunford-Pettis property. Since λ_∞(A) is Montel if and only if it coincides with λ₀(A), this identifies an extensive class of non-normable, non-Montel Fréchet spaces having these two properties. Even though the canonical unit vectors in λ_∞(A) fail to form an unconditional basis whenever λ_∞(A) ≠ λ₀(A), it is shown, nevertheless, that in this case λ_∞(A) still admits unconditional Schauder decompositions (provided it satisfies the density condition). This is in complete contrast to the Banach space setting, where Schauder decompositions never exist. Consequences for spectral measures are also given.     

An Extension of the Notion of Orthogonality to Banach Spaces

We extend the usual notion of orthogonality to Banach spaces. We show that the extension is quite rich in structure by establishing some of its main properties and consequences. Geometric characterizations and comparison results with other extensions are established. Also, we establish a characterization of compact operators on Banach spaces that admit orthonormal Schauder bases. Finally, we characterize orthogonality in the spaces l₂^p(C).     

On shrinking and boundedly complete Schauder frames of Banach spaces

This paper studies Schauder frames in Banach spaces, a concept which is a natural generalization of frames in Hilbert spaces and Schauder bases in Banach spaces. The associated minimal and maximal spaces are introduced, as are shrinking and boundedly complete Schauder frames. Our main results extend the classical duality theorems on bases to the situation of Schauder frames. In particular, we will generalize James' results on shrinking and boundedly complete bases to frames. Secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames.     

Some remarks on James-Schreier spaces

The James-Schreier spaces Vp, where 1?p<∞, were recently introduced by Bird and Laustsen (in press) [5] as an amalgamation of James' quasi-reflexive Banach space on the one hand and Schreier's Banach space giving a counterexample to the Banach-Saks property on the other. The purpose of this note is to answer some questions left open in Bird and Laustsen (in press) [5]. Specifically, we prove that (i) the standard Schauder basis for the first James-Schreier space V₁ is shrinking, and (ii) any two Schreier or James-Schreier spaces with distinct indices are non-isomorphic. The former of these results implies that V₁ does not have Pe?czyński's property (u) and hence does not embed in any Banach space with an unconditional Schauder basis.     

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L¹(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Locally Asplund Preduals of Spaces of Holomorphic Functions

Defant [5] introduced the local Radon–Nikodym property for duals of locally convex spaces. This is a generalization of Asplund spaces as defined in Banach space theory. In this paper we generalise Dunford"s Theorem [7] to Banach spaces with Schauder decompositions and apply this result to spaces of holomorphic functions on balanced domains in a Banach space. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Characterization of Schauder Frames Which Are Near-Schauder Bases

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we give a solution to this problem using the notion of the minimal-associated sequence spaces and the minimal-associated reconstruction operators for Schauder frames. We prove that a Schauder frame is a near-Schauder basis if and only if the kernel of the minimal-associated reconstruction operator contains no copy of c ₀. In particular, a Schauder frame of a Banach space with no copy of c ₀ is a near-Schauder basis if and only if the minimal-associated sequence space contains no copy of c ₀. In these cases, the minimal-associated reconstruction operator has a finite dimensional kernel and the dimension of the kernel is exactly the excess of the near-Schauder basis. Using these results, we make related applications on Besselian frames and near-Riesz bases.     

A generalization of de Boor’s stability result and symmetric preconditioning

We provide a symmetric preconditioning method based on weighted divided differences which can be applied in order to solve certain ill-conditioned scattered-data interpolation problems in a stable way; more precisely, our method applies to cases where the ill-conditioning comes from the fact that the basis function is slowly growing, and the number of interpolation points is large. Concerning the theoretical background, an a priori unbounded operator on ?₂(?) is preconditioned so as to get a bounded and coercive operator. The method has another and probably even more interesting interpretation in terms of constructing certain Riesz bases of appropriate closed subspaces ofL ₂(?). In extending Mallat’s multiresolution analysis to the scattered data case, we construct nested sequences of spaces giving rise to orthogonal decompositions of functions inL ₂(?); in this way the idea of wavelet decompositions is (theoretically) carried over to scattered-data methods.     

Unconditional Schauder decompositions and stopping times in the Lebesgue-Bochner spaces

We extend Troitsky's study of martingales in Banach lattices to include stopping times. Results from the theory of unconditional Schauder decompositions and multipliers are used to derive an optional stopping theorem for unbounded stopping times. We also apply these techniques to convergent nets of stopped processes, as well as to unconditional Schauder decompositions in vector-valued Lp-spaces (1<p<∞).     

Atomic decompositions for tensor products and polynomial spaces

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product , for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X.     

Schauder decomposition and its application to integral equations of the second kind

The rates of convergence of Schauder decompositions in L_p and W_p⁽ⁿ⁾ are established. An application of Schauder decompositions to numerical solutions for integral of the second kind is made.     

Holomorphic automorphisms of complex Banach spaces

We consider holomorphic automorphisms of infinite dimensional complex Banach spaces. First we look at automorphisms with an attracting fixed point to construct Fatou–Bieberbach domains in Banach spaces. Second, we look tame sets in Banach spaces. Recall that a discrete set in X is tame if it can be mapped onto an arithmetic progression via an automorphism of X. We show that bounded discrete sets of Banach spaces allowing a Schauder basis are tame. In contrast, \(l_\infty \) has several bounded discrete sets which are not tame.     

Approximation Properties AP s and p-Nuclear Operators (the Case 0 < s ≤ 1)

We investigate Banach spaces possessing (or not possessing) the approximation properties AP _s, 0 < s ≤ 1, in connection with the following known question in the geometric theory of operators: under which conditions on Banach spaces X and Y and on positive numbers r and p does the p-nuclearity of the second adjoint of a continuous operator T from X to Y imply the p-nuclearity of T? Actually, we give necessary and sufficient conditions under which this question is answered affirmatively. In addition, the corresponding counterexamples are obtained in the maximally strong form. For instance, it is shown (and this statement is a significant strengthening of the previous results of that sort) that there exists a pair of separable Banach spaces Z and W such that the spaces Z ^** and W have Schauder bases, while for every p, 1 ≤ p < 2, there is a non-p-nuclear operator from W to Z with a p-nuclear second adjoint. Earlier, in similar examples, the corresponding spaces did not possess even the Grothendieck approximation property. The technique developed in this paper does not allow us to treat the case p > 2. That case will be studied in a forthcoming paper of the author. Bibliography: 11 titles.     

On the Banach spaces with the property (V*) of Pelczynski

Summary We consider the Banach spaces with the property (V*) of Pelczynski giving a sufficient condition for a Banach space to have this property as well as a characterization of Banach lattices with the same property. Several other results are given which are concerning relationships among that property and other famous isomorphic properties of Banach spaces. Also a characterization of Banach spaces with property (V*) using Schauder decompositions is given. Some result concerning lifting of that property from a Banach space E to L¹(, E) is presented, too.Work performed under the auspices of G.N.A.F.A. of C.N.R. and partially supported by M.P.I. of Italy (40%).     

Norm attaining operators and renormings of Banach spaces

We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc _o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+ε)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+ε)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.     

Two results in metric fixed point theory

We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space X into X, and the second with an application of the continuation method to the case where they satisfy the Leray–Schauder boundary condition 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.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

On certain Banach spaces in connection with interpolation theory

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper certain Banach spaces. A relation between these spaces and the space (C₀,S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.Finally, certain inequalities for the minimax series of a product are obtained which permit us to define a product over the space (C₀,S). With such a product (C₀,S) is a Banach Algebra.