Embedding Banach spaces with unconditional bases into spaces with symmetric bases

Every reflexive Banach space with unconditional basis is isomorphic to a complemented subspace of a reflexive Banach space with symmetric basis.     

On unconditional bases in certain Banach function spaces

Let X be a Banach space, L ([0,1])XL ¹([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f _n} _m=1 , where f _n in C([0,1]), nN. In this paper, we introduce the notion that X ^*has the singularity property of X ^*at a point t ₀[0,1]. It is proved that if X ^*has the singularity property at a point t ₀ [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.     

Polynomials on Banach spaces with unconditional bases

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

     

The C-convexity of Banach spaces with unconditional bases

A Banach space is called C-convex if the space c₀ cannot be represented finitely in it. Necessary and sufficient conditions for the C-convexity of a space with an unconditional basis and of the product of a space Y with respect to the unconditional basis of a space X are obtained. These conditions are rendered concrete for two classes of spaces: The Orlich space of sequences is C-convex if and only if its normalizing function satisfies the δ₂-condition; the Lorentz space of sequences is C-convex if and only if its normalizing sequence satisfies the condition \(\mathop {\underline {\lim } }\limits_{n \to \infty } {{\sum\nolimits_{i = 1}^{2n} {c_i } } \mathord{\left/ {\vphantom {{\sum\nolimits_{i = 1}^{2n} {c_i } } {\sum\nolimits_{i = 1}^n {c_i > 1} }}} \right. \kern-0em} {\sum\nolimits_{i = 1}^n {c_i > 1} }}\) . We call a Banach space X a C-convex space if the following condition is fulfilled: $$\mathop {\sup }\limits_n \inf d\left( {X_n , l_\infty ^n } \right) = \infty $$ ,     

Uniqueness of unconditional bases in Banach spaces

We prove a general results on complemented unconditional basic sequences in Banach lattices and apply it to give some new examples of spaces with unique unconditional basis. We show that Tsirelson space and certain Nakano spaces have unique unconditional bases. We also construct an example of a space with a unique unconditional basis with a complemented subspace failing to have a unique unconditional basis. Both authors were supported by NSF Grant DMS-9201357.     

On banach spaces with unconditional bases

It is shown that for every space with an unconditional basis there exists a uniformly bounded sequence of projectionsP _n;n=1, 2, ... whose ranges are uniformly isomorphic tol _p ⁿ ;n=1, 2, ... either forp=1, orp=2, or forp=∞.     

Subspaces and quotients of Banach spaces with shrinking unconditional bases

The main result is that a separable Banach space with the weak* unconditional tree property is isomorphic to a subspace as well as a quotient of a Banach space with a shrinking unconditional basis. A consequence of this is that a Banach space is isomorphic to a subspace of a space with a shrinking unconditional basis if and only if it is isomorphic to a quotient of a space with a shrinking unconditional basis, which solves a problem dating to the 1970s. The proof of the main result also yields that a uniformly convex space with the unconditional tree property is isomorphic to a subspace as well as a quotient of a uniformly convex space with an unconditional finite dimensional decomposition.     

Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

LetX be a Banach space with an unconditional finite-dimensional Schauder decomposition (E _n). We consider the general problem of characterizing conditions under which one can construct an unconditional basis forX by forming an unconditional basis for eachE _n. For example, we show that if sup _n dimE _n<∞ andX has Gordon-Lewis local unconditional structure thenX has an unconditional basis of this type. We also give an example of a non-Hilbertian spaceX with the property that wheneverY is a closed subspace ofX with a UFDD (E _n) such that sup _n dimE _n<∞ thenY has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved. Both authors were supported by NSF Grant DMS-9201357.     

Embedding spaces with unconditional bases

Every super-reflexive space with an unconditional basis is isomorphic to a complemented subspace of a super-reflexive space with a symmetric basis.     

On Zippin's Embedding Theorem of Banach spaces into Banach spaces with bases

We present a new proof of Zippin's Embedding Theorem, that every separable reflexive Banach space embeds into one with shrinking and boundedly complete basis, and every Banach space with a separable dual embeds into one with a shrinking basis. This new proof leads to improved versions of other embedding results.     

On equivalent bases in Banach spaces

We present several generalizations of the classical Bari theorem on the Riesz basis property of close systems in Hilbert spaces to Banach spaces. We introduce the corresponding definitions and formulate theorems on the basis property of close systems in Banach spaces. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 551–554, April, 2007.     

On unconditional bases of reproducing kernels in Fock-type spaces

The existence of unconditional bases of reproducing kernels in the Fock-type spaces F _φ with radial weights φ is studied. It is shown that there exist functions φ(r) of arbitrarily slow growth for which ln r = o(φ(r)) as r → ∞ and there are no unconditional bases of reproducing kernels in the space F _φ . Thus, a criterion for the existence of unconditional bases cannot be given only in terms of the growth of the weight function.     

Banach spaces without local unconditional structure

For a large class of Banach spaces, a general construction of subspaces without local unconditional structure is presented. As an application it is shown that every Banach space of finite cotype contains eitherl ₂ or a subspace without unconditional basis, which admits a Schauder basis. Some other interesting applications and corollaries follow. The contribution of this author is a part of his Ph.D. Thesis written at the University of Alberta under the supervision of the second author. An erratum to this article is available at .     

On unconditional polynomial bases inL p and bergman spaces

In this paper we consider unconditional bases inL _p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH _p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.     

On perfectly homogeneous bases in Banach spaces

It is proved that a Banach space is isomorphic toc _o or tol _p if and only if it has a normalized basis {χ_i i _{} i=1} ^∞ which is equivalent to every normalized block-basis with respect to {χ_i i _{} i=1} ^∞ . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful guidance and for the interest he showed in the paper, and the referee for his valuable remakrs.     

Absolutely summing polynomials on Banach spaces with unconditional basis

If X is a Banach space with a normalized unconditional Schauder basis (xn), we define whenever and obtain estimates for μX,(xn) when every continuous m-homogeneous polynomial from X into Y is absolutely (q,1) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m=1, we obtain new contributions to the linear theory of absolutely summing operators.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

On unconditional bases in spaces of holomorphic functions in infinite dimensions

This paper was written during the first author's stay at the Department of Mathematics of the Universidade Federal do Rio de Janeiro. He wants to thank the Department for the kind invitation and the CAPES/DAAD for its financial support.     

A class of Banach spaces with no unconditional basic sequence

We give a construction of a reflexive Banach space X_ω1 with a transfinite basis of length ω₁ and with no unconditional basic sequence. In addition every bounded operator from a subspace of X_ω1 into the space X_ω1 is a sum of a simple diagonal operator and a strictly singular one. To cite this article: S.A. Argyros et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).