On property (β) in Banach lattices, Calderón-Lozanowskii and Orlicz-Lorentz spaces

The geometry of Calderón-Lozanowskii spaces, which are strongly connected with the interpolation theory, was essentially developing during the last few years (see [4, 9, 10, 12, 13, 17]). On the other hand many authors investigated property (β) in Banach spaces (see [7, 19, 20, 21, 25, 26]). The first aim of this paper is to study property (β) in Banach function lattices. Namely a criterion for property (β) in Banach function lattice is presented. In particular we get that in Banach function lattice property (β) implies uniform monotonicity. Moreover, property (β) in generalized Calderón-Lozanowskii function spaces is studied. Finally, it is shown that in Orlicz-Lorentz function spaces property (β) and uniform convexity coincide.     

Multilevel decompositions of functional spaces

A unified abstract framework for the multilevel decomposition of both Banach and quasi-Banach spaces is presented. The characterization of intermediate spaces and their duals is derived from general Bernstein and Jackson inequalities. Applications to compactly supported biorthogonal wavelet decompositions of families of Besov spaces are also given. The first author was partially supported by grants from MURST (40% Analisi Numerica) and ASI (Contract ASI-92-RS-89), whereas the second author was partially supported by grants from MURST (40% Analisi Funzionale) and CNR (Progetto Strategico “Applicazioni della Matematica per la Tecnologia e la Società”).     

Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L_X^p, where X is a Banach space and 1≤ p<∞, and extend the result to vector-valued Banach function spaces E_X, where E is a Banach function space with order continuous norm. The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281.     

On the Lindenstrauss-Rosenthal theorem

We present a homological principle that governs the behaviour of couples of exact sequences of quasi-Banach spaces. Three applications are given: (i) A unifying method of proof for the results of Lindenstrauss, Rosenthal, Kalton, Peck and Kislyakov about the extension and lifting of isomorphisms inc ₀,ι _∞,ι _p andL _pfor 0<p≤1; (ii) A study of the Dunford-Pettis property in duals of quotients ofL _∞-spaces; and (iii) New results on the extension ofC(K)-valued operators. The research has been supported in part by DGICYT project BFM 2001-0813.     

Affine Approximation of Lipschitz Functions and Nonlinear Quotients

New concepts related to approximating a Lipschitz function between Banach spaces by affine functions are introduced. Results which clarify when such approximations are possible are proved and in some cases a complete characterization of the spaces X, Y for which any Lipschitz function from X to Y can be so approximated is obtained. This is applied to the study of Lipschitz and uniform quotient mappings between Banach spaces. It is proved, in particular, that any Banach space which is a uniform quotient of L _p , 1 < p < , is already isomorphic to a linear quotient of L _p . Submitted: June 1998, revised: December 1998.     

Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and f ∈ L _p(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No. M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No. T043769, T047128, T047132.     

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch?s (p,r,s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q,p)-dominated operators.     

Fixräume regulärer operatoren und ein Korovkin-Satz

S. J. Bernau has introduced the notion of an exchange subspace of an L_p-space and has shown that the range of a contractive linear projection on an L_p-space (1 ? p < ∞, p ≠ 2) is an exchange subspace. In the present paper we define this notion for real Banach lattices with order continuous norm and prove among other things that fixed spaces of special regular operators on these spaces are exchange subspaces. As application we give a Korovkin theorem for sequences of contractions on real Banach lattices with an uniformly monotone norm.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

Exposed and denting points in duals of operator spaces

We study the extremal structure of the dual unit balls of various operator spaces. Mainly, we show that the classes of [w*-] strongly exposed, [w*-] exposed, and denting points in the dual unit balls of spaces of compact operators between Banach spacesX andY are completely — and in a canonical way — determined by the corresponding classes of points in the unit balls of the (bi-)duals of the factor spacesX andY. Applications to the duality of operator spaces and differentiability properties of the norm in operator spaces are given.     

A class of Banach sequence spaces analogous to the space of Popov

Hagler and the first named author introduced a class of hereditarily l ₁ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily l _p Banach spaces for 1 ⩽ p < ∞. Here we use these spaces to introduce a new class of hereditarily l _p (c ₀) Banach spaces analogous of the space of Popov. In particular, for p = 1 the spaces are further examples of hereditarily l ₁ Banach spaces failing the Schur property.     

Some approximation properties of Banach spaces and Banach lattices

The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is a ℒ _∞-space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a surjection, X is a ℒ ₁-space and Z is a ℒ _p -space (1 ≤ p ≤ ∞), then ker Q has the UAP. A complemented subspace of a weakly sequentially complete Banach lattice has the separable complementation property = SCP. A criterion for a space with GL-l.u.st. to have the SCP is given. Spaces which are quotients of weakly sequentially complete lattices and are uncomplemented in their second duals are studied. Examples are given of spaces with the SCP which have subspaces that fail the SCP. The results are applied to spaces of measures on a compact Abelian group orthogonal to a fixed Sidon set and to Sobolev spaces of functions of bounded variation on ℝ ⁿ .     

Extension of multilinear maps defined on subspaces

We study the problem of whether every multilinear form defined on the product of n closed subspaces has an extension defined on the product of the entire Banach spaces. We prove that the property derived from this condition (the Multilinear Extension Property) is local. We use this to prove that, for a wide variety of Banach spaces, there exist a product of closed subspaces and a multilinear form defined on it, which has no extension to the product of the entire spaces. We show that the ℓ _p spaces, with 1 ≤p ≤ ∞ and p ≠ 2, are among them and, more generally, every Banach space which fails to have type p for some p < 2 or cotype q for some q > 2.     

An alternative approach to the uniqueness of unconditional basis of for

We give an alternative and much simpler proof of the uniqueness of unconditional basis (up to equivalence and permutation) in the quasi-Banach spaces ℓ_p(c₀) for 0<p<1 and its complemented subspaces with unconditional basis. The new approach uses the fact that the Banach envelope of these spaces is not sufficiently Euclidean with the lattice structure induced by its unconditional basis.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Metrical characterization of super-reflexivity and linear type of Banach spaces

We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain’s result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniform embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with the metrics d _p induced by the ℓ _p norms. Received: 2 August 2006, Revised: 10 April 2007     

Summability Properties for Multiplication Operators on Banach Function Spaces

Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized Köthe duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with ${x \in X}Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized K?the duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with x ? X{x \in X} and y ? Y{y \in Y} .     

Algebra descent spectrum of operators

We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ₁, all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular, by the hereditarily indecomposable Banach spaces [8]), but not by ℓ _p , for 1 < p ≤ ∞ with p ≠ 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE.     

Universal K?the Sequence Spaces

 A characterization is given for the K?the matrices B such that the K?the sequence space , with , contains all K?the sequence spaces of order p as subspaces. It follows that the class of K?the sequence spaces of order p has a universal element which is quasinormable. In particular, there is a quasinormable space (respectively, which contains every nuclear Fréchet space with basis (respectively, every countably normed Fréchet Schwartz space). Only Fréchet spaces with continuous norm are considered in this note.