Antiproximinal Norms in Banach Spaces

We prove that every Banach space containing a complemented copy of c₀ has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast, it is shown that there are no antiproximinal norms in Banach spaces with the convex point of continuity property (CPCP). Other questions related to the existence of antiproximinal bodies are also discussed.     

A renorming of some nonseparable Banach spaces with the Fixed Point Property

We prove that for every Banach space which can be embedded in c₀(Γ) (for instance, reflexive spaces or more generally spaces with M-basis) there exists an equivalent renorming which enjoys the (weak) Fixed Point Property for non-expansive mappings. As a consequence, we solve a longtime open question in Metric Fixed Point Theory: Every reflexive Banach can be renormed to satisfy the Fixed Point Property. Furthermore, this norm can be chosen arbitrarily closed to the original norm.     

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.     

Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

We provide a short characterization of p-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for \(1<p<\infty \), the class of p-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal.     

Some estimates in spaces

We consider a class of rearrangement invariant Banach Function Spaces recently appeared in a paper by the same authors, containing at the same time some Lorentz spaces Γ(ν), classical Lebesgue spaces and small Lebesgue spaces. We discuss the main properties coming directly from the norm, and, for certain values of the involved parameters, we prove some estimates of the norm of the associate space.     

Additions to the Periodic Decomposition Theorem

A pair of linear bounded commuting operators T₁, T₂ in a Banach space is said to possess a decomposition property (DePr) if Ker (I-T₁)(I-T₂) = Ker (I-T₁) + Ker (I-T₂). A Banach space X is said to possess a 2-decomposition property (2-DePr) if every pair of linear power bounded commuting operators in X possesses the DePr. It is known from papers of M. Laczkovich and Sz. Révész that every reflexive Banach space X has the 2-DePr. In this paper we prove that every quasi-reflexive Banach space of order 1 has the 2-DePr but not all quasi-reflexive spaces of order 2. We prove that a Banach space has no 2-DePr if it contains a direct sum of two non-reflexive Banach spaces. Also we prove that if a bounded pointwise norm continuous operator group acts on X then every pair of operators belonging to it has a DePr. A list of open problems is also included. This revised version was published online in August 2006 with corrections to the Cover Date.     

Extension of vector-valued integral polynomials

We study the extendibility of integral vector-valued polynomials on Banach spaces. We prove that an X-valued Pietsch-integral polynomial on E extends to an X-valued Pietsch-integral polynomial on any space F containing E, with the same integral norm. This is not the case for Grothendieck-integral polynomials: they do not always extend to X-valued Grothendieck-integral polynomials. However, they are extendible to X-valued polynomials. The Aron-Berner extension of an integral polynomial is also studied. A canonical integral representation is given for domains not containing ?₁.     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Geometry of Banach spaces with property β

We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L ₁[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact space To the memory of A. Plans Supported in part by DGICYT grant PB 94-0243 and DGICYT PB 96-0607.     

Banach Spaces with Property (M) and their Szlenk Indices

We show that a separable Banach space with property (M*) has a Szlenk index equal to ω₀, and a norm with an optimal modulus of asymptotic uniform smoothness. From this we derive a condition on the Szlenk functions of the space and its dual which characterizes embeddability into c ₀ or an ℓ _p -sum of finite dimensional spaces. We also prove that two Lipschitz-isomorphic Orlicz sequence spaces contain the same ℓ _p -spaces.      

A characterization of reflexive spaces

We prove that a Banach space is reflexive if for every equivalent norm, the set of norm attaining functionals has non-empty norm-interior in the dual space. It is also proved that the set of norm attaining functionals on a Banach space that is not a Grothendieck space is not a w*-G _δ subset of the dual space.     

On Minimal Subspaces in Tensor Representations

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.     

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.     

Banach spaces with many boundedly complete basic sequences failing PCP

We prove that there exist Banach spaces not containing ?₁, failing the point of continuity property and satisfying that every semi-normalized basic sequence has a boundedly complete basic subsequence. This answers in the negative the problem of Remark 2 in Rosenthal (2007) [12].     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.     

Octahedral norms and convex combination of slices in Banach spaces

We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of w^?$w^{?}$-slices in the dual unit ball has diameter 2, which answers an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing _?1${?}_1$ and for every ε>0$\epsilon >0$ there is an equivalent norm so that every convex combination of w^?$w^{?}$-slices in the dual unit ball has diameter at least 2−ε$2-\epsilon$.     

Weakly stable Banach spaces

The class of stable Banach spaces, inspired by the stability theory in mathematical logic, was introduced by Krivine and Maurey and provided the proper context for the abstract formulation of Aldous’ result of subspaces ofL ¹. In this paper we study the wider class of weakly stable Banach spaces, where the exchangeability of the iterated limits occurs only for sequences belonging to weakly compact subsets, introduced independently by Garling (in an earlier unpublished version of his expository paper on stable Banach spaces brought recently to our attention) and by the authors. Taking into account Rosenthal’s application of the study of pointwise compact sets of Baire-1 functions (Rosenthal compact spaces) in the study of Banach spaces (for whichl ¹ does not embed isomorphically) and of the study of Rosenthal compact sets by Rosenthal and Bourgain-Fremlin-Talagrand, we prove the following analogue of the Krivine-Maurey theorem for weakly stable spaces:If X is infinite dimensional and weakly stable then either l ^p for some p≧1or c_o embeds isomorphically in X (§1). Garling (in the above reference) proved this result under the additional assumption thatX* is separable. We also construct an example of a Banach spaceX which is weakly stable, without an equivalent stable norm, and such thatl ² embeds isomorphically in every infinite dimensional subspace ofX (§3).     

Composition and nuclearity of kernel operators

Let T and S be kernel operators of finite double norm, respectively finite inverse double norm. We prove that the compositions TS and ST are nuclear operators, provided one of the function norms is order continuous. Conversely we prove theorems about factorizations of nuclear kernel operators on Banach function spaces. These results are then used to derive optimal eigenvalue estimates for nuclear operators on Banach lattices by relating Pisier's results with the results of König and Weis.     

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.     

Asymptotic Behavior of Factorization and Projection Constants

This paper is concerned with estimates of important factorization constants that appear in Banach space theory. We prove upper bounds of the Hilbertian norm of projections on finite-dimensional spaces of interpolation spaces generated by certain abstract interpolation functors and show applications to Calderón–Lozanovskii spaces. We also prove estimates of the p-factorization norm and projection constants for finite-dimensional Banach lattices. We show as a consequence of our results that in a large class of n-dimensional Banach sequence lattices \(E_n\) the projection constants \(\lambda (E_n)\) satisfy \(\lim _{n\rightarrow \infty }\lambda (E_n)/\sqrt{n} = c\), where \(c=\sqrt{2/\pi }\) in the real case and \(c= \sqrt{\pi }/2\) in the complex case. Applications are given to vector-valued sequence spaces.