Let X1, X2, …, XN be Banach spaces and ψ a continuous convex function with some appropriate conditions on a certain convex set in RN−1. Let (X1⊕X2⊕…⊕XN)Ψ be the direct sum of X1, X2, …, XN equipped with the norm associated with Ψ. We characterize the strict, uniform, and locally uniform convexity of (X1 ⊕ X2 ⊕ … ⊕ XN)Ψ; by means of the convex function Ψ. As an application these convexities are characterized for the ℓp, q-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p, q (1 < q ≤ p ≤ ∈, q < ∞), which includes the well-known facts for the ℓp-sum (X1 ⊕ X2 ⊕ … ⊕ XN)p in the case p = q.

Let Β1, Β2 be a pair of Banach spaces and T be a vector valued martingale transform (with respect to general filtration) which maps Β1-valued martingales into Β2-valued martingales. Then, the following statements are equivalent: T is bounded from into for some p (or equivalently for every p) in the range 1 < p < ∞; T is bounded from into BMOB2; T is bounded from BMOB1 into BMOB2; T is bounded from into . Applications to UMD and martingale cotype properties are given. We also prove that the Hardy space defined in the case of a general filtration has nice dense sets and nice atomic decompositions if and only if Β has the Radon-Nikodým property.

Let X be a complex Banach space, G a compact abelian group and Λ a subset of Ĝ, the dual group pf G. Then LΛ1(G, X) has the Radon-Nikodym property if and only if X has the Radon-Nikodym property and Λ is Riesz set. In particular, H1 (T, X) has the Radon-Nikodym property if and only if X has the Radon-Nikodym property. This solves a problem of Hensgen.

The stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that ℳ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C[0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.

We present an operator algebraic approach to Wigner's unitary-antiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wigner-type result for maps on prime C*-algebras.

Recently the concept of uniform rotundity was generalized for real Banach spaces by using a type of “area” devised for these spaces. This paper modifies the methods used for uniform rotundity and applies them to weak rotundity in real and complex spaces. This leads to the definition of k-smoothness, k-very smoothness and k-strong smoothness. As an application, several sufficient conditions for reflexivity are obtained.

Composition operators Cτ between Orlicz spaces Lϕ (Ω, Σ, μ) generated by measurable and nonsingular transformations τ from Ω into itself are considered. We characterize boundedness and compactness of the composition operator between Orlicz spaces in terms of properties of the mapping τ, the function ϕ and the measure space (Ω, Σ, μ). These results generalize earlier results known for Lp-spaces.

Recall a closed convex set C is said to have the weak drop property if for every weakly sequentially closed set A disjoint from C there exists x ∈ A such that co({x} ∩ C) ∪ A = {x}. Giles and Kutzarova proved that every bounded closed convex set with the weak drop property is weakly compact. In this article, we show that if C is an unbounded closed convex set of X with the weak drop property, then C has nonempty interior and X is a reflexive space.

For normed linear spaces two similar characterizations of strong differentiability of the norm and rotundity of the dual space are established, but it is shown that in general there is no causal relation between these two concepts.

Let X be a complex Banach space, G a compact abelian metrizable group and Λ a subset of Ĝ, the dual group of G. If X has the Radon-Nikodym property and is separable then has the Radon-Nikodym property. One consequence of this is that CΛ(G, X) has the Radon-Nikodym property whenever X has the Radon-Nikodym property and the Schur property and Λ is a Rosenthal set. A partial stability property for products of Rosenthal sets is also obtained.

A Banach space (X, ∥ · ∥) is said to be a dual differentiation space if every continuous convex function defined on a non-empty open convex subset A of X* that possesses weak* continuous subgradients at the points of a residual subset of A is Fréchet differentiable on a dense subset of A. In this paper we show that if we assume the continuum hypothesis then there exists a dual differentiation space that does not admit an equivalent locally uniformly rotund norm.

This note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

It is shown that a normed linear space admitting (Chebyshev) centers is complete. Then the ideas in the proof of this fact are used to show that every incomplete CLUR (compactly locally uniformly rotund) normed linear space contains a closed bounded convex subset B with the following properties: (a) B does not contain any farthest point; (b) B does not contain any nearest point (to the elements of its complement).

Let B be an ordered Banach space with ordered Banach dual space. Let N denote the canonical half-norm. We give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is α-monotone (α ≥ 1) if and only if for each f in B* there exists g ∈ B* with g ≥ 0, f and ∥g∥ ≤ α N(f). We also establish a dual result characterizing α-monotonicity of B*.

Given an ordered Banach space ℬ equipped with an order-norm we construct a larger space ℬ¯ with an order-norm and order-identity such that B is isometrically order-isomorphic to a Banach subspace of B. We also discuss the extension of positive operators from ℬto ℬ¯.

Examples are given that show the following: (1) normal structure need not be inherited by quotient spaces; (2) uniform normal structure is not a self-dual property; and (3) no degree of k–uniform rotundity need be present in a space with uniform normal structure.

We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that l1, c0 and l∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class ∧. We also present examples of weighted Orlicz spaces with the Schur property which are not L1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.

Some results on fixed points of certain involutions in Banach spaces have been obtained, and whence a few coincidence theorems are also derived. These are indeed generalization of previously known results due to Browder, Goebel-Zlotkiewicz and Iséki. Illustrative examples are also given.

Some geometric properties of Lp spaces are studied which shed light on the prediction of infinite variance processes. In particular, a Pythagorean theorem for Lp is derived. Improved growth rates for the moving average parameters are obtained.

A flat spot in a Banach space X is an element x ∈ Sx = {x ∈ X: ‖x‖ = 1} with the property that the infimum m(x) of the lengths of all curves in Sx joining x to −x is 2. Flat spots occur in every non-superreflexive space when suitably renormed. A study is made of the geometric implications of the existence of flat spots. Connections with other notions such as differentiability, decomposition constants and Kadec-Klee norms are explored and some renorming results for non-superreflexive spaces are proved.