A characterization of reflexive Banach spaces

A Banach space is not reflexive if and only if there exist a closed separable subspace of and a convex closed subset of with empty interior which contains translates of all compact sets in . If, moreover, is separable, then it is possible to put .

     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

A theorem on reflexive large rank operator spaces

If every nonzero operator in an -dimensional operator space has rank , then is reflexive.

     

-contingent epiderivatives in reflexive spaces

In this paper we introduce a notion of the τ^w-contingent epiderivative of a set-valued map by considering the weak topology in the image space. We establish a chain rule formula for families of τ^w-contingent and contingent epiderivatives. When the image space is reflexive and the ordering cone a half-space, we give conditions for the existence of the family of τ^w-contingent epiderivatives of a stable set-valued map; moreover we obtain an interesting formula that connects this family with the contingent epiderivative of an associated scalar set-valued map. Finally we apply the previous results in order to establish optimality conditions for a set-valued optimization problem via epiderivatives with respect to half-spaces associated with the ordering cone. In particular we provide a scalarization method for computing these conditions in reflexive spaces.     

A universal property of reflexive hereditarily indecomposable Banach spaces

It is shown that every separable Banach space universal for the class of reflexive Hereditarily Indecomposable space contains isomorphically and hence it is universal for all separable spaces. This result shows the large variety of reflexive H.I. spaces.

     

A note on causality in reflexive Banach spaces.

In this short note we discuss the causality of linear closable operators in reflexive Banach spaces. We show that in general causality is not preserved under closure procedures. We shall give an alternative definition of causality for closable operators by means of a certain continuity poperty that carries over to the closure. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous selections and reflexive Banach spaces

Every l.s.c. mapping from a space into the non-empty closed convex subsets of a reflexive Banach space admits a continuous selection provided it satisfies a ``weak' u.s.c. condition. This result partially generalizes some known selection theorems. Also, it is successful in solving a problem concerning the set of proper lower semi-continuous convex functions on a reflexive Banach space.

     

Weighted projective spaces and reflexive simplices

 We derive a classification algorithm for reflexive simplices in arbitrary fixed dimension. It is based on the assignment of a weight Q ? ℕ ⁿ⁺¹ to an integral n-simplex, the construction, up to an isomorphism, of all simplices with given weight Q, and the characterization in terms of the weight as to when a simplex with reduced weight is reflexive. This also yields a convex-geometric reproof of the characterization in terms of weights for weighted projective spaces to have at most Gorenstein singularities. Received: 30 March 1999 / Revised version: 18 October 2001     

Trigonometric approximation in reflexive Orlicz spaces

The Lipschitz classes Lip(α, M) , 0 α≤ 1 are defined for Orlicz space generated by the Young function M, and the degree of approximation by matrix transforms of f ∈ Lip (α, M) is estimated by n-α .     

A coupled system of integral equations in reflexive banach spaces

We present an existence theorem for at least one weak solution for a coupled system of integral equations of Volterra type in a reflexive Banach spaces relative to the weak topology.     

Product integration in reflexive Banach spaces

LetX denote a reflexive Banach space and {A(t)|t[0,T]} a time dependent family of accretive operators defined onX. Conditions are placed on {A(t)|t[0,T]} which are sufficient to guarantee the existence of solutions to the Cauchy initial value problem:u(t,x)+A(t)u(t,x)=0; u(0,x)=x. These solutions are obtained via the method of product integration; however limits for the infinite product are taken with respect to the weak topology.     

On a pattern of reflexive operator spaces

A linear subspace is a separating subspace for an operator space if the only member of annihilating is 0. It is proved in this paper that if has a strictly separating vector and a separating subspace satisfying , then is reflexive. Applying this to finite dimensional leads to more results on reflexivity. For example, if dim , and every nonzero operator in has rank , then is reflexive.

     

Hammerstein integral inclusions in reflexive Banach spaces

In this paper we examine multivalued Hammerstein integral equations defined in a separable reflexive Banach space. We prove existence theorems for both the ``convex' problem (the multifunction is convex-valued) and the ``nonconvex' problem (the multifunction is not necessarily convex-valued). We also show that the solution set of the latter is dense in the solution set of the former (relaxation theorem). Finally we present some examples illustrating the applicability of our abstract results.

     

Proximal analysis in reflexive smooth Banach spaces

In the present paper, we introduce and study a new proximal normal cone in reflexive Banach spaces in terms of a generalized projection operator. Two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth Banach spaces. The density theorem for both proximal subdifferentials has been proved in p-uniformly convex and q-uniformly smooth Banach spaces. Various important properties and applications of our concepts are also proved.     

Economic equilibrium problems in reflexive Banach spaces

The problem is to minimize a finite collection of objective functions over admissible sets depending on the so-called price vector. The minima in question and the price vector are linked together by a subdifferential governing law. The problem stated as a system of variational–hemivariational inequalities, defined on a nonconvex feasible set, is reduced to one variational–hemivariational inequality involving nonmonotone multivalued mapping. The existence of solutions is examined under the assumption that the constrained functions are positive homogeneous of degree one. The study has been inspired by economic issues and leads to new results concerning the existence of competitive equilibria.     

A projection method for solving nonlinear problems in reflexive Banach spaces

We study the convergence of an iterative algorithm for finding common fixed points of finitely many Bregman firmly nonexpansive operators in reflexive Banach spaces. Our algorithm is based on the concept of the so-called shrinking projection method and it takes into account possible computational errors. We establish a strong convergence theorem and then apply it to the solution of convex feasibility and equilibrium problems, and to finding zeroes of two different classes of nonlinear mappings.