Regularization for unconstrained vector optimization of convex functionals in Banach spaces

An operator regularization method is considered for ill-posed vector optimization of weakly lower semicontinuous essentially convex functionals on reflexive Banach spaces. The regularization parameter is chosen by a modified generalized discrepancy principle. A condition for the estimation of the convergence rate of regularized solutions is derived. This article was submitted by the author in English.     

On support points and support functionals of convex sets

Let K be a bounded closed convex subset of a real Banach space of dimension at least two. Then the set of the support points of K is pathwise connected and the set Σ₁(K) of the norm-one support functionals of K is uncountable in each nonempty open set that intersects the dual unit sphere. In particular, the set Σ ₁(K) is always uncountable, which answers a question posed by L. Zajíček.     

Invariant functionals and the uniqueness of invariant norms

Let τ be a representation of a compact group G on a Banach space (X,||·||). The question we address is whether X carries a unique invariant norm in the sense that ||·|| is the unique norm on X for which τ is a representation. We characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when X is a dual Banach space and τ is a -continuous representation of G on X such that τ(G) consists of -continuous operators. We illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces Lp(Ω), where Ω is a locally compact Hausdorff space equipped with a positive Radon measure and G acts on Ω as a group of continuous invertible measure-preserving transformations.     

The sets of monomorphisms and of almost open operators between locally convex spaces

If the set of monomorphisms between locally convex spaces is not empty, then it is an open subset of the space of all continuous and linear operators endowed with the topology of the uniform convergence on the bounded sets if and only if the domain space is normable. The corresponding characterization for the set of almost open operators is also obtained; it is related to the lifting of bounded sets and to the quasinormability of the domain space. Other properties and examples are analyzed.

     

A note on convex renorming and fragmentability

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.     

Characterizations of strictly convex sets by the uniqueness of support points

Abstract

This short paper characterizes strictly convex sets by the uniqueness of support points (such points are called unique support points or exposed points) under appropriate assumptions. A class of so-called regular sets, for which every extreme point is a unique support point, is introduced. Closed strictly convex sets and their intersections with some other sets are shown to belong to this class. The obtained characterizations are then applied to set-valued maps and to the separation of a convex set and a strictly convex set. Under suitable assumptions, so-called set-valued maps with path property are characterized by strictly convex images of the considered set-valued map.     

Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space

We show that any fixed point of a Lipschitzian, strictly pseudocontractive mapping on a closed, convex subset of a Banach space is necessarily unique, and may be norm approximated by an iterative procedure. Our argument provides a convergence rate estimate and removes the boundedness assumption on , generalizing theorems of Liu.

     

Noncontinuity of spectrum for the adjoint of an operator

This paper deals with the connection between continuity of spectrum at an element of the Banach algebra of all bounded linear operators on a Banach space and at the adjoint of . In particular, we show that, if is not reflexive, the spectrum function may be continuous at and discontinuous at .

     

Boundaries of compact convex sets and fragmentability

If X is a compact convex set in a real locally convex space, B⊂X is said to be its boundary if every affine continuous function on X attains its maximum at some point of B. We study relations between fragmentability of B and the whole set X. As a byproduct we obtain a characterization of separable Asplund spaces. We also study the possibility of finding the Haar system in a boundary of a metrizable compact convex set.     

Lipschitz泛函的本质临界点

本文指出Ｂａｎａｃｈ空间中Ｃ^1－0（即局部Ｌｉｐｓｃｈｉｔｚ）泛函的Ｃｌａｒｋｅ－临界点一般不具有在Ｃ^1－0局部坐标变换下的不变性。据此提出了Ｃ^1－0泛函的本质临界点的概念，它能适应研究Ｃ^1－0流形上的Ｃ^1－0泛函的临界点的需要．还给出了有关本质临界点的几个结果。     

Methods of Chebyshev points of convex sets and their applications

Chebyshev points of bounded convex sets, search algorithms for them, and various applications to convex programming are considered for simple approximations of reachable sets, optimal control, global optimization of additive functions on convex polyhedra, and integer programming. The problem of searching for Chebyshev points in multicriteria models of development and operation of electric power systems is considered.     

Virial inversion and density functionals

We prove a novel inversion theorem for functionals given as power series in infinite-dimensional spaces. This provides a rigorous framework to prove convergence of density functionals for inhomogeneous systems with applications in classical density function theory, liquid crystals, molecules with various shapes or other internal degrees of freedom. The key technical tool is the representation of the inverse via a fixed point equation and a combinatorial identity for trees, which allows us to obtain convergence estimates in situations where Banach inversion fails.     

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L^(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L^(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L^(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L^(1,∞) if and only if the sequence of singular numbers {s_n(x)}_n?1 (in the descending order and counting the multiplicities) satisfies . In this case, our characterization amounts to saying that a positive element x∈L^(1,∞) is measurable if and only if exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space , where the space is the closure of all finite rank operators in L^(1,∞) in the norm ∥.∥_(1,∞).     

Measures of noncompactness and essential pseudospectra on Banach space

In this research article, we work with the notion of the measures of noncompactness in order to establish some results concerning the essential pseudospectra of closed, densely defined linear operators in the Banach space. We start by giving a refinement of the definition of the essential pseudospectra by means of the measure of noncompactness, and we give sufficient conditions on the perturbed operator to have its invariance. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces

In this paper, we consider an Ishikawa type iteration process with errors, which converges to the unique common fixed point of a pair of contractive type mappings in complete generalized convex metric spaces. Furthermore, we obtain the corresponding results in Banach spaces. Our results extend and generalize many known results.     

A note on non-support points, negligible sets, Gâteaux differentiability and Lipschitz embeddings

This paper first presents a characterization of three classes of negligible closed convex sets (i.e., Gauss null sets, Aronszajn null sets and cube null sets) in terms of non-support points; then gives a generalization of Gâteaux differentiability theorems of Lipschitz mapping from open sets to those closed convex sets admitting non-support points; and as their application, finally shows that a closed convex set in a separable Banach space X can be Lipschitz embedded into a Banach space Y with the Radon–Nikodym property if and only if the closure of its linear span is linearly isomorphic to a closed subspace of Y.