The p-approximation property in terms of density of finite rank operators

We characterize the p-approximation property (p-AP) introduced by Sinha and Karn [D.P. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ?p, Studia Math. 150 (2002) 17-33] in terms of density of finite rank operators in the spaces of p-compact and of adjoints of p-summable operators. As application, the p-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi-p-nuclear operators. This relates the p-AP to Saphar's approximation property APp^′. As another application, the p-AP is characterized via a trace condition, allowing to define the trace functional on certain subspaces of the space of nuclear operators.     

Factorizing τ-spectra of mappings

In this paper by a spectrum of mappings we mean a morphism of spectra of spaces. However, using the notion of a mapping of mappings, we give the definition of a spectrum of mappings similar to that of a spectrum of spaces. In this case, the formulations of the given results are also similar to the formulations of the corresponding results concerning the spectra of spaces.For the spectra of mappings we define the notion of a τ-spectrum of mappings factorizing in a special sense and prove a version of the Spectral Theorem for such spectra. Furthermore, to a given indexed collection F of mapping we associate a τ-spectrum factorizing in the above special sense whose mappings are Containing Mappings for F constructed in Iliadis (2005) [4]. These associated τ-spectra and the corresponding version of the Spectral Theorem imply that for a given indexed collection F of mappings any so-called “natural” τ-spectrum for F factorizing in the special sense contains a cofinal and τ-closed subspectrum whose mappings are Containing Mapping for F. Thus, Containing Mappigs for F appear here without any concrete construction. The associated τ-spectra are used also in order to define and characterize the so-called second-type saturated classes of mappings (which are “saturated” by universal elements).     

Hardy-Littlewood maximal function of τ-measurable operators

We define the Hardy-Littlewood maximal function of τ-measurable operators and obtain weak (1,1)-type and (p,p)-type inequalities for the Hardy-Littlewood maximal function.     

Chebyshev Centers in Hyperplanes of c 0

We give a full characterization of the closed one-codimensional subspaces of c ₀, in which every bounded set has a Chebyshev center. It turns out that one can consider equivalently only finite sets (even only three-point sets) in our case, but not in general. Such hyperplanes are exactly those which are either proximinal or norm-one complemented.     

On Downing-Kirk's theorem

In this paper, using the notion of τ-distance, we prove a generalization of Downing-Kirk's theorem.     

On normal τ-measurable operators affiliated with semifinite von Neumann algebras

Let τ be a faithful normal semifinite trace on the von Neumann algebra M, 1 ≥ q > 0. The following generalizations of problems 163 and 139 from the book [1] to τ-measurable operators are obtained; it is established that: 1) each τ-compact q-hyponormal operator is normal; 2) if a τ-measurable operator A is normal and, for some natural number n, the operator A ⁿ is τ-compact, then the operator A is also τ-compact. It is proved that if a τ-measurable operator A is hyponormal and the operator A ² is τ-compact, then the operator A is also τ-compact. A new property of a nonincreasing rearrangement of the product of hyponormal and cohyponormal τ-measurable operators is established. For normal τ-measurable operators A and B, it is shown that the nonincreasing rearrangements of the operators AB and BA coincide. Applications of the results obtained to F-normed symmetric spaces on (M, τ) are considered.     

Applications of Ky Fan's inequality on σ-compact set to variational inclusion and n-person game theory

In this paper, Ky Fan's inequality on σ-compact set is applied to variational inclusions and n-person game theory. We give results of some variational inclusions and existence of non-cooperative equilibrium in n-person game on σ-compact set.     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.     

Singular and nonsingular modules relative to a torsion theory

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept of τ-essential submodule for a hereditary torsion theory τ. We introduce two new module classes called τ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization of τ-semisimple rings via non-τ-singular modules. We prove that if M∕τ(M) is non-τ-singular for a module M, then every submodule of M has a unique τ-closure. We give some properties of the torsion theory generated by the class of all τ-singular modules. We obtain a decomposition theorem for a strongly τ-extending module by using non-τ-singular modules.     

Reflexivity of prodiscrete topological groups

We study the duality properties of two rather different classes of subgroups of direct products of discrete groups (protodiscrete groups): P-groups, i.e., topological groups such that countable intersections of its open subsets are open, and protodiscrete groups of countable pseudocharacter (topological groups in which the identity is the intersection of countably many open sets). It was recently shown by the same authors that the direct product Π of an arbitrary family of discrete Abelian groups becomes reflexive when endowed with the ω-box topology. This was the first example of a non-discrete reflexive P-group. Here we present a considerable generalization of this theorem and show that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is reflexive. In particular, every locally compact Abelian group with the P-modified topology is reflexive. We also examine the reflexivity of dense subgroups of products Π with the P-modified topology and obtain the first examples of non-complete reflexive P-groups. We find as well that the better behaved class of prodiscrete groups (complete protodiscrete groups) of countable pseudocharacter contains non-reflexive members—any uncountable bounded torsion Abelian group G of cardinality ω² supports a topology τ such that (G,τ) is a non-reflexive prodiscrete group of countable pseudocharacter.     

Convergence theorem for the common solution for a finite family of ?-strongly accretive operator equations

The purpose of this paper is to study a strong convergence of multi-step iterative scheme to a common solution for a finite family of uniformly continuous ?-strongly accretive operator equations in an arbitrary Banach space. As a consequence, the strong convergence theorem for the multi-step iterative sequence to a common fixed point for finite family of ?-strongly pseudocontractive mappings is also obtained. The results presented in this paper thus improve and extend the corresponding results of Inchan [6], Kang [8] and [9] and many others.     

Lacunary Sets and Function Spaces with Finite Cotype

We investigate translation invariant subspaces of the space of uniformly convergent Fourier series and Orlicz spaces, with finite cotype. In the case of Orlicz spaces, this leads to some new characterizations of p-Rider sets.     

Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb^′(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever g○f=0 for any f∈Cb(X,E), g∈E^′.     

Spaces of holomorphic functions on non-balanced domains

This paper studies the coincidence of the _τω${\tau }_{\omega }$ and _τδ${\tau }_{\delta }$ topologies on the space of holomorphic functions defined on an open subset U   of a Banach space. Dineen and Mujica proved that _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ when U   is a balanced open subset of a separable Banach space with the bounded approximation property. Here, we study the _τω=_τδ${\tau }_{\omega }={\tau }_{\delta }$ problem for several types of non-balanced domains U.     

Characterization of the closedness of the sum of two shift-invariant spaces

We first present a formula for the supremum cosine angle between two closed subspaces of a separable Hilbert space under the assumption that the ‘generators’ form frames for the subspaces. We then characterize the conditions that the sum of two, not necessarily finitely generated, shift-invariant subspaces of L²(Rd) be closed. If the fibers of the generating sets of the shift-invariant subspaces form frames for the fiber spaces a.e., which is satisfied if the shift-invariant subspaces are finitely generated or if the shifts of the generating sets form frames for the respective subspaces, then the characterization is given in terms of the norms of possibly infinite matrices. In particular, if the shift-invariant subspaces are finitely generated, then the characterization is given wholly in terms of the norms of finite matrices.     

Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients

We consider the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients of order 2m in . If the coefficients of top order are Hölder continuous of exponent τ∈(0,1], we can derive the remainder estimate of the form O(t^(n−θ)/2m) with any θ∈(0,τ). This result holds without the condition 2m>n, which was always assumed in many papers. We also show that the spectral function is differentiable up to order <m.     

Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

In this paper, we introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland's variational principle, we derive generalized Caristi's (common) fixed point theorems, a generalized Takahashi's nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland's variational principle.     

Onm-compact spaces

We give characterizations ofm-compact andm-closed spaces by means of nets and filterbases. We study the images ofm-compact andm-closed sets under various modifications ofM-continuous functions. We introduce the notion ofmc-compact space and investigate some relations betweenmc-compact spaces andM-continuous functions.