Some fundamental properties for duals of Orlicz spaces

In this paper some basic properties of Orlicz spaces are extended to their dual spaces, and finally, criteria for extreme points in these spaces are given.     

The stabilized set of p's in Krivine's theorem can be disconnected

For any closed subset F   of [1,∞]$[1,\infty ]$ which is either finite or consists of the elements of an increasing sequence and its limit, a reflexive Banach space X with a 1-unconditional basis is constructed so that in each block subspace Y of X  , _?p${?}_p$ is finitely block represented in Y   if and only if p∈F$p\in F$. In particular, this solves the question as to whether the stabilized Krivine set for a Banach space had to be connected. We also prove that for every infinite dimensional subspace Y of X there is a dense subset G of F such that the spreading models admitted by Y   are exactly the _?p${?}_p$ for p∈G$p\in G$.     

Local rotundity structure of generalized Orlicz–Lorentz sequence spaces

We give some criteria for extreme points and strong U-points in generalized Orlicz–Lorentz sequence spaces, which were introduced in [P. Foralewski, H. Hudzik, L. Szymaszkiewicz, On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces, Math. Nachr. (in press)] (cf. [G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176–179; M. Nawrocki, The Mackey topology of some F-spaces, Ph.D. Dissertation, Adam Mickiewicz University, Poznań, 1984 (in Polish)]). Some examples show that in these spaces the notion of the strong U-point is essentially stronger than the notion of the extreme point. This paper is related to the results from [A. Kamińska, Extreme points in Orlicz–Lorentz spaces, Arch. Math. 55 (1990) 173–180] (see Remark 1).     

Distributional Taylor formula

We derive a distributional Taylor formula with precise integral remainder. We give applications of it and estimates for the remainder.     

Random strict convexity and random uniform convexity in random normed modules

Based on the analysis of stratification structure on random normed modules, we first present random strict convexity and random uniform convexity in random normed modules. Then, we establish their respective relations to classical strict and uniform convexity: in the process some known important results concerning strict convexity and uniform convexity of Lebesgue-Bochner function spaces can be obtained as a special case of our results. Further, we also give their important applications to the theory of random conjugate spaces as well as best approximation. Finally, we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces.     

The criteria of strict monotonicity and rotundity points in generalized Calderón–Lozanovskiǐ spaces

In this paper, some basic properties of the general modular space are proven. Criteria for strictly monotone points, extreme points and SU$SU$-points in generalized Calderón–Lozanovskiǐ spaces are obtained. Consequently, the sufficient and necessary conditions for the rotundity properties of such spaces are given.     

Thin and thick sets in locally convex spaces

We introduce in this work some normed space notions such as norming, thin and thick sets in general locally convex spaces. We also study some effects of thick sets on the uniform boundedness-like principles in locally convex spaces such as “weak*-bounded sets are strong*-bounded if and only if the space is a Banach–Mackey space”. It is proved that these principles occur under some weaker conditions by means of thick sets. Further, we show that the thickness is a duality invariant, that is, all compatible topologies for some locally convex space have the same thick sets.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

On some geometric properties of generalized Orlicz–Lorentz function spaces

In this article we continue investigations concerning generalized Orlicz–Lorentz function spaces _Λφ${\Lambda }_{\phi }$ initiated in the papers (Foralewski, 2011)  and  (cf. also (Foralewski, 2008)  and ). First, it is shown that modular _?φ${?}_{\phi }$ is orthogonally subadditive. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space _Λφ${\Lambda }_{\phi }$ a strong condition of _Δ2${\Delta }_2$ type and the notion of regularity of the generated Musielak–Orlicz function φ$\phi$ are introduced. Finally, criteria for non-squareness of _Λφ${\Lambda }_{\phi }$ are presented.     

A basic strict separation theorem in random locally convex modules

The central idea of this paper is to make full use of the recently developed theory of random conjugate spaces to establish a basic strict separation theorem that is universally suitable in an arbitrary random locally convex module. A series of interesting corollaries of the basic theorem are also included.     

Derivations on the Algebra of τ-Compact Operators Affiliated with a Type I von Neumann Algebra

Let M be a type I von Neumann algebra with the center Z, and a faithful normal semi-finite trace τ. Consider the algebra L(M, τ) of all τ-measurable operators with respect to M and let S ₀(M, τ) be the subalgebra of τ-compact operators in L(M, τ). We prove that any Z-linear derivation of S ₀(M, τ) is spatial and generated by an element from L(M, τ).      

Some properties of the injective tensor product of L[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1<p,p′<∞ such that 1/p+1/p′=1. Then , the injective tensor product of L^p[0,1] and X, has the Radon-Nikodym property (resp. the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of c₀, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L^p′[0,1] to X is compact.     

Average locally uniform rotundity and a class of nonlinear maps

We consider some topological characterizations of dual Banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of σ-slicely continuous maps.     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

Interpolation of bilinear operators and compactness

The behavior of bilinear operators acting on interpolation of Banach spaces for the ρ method in relation to the compactness is analyzed. Similar results of Lions-Peetre, Hayakawa and Persson’s compactness theorems are obtained for the bilinear case and the ρ method.     

Compactness by the Hausdorff measure of noncompactness

In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary BK-space into the sequence spaces c₀, c, ?_∞ and ?₁, and into the matrix domains of triangles in these spaces. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on the BK-spaces.