Smooth points in some spaces of bounded operators

We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel ^p -orc ₀-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y).     

A generalization of Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> by using atoms

Let X = （X, d,μ） The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P（X） and prove that the generalized Hardy spaces have the same property as H^P（X）. Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let （X, δ, μ） be the normalized space of （X, d, μ） in the sense of Macias and Segovia. We also study the relations of our function spaces for （X, d, μ） and （X, δ,μ）.     

Density and unique decomposition theorems for the lattice of cellular classes

A classC of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy pushouts. The smallest cellular class containingX is denoted byC(X), and a partial order relation ≪ is defined by:X ≪Y ifY εC(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes.     

强S-仿紧空间

在一般拓扑空间中给出了强S-仿紧空间的概念,研究了它的性质并给出了Hausdorff空间下它的等价刻画,同时讨论了它的相对性质.     

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification ${ \overline{X} = X \cup \partial X}We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification [`(X)] = X è?X{ \overline{X} = X \cup \partial X} . Using this fact, several results known for proper CAT(0) spaces may be extended to finite-dimensional spaces, including the existence of canonical fixed points at infinity for parabolic isometries, algebraic and geometric restrictions on amenable group actions, and geometric superrigidity for non-elementary actions of irreducible uniform lattices in products of locally compact groups.     

Towards an L^p Potential Theory for Sub-Markovian Semigroups： Kernels and Capacities

We study in fairly general measure spaces （X,μ） the （non-linear） potential theory of L^p sub-Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence （and regularity） of such densities. We give various （dual） representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of R^n where abstract Bessel potential spaces can be identified with concrete function spaces.     

Ideal properties of the Dunford integration operator

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F‐valued Dunford integrable functions to the bidual of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach‐Saks property, separable Banach spaces not containing c₀, Banach spaces not containing c₀ or ?₁ and Asplund spaces not containing c₀.     

The controlled separable projection property for Banach spaces

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y.     

Products of Compacta with Plyhedra and Topological Spaces in the Shape Category

In the literature there exist examples of separable metric spaces X,Y whose Cartesian product X × Y is not a product in the shape category Sh(Top). It is an open question whether, for X a compact Hausdorff space, X × Y is a product in Sh(Top), for every topological spaces Y. The main result of the paper asserts that the answer is positive provided X × P is a product in Sh(Top), for every polyhedron P.     

Grothendieck Spaces of Compact Operators

In this paper we study the Grothendieck spaces among the operator spaces L_e(E'_c, F). Conditions under which L_e(E'_c, F) contains complemented copy of c₀ are given. We apply these results to spaces of the type C_b(X; F) endowed with strict topologies.     

Continuous operators on asymmetric normed spaces

If (X, p) and (Y, q) are two asymmetric normed spaces, the set LC(X, Y) of all continuous linear mappings from (X, p) to (Y, q) is not necessarily a linear space, it is a cone. If X and Y are two Banach lattices and p and q are, respectively, their associated asymmetric norms (p(x) = ‖⁺‖, q(y) = ‖y ⁺‖), we prove that the positive operators from X to Y are elements of the cone LC(X, Y). We also study the dual space of an asymmetric normed space and finally we give open mapping and closed graph type theorems in the framework of asymmetric normed spaces. The classical results for normed spaces follow as particular cases. The author acknowledges the support of the Ministerio de Educación y Ciencia of Spain and FEDER, under grant MTM2006-14925-C02-01 and Generalitat Valenciana under grant GV/2007/198.     

Factorization Theorems on Symmetric Spaces of Noncompact Type

We prove the analogs of the Khinchin factorization theorems for K-invariant probability measures on symmetric spaces X=G/K with G semisimple noncompact. We use the Kendall theory of delphic semigroups and some properties of the spherical Fourier transform and spherical functions on X.     

Holomorphic Jackson's theorems in polydiscs

The purpose of this article is to establish Jackson-type inequality in the polydiscs U^N of for holomorphic spaces X, such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely,

where is the deviation of the best approximation of fX by polynomials of degree at most k_j about the jth variable z_j with respect to the X-metric and is the corresponding modulus of continuity.     

Vector-valued Fourier multipliers on symmetric spaces of the noncompact type

LetX be a Riemannian symmetric space of the noncompact type. We prove the multiplier theorem for the Helgason-Fourier transform and the vector valued function spacesL _p (X, l _q ). As a consequence we get the inequalities of the Littlewood-Paley type forL _p (X) spaces.Research supported by K.B.N. Grant 210519101 (Poland).     

Invariant subspaces of X** under the action of biconjugates

We study conditions on an infinite dimensional separable Banach space X implying that X is the only non-trivial invariant subspace of X** under the action of the algebra of biconjugates of bounded operators on . Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of c ₀, and show in particular that any space which does not contain ℓ₁ and has property (u) of Pelczynski is simple.     

Spaces in which every dense subset is a Gδ

A topological space X is called a DG_δ-space if every subset of X is a G_δ-set in its closure. In this paper we study DG_δ-spaces that contains subspaces in which every dense subset is open and spaces in which every subset is a G_δ . We give some new results in these classes of topological spaces.     

A note on operator Banach spaces containing a complemented copy of c0

Let X and Y Banach spaces. Two new properties of operator Banach spaces are introduced. We call these properties "boundedly closed" and "d-boundedly closed". Among other results, we prove the following one. Let U(X, Y){\cal U}(X, Y) an operator Banach space containing a complemented copy of c₀. Then we have: 1) If U(X, Y){\cal U}(X, Y) is boundedly closed then Y contains a copy of c₀. 2) If U(X, Y){\cal U}(X, Y) is d-boundedly closed, then X^* or Y contains a copy of c₀.     

Radial maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let ${{\mathcal X}}Let X{{\mathcal X}} be an RD-space with m(X)=￥{\mu({\mathcal X})=\infty}, which means that X{{\mathcal X}} is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this paper, we characterize the atomic Hardy spaces H^p_at(X){H^p_{\rm at}({\mathcal X})} of Coifman and Weiss for p ? (n/(n+1),1]{p\in(n/(n+1),1]} via the radial maximal function, where n is the “dimension” of X{{\mathcal X}}, and the range of index p is the best possible. This completely answers the question proposed by Ronald R. Coifman and Guido Weiss in 1977 in this setting, and improves on a deep result of Uchiyama in 1980 on an Ahlfors 1-regular space and a recent result of Loukas Grafakos et al in this setting. Moreover, we obtain a maximal function theory of localized Hardy spaces in the sense of Goldberg on RD-spaces by generalizing the above result to localized Hardy spaces and establishing the links between Hardy spaces and localized Hardy spaces. These results have a wide range of applications. In particular, we characterize the Hardy spaces H^p_at(M){H^p_{\rm at}(M)} via the radial maximal function generated by the heat kernel of the Laplace-Beltrami operator Δ on complete noncompact connected manifolds M having a doubling property and supporting a scaled Poincaré inequality for all p ? (n/(n+a),1]{p\in(n/(n+\alpha),1]}, where α represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ.     

THE WEAK RADON-NIKODYM PROPERTY IN SPACES OF NUCLEAR OPERATORS

Abstract

We prove that if X and Y are Banach spaces such that X* has the weak Radon-Nikodym property (WRNP), Y has the Radon-Nikodym property (RNP) and Y is complemented in its bidual, then the space N(X,Y) of all nuclear operators from X to Y has the WRNP. If moreover X* has the RNP, then N(X,Y) has the RNP.