关于凸算子的广义Frechet可微性

本文得到如下结果：设F为从自反的Banaeh空间层到Banaeh格空间X的育有界凸算子，假若(i)X与X^*皆弱紧的区间；(ii)X有范紧区间，则F在层的某个稠子集Gδ的每一点都是Frechet可微的，推广了实值凸函数的有关结论．     

About positive Dunford-Pettis operators on Banach lattices

We give several characterizations of Banach lattices on which each positive Dunford-Pettis operator is compact. As consequences, we obtain new sufficient and necessary conditions under which a norm of a Banach lattice is order continuous, a positive weakly compact operator is compact and the dual operator of a positive Dunford-Pettis operator is Dunford-Pettis.     

Banach f-algebras and Banach lattice algebras with unit

We give a necessary and sufficient condition for a Banach lattice algebra to be representable as a Banach lattice of continuous realvalued functions on a compact space, endowed with the pointwise defined multiplication. Moreover, we give a characterisation of Banach lattice algebras possessing an algebraic unit.     

A weak Grothendieck compactness principle

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in the closed convex hull of a weakly null sequence if and only if the Banach space has the Schur property.     

On dual locally uniformly rotund norms

We show that the existence of an equivalent dual LUR norm on a dual Banach space can be characterized by a topological property similar to the fragmentability. The compact spaces homeomorphic to weak* compact subsets of a dual LUR Banach space have the same properties as the class of Radon-Nikodym compact spaces. Research supported by the DGICYT PB 95-1025.     

A Grothendieck compactness principle for the Mackey dual topology

Grothendieck [6] proved that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In a recent paper [3], an analogous result for weak compactness in a Banach space is shown to be equivalent to the Schur property. In this article, we obtain a similar type result in the Mackey dual of a Banach space. A related result for weak^? compactness is also obtained.     

Regular-Norm Balls can be Closed in the Strong Operator Topology

The main results obtained are:– A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T Lr(X,Y): ||T||r 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y).– A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y).– A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm.– A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm.Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance:– If X = Lp() and Y = Lq(), where 1 < p,q < , then Lr(X,Y) is a first category subset of L(X,Y).     

Incompleteness of the linear span of the positive compact operators

We show that even in the case of a Banach lattice with an order continuous norm (or whose dual has an order continuous norm) the linear span of the positive compact operators on need not be complete under the regular norm.

     

Some loose ends on unbounded order convergence

The notion of almost everywhere convergence has been generalized to vector lattices as unbounded order convergence, which proves to be a very useful tool in the theory of vector and Banach lattices. In this short note, we establish some new results on unbounded order convergence that tie up some loose ends. In particular, we show that every norm bounded positive increasing net in an order continuous Banach lattice is uo-Cauchy and that every uo-Cauchy net in an order continuous Banach lattice has a uo-limit in the universal completion.     

Boundaries for Banach spaces determine weak compactness

A boundary for a real Banach space is a subset of the dual unit sphere with the property that each element of the Banach space attains its norm on an element of that subset. Trivially, the pointwise convergence with respect to such a boundary is coarser than the weak topology on the Banach space. The boundary problem asks whether nevertheless both topologies have the same norm bounded compact sets.The main theorem of this paper states the equivalence of countable and sequential compactness of norm bounded sets with respect to an appropriate topology. This result contains, as a special case, the positive answer to the boundary problem and it carries James’ sup-characterization as a corollary.     

LW-compact operators and domination problem

Positivity - This paper introduces a ‘new’ class of operators. An operator T from a Banach lattice E into a Banach space Y will be called an LW-compact if T carries L-weakly compact...     

Convolution Operators on Banach Lattices with Shift-Invariant Norms

Let G be a locally compact abelian group and let μ be a complex valued regular Borel measure on G. In this paper we consider a generalisation of a class of Banach lattices introduced in Johansson (Syst Control Lett 57:105–111, 2008). We use Laplace transform methods to show that the norm of a convolution operator with symbol μ on such a space is bounded below by the L ^∞ norm of the Fourier–Stieltjes transform of μ. We also show that for any Banach lattice of locally integrable functions on G with a shift-invariant norm, the norm of a convolution operator with symbol μ is bounded above by the total variation of μ.     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Almost Dunford–Pettis sets in Banach lattices

We introduce and study the class of almost Dunford–Pettis sets in Banach lattices. It also discusses some of the consequences derived from this study. As an application, we characterize Banach lattices whose relatively weakly compact sets are almost Dunford–Pettis sets. Also, we establish some necessary and sufficient conditions on which an almost Dunford–Pettis set is L-weakly compact (respectively, relatively weakly compact). In particular, we characterize Banach lattices under which almost Dunford–Pettis sets in the topological dual of a Banach lattice coincide with that of L-weakly compact (respectively, relatively weakly compact) sets. As a consequences we derive some results.     

L-weakly and M-weakly compact operators and the centre

We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide.     

Norm-attaining compact operators

We show examples of compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. This is the negative answer to an open question posed in the 1970s. Actually, any strictly convex Banach space failing the approximation property serves as the range space. On the other hand, there are examples in which the domain space has a Schauder basis.     

Order Norm Completions of Cone Metric Spaces

In this article, a completion theorem for cone metric spaces and a completion theorem for cone normed spaces are proved. The completion spaces are constructed by means of an equivalence relation defined via an ordered cone norm on the Banach space E whose cone is strongly minihedral and ordered closed. This order norm has to satisfy the generalized absolute value property. In particular, if E is a Dedekind complete Banach lattice, then, together with its absolute value norm, satisfy the desired properties.     

The Norm of a Complex Banach Lattice

In this note we study the problem how the complexification of a real Banach space can be normed in such a way that it becomes a complex Banach space from the point of view of the theory of cross-norms on tensor products of Banach spaces. In particular we show that the norm of a complex Banach lattice can be interpretated in terms of the l-tensor product of real Banach lattices.     

Banach Lattices on Which Every Power-Bounded Operator is Mean Ergodic

Given a Banach lattice E that fails to be countably order complete, we construct a positive compact operator A: E E for which T = I - A is power-bounded and not mean ergodic. As a consequence, by using the theorem of R. Zaharopol, we obtain that if every power-bounded operator in a Banach lattice is mean ergodic then the Banach lattice is reflexive.     

Operator martingale decompositions and the Radon-Nikodým property in Banach spaces

We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon-Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney-Schaefer l-tensor product , where E is a suitable Banach lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach lattices Y with the Radon-Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on . Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach lattice and a Banach space. This result is used to characterize Banach spaces with the Radon-Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1<p<∞, our results yield Lp(μ,Y)-space analogues of some of the well-known results on uniform amarts in L¹(μ,Y)-spaces.