Vector lattices of weakly compact operators on Banach lattices

A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.

     

A Theorem of the Alternative in Banach Lattices

We consider a linear sytem in a Banach lattice and provide a simple theorem of the alternative (or Farkas lemma) without the usual closure condition.

     

On positive unipotent operators on Banach lattices

Let be a positive operator on a complex Banach lattice. We prove that is greater than or equal to the identity operator if

     

and interpolation between Banach lattices

We study the possibility of obtaining the -norm by an interpolation method starting from a couple of Banach lattice norms. We describe all couples of Banach lattice norms in such that the -norm is a strict interpolation norm between them. Further we consider the possibility of obtaining the -norm by any method which guarantees interpolation of not only linear operators (= bilinear forms on but also of all polylinear forms. Here we show that either one of the initial norms has to be proportional to the -norm, or both have to be weighted -norms.

     

Banach 空间上算子的几种不可约性

本文根据Cowen-Douglas 算子的定义引入两类与强不可约算子有紧密联系的算子类— B_n 算子与B 算子. 说明了在可分Banach 空间上存在B_n 算子与B 算子. 文章详细讨论了几种具有不可约性的算子类之间的关系并得到了一个关系图. 本文还给出这些算子类的一些性质, 包括算子的(拟) 相似不变性等.     

Martingales in Banach Lattices

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (E_n) on a Banach lattice F is said to be a filtration if E_nE_m = E_{n∧ m}. A sequence (x_n) in F is a martingale if E_nx_m = x_n whenever n ≤ m. Denote by M = M(F, (E_n)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn’t contain a copy of c₀ or if every E_n is of finite rank then M is itself a Banach lattice. Convergence of martingales is investigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. Mathematics Subject Classification (2000). 60G48, 46B42     

The multisublinear maximal type operators in Banach function lattices

The main aim of this paper is to study a general multisublinear operators generated by quasi-concave functions between weighted Banach function lattices. These operators, in particular, generalize the Hardy–Littlewood and fractional maximal functions playing an important role in harmonic analysis. We prove that under some general geometrical assumptions on Banach function lattices two-weight weak type and also strong type estimates for these operators are true. To derive the main results of this paper we characterize the strong type estimate for a variant of multilinear averaging operators. As special cases we provide boundedness results for fractional maximal operators in concrete function spaces.     

关于巴拿赫格中的l∞渐进等距拷贝

In this paper, we show that any σ-complete Banach lattice, with a σ-order semicontinuous but not σ-order continuous norm, contains an asymptotically isometric copy of l∞.We also get that the Fenchel-Orlicz space with the Orlicz norm may not contain an asymptotically isometric copy of l∞.     

Matrix multiplication operators on Banach function spaces

In this paper, we study the matrix multiplication operators on Banach function spaces and discuss their applications in semigroups for solving the abstract Cauchy problem.     

关于巴拿赫格中的$l^\infty$渐进等距拷贝

本文证明了具有$\sigma$-序半连续但非$\sigma$-序连续范数的$\sigma$-完备巴拿赫格包含$l^\infty$的渐进等距拷贝.并且证明了具有Orlicz范数的Fenchel-Orlicz空间不一定含有$l^\infty$的渐进等距拷贝.     

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.     

A Complete Convergence Theorem for Row Sums from Arrays of Rowwise Independent Random Elements in Rademacher Type p Banach Spaces

We extend in several directions a complete convergence theorem for row sums from an array of rowwise independent random variables obtained by Sung, Volodin, and Hu [8 Sung , S.H. , Volodin , A.I. , and Hu , T.-C. ( 2005 ). More on complete convergence for arrays. Statist. Probab. Lett. 71:303–311.  [Google Scholar]] to an array of rowwise independent random elements taking values in a real separable Rademacher type p Banach space. An example is presented which illustrates that our result extends the Sung, Volodin, and Hu result even for the random variable case.     

A Minimax Theorem in Banach Lattices

This paper is mainly concerned with measures which take their values in a Banach lattice. Examples of capacities are given and a type of minimax theorem for vector capacities is proved.     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

Composition operators on Banach function spaces

We study the boundedness and the compactness of composition operators on some Banach function spaces such as absolutely continuous Banach function spaces on a -finite measure space, Lorentz function spaces on a -finite measure space and rearrangement invariant spaces on a resonant measure space. In addition, we study some properties of the spectra of a composition operator on the general Banach function spaces.

     

Some estimations of Banach limits

In the first part of this paper, we introduce the notions of upper weight, lower weight and weight of subsequences of natural numbers and investigate some new estimations about Banach limits by using some results from Sucheston [L. Sucheston, On existence of finite invariant measures, Math. Z. 86 (1964) 327-336; L. Sucheston, Banach limit, Amer. Math. Monthly 74 (1967) 308-311]. In the second part of this paper, we study the connections between weights and densities of subsequences of natural numbers, and give a familiar formula to find some values of Banach limits on almost convergent sequences.     

Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

For a sequence of constants {a _n,n1}, an array of rowwise independent and stochastically dominated random elements { V _nj, j1, n1} in a real separable Rademacher type p (1p2) Banach space, and a sequence of positive integer-valued random variables {T _n, n1}, a general weak law of large numbers of the form is established where {c _nj, j1, n1}, _n , b _n are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V _nj, j1, n1}. Illustrative examples include one wherein the strong law of large numbers fails.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.