Invariant subspaces of positive strictly singular operators on Banach lattices

It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators.     

Some characterizations of compact operators on Banach lattices

We study the compactness of the class of operators which are AM-compact and semi-compact on Banach lattices and as consequences, we obtain some characterizations of order continuous norms.      

About positive weak Dunford-Pettis operators on Banach lattices

We characterize Banach lattices for which each positive weak Dunford-Pettis operator from a Banach lattice into another dual Banach lattice is almost Dunford-Pettis. Also, we give some sufficient and necessary conditions for which the class of positive weak Dunford-Pettis operators coincides with that of positive Dunford-Pettis operators, and we derive some consequences.     

On the duality problem of positive Dunford-Pettis operators on Banach lattices

We give some sufficient and necessary conditions for that a positive Dunford-Pettis operator admits a dual operator which is also Dunford-Pettis, and conversely.      

Disjointly homogeneous Banach lattices: Duality and complementation

We study several properties of disjointly homogeneous Banach lattices with a special focus on two questions: the self-duality of this class and the existence of disjoint sequences spanning complemented subspaces. Various results around these problems are given. In particular, we provide examples of reflexive disjointly homogeneous spaces whose dual spaces are not disjointly homogeneous.     

Banach格上的b-几乎Dunford-Pettis算子

首先给出了Banah格上的b-几乎Dunford-Pettis算子的定义;其次,研究了b-几乎Dunford-Pettis算子的相关性质,如b-几乎Dunford-Pettis算子的等价刻画,构成空间的性质,以及控制性;最后,研究了b-几乎DunfordPettis算子与相关算子(b-弱紧算子,弱紧算子,几乎Dunford-Pettis算子)间的关系.     

On some vector lattices of operators and their finite elements

If the vector space of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for to be a vector lattice. and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space or, in the case that is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices , where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in . The paper contains also some generalization of results obtained for the case in [10].      

On extensions of compact positive operators on Banach lattices

Extension properties of compact positive operators on Banach lattices are investigated. The following results are obtained:

• 1. 

  (1) Any compact positive operator (any compact lattice homomorphism, resp.) from a majorizing sublattice G of a Banach lattice E into another Banach lattice F can be extended to a compact positive operator (a compact lattice homomorphism, resp.) from E into F;

• 2. 

  (2) Any compact positive operator defined on a closed majorizing sublattice G of a Banach lattice E has a compact positive extension on E that preserves the spectrum (a necessary modification is needed).

Related extension problems are also studied.     

Compactness of L-weakly and M-weakly compact operators on Banach lattices

We investigate conditions under which L-weakly compact operators and M-weakly compact operators must be compact.     

Compact groups of positive operators on Banach lattices

In this paper, we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.     

On the structure of non-weakly compact operators on Banach lattices

Mathematische Annalen -     

A Perron-Frobenius theorem for positive polynomial operators in Banach lattices

In this paper, we extend the Perron-Frobenius theorem for positive polynomial operators in Banach lattices. The result obtained is applied to derive necessary and sufficient conditions for the stability of positive polynomial operators. Then we study stability radii: complex, real and positive radii of positive polynomial operators and show that in this case the three radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.     

本性不可比的Banach空间

通过引入本性严格奇异算子和投影严格奇异算子,讨论本性不可比的Banach空间,对M.Gonzalez提出的猜想作出部分的肯定回答,最后,利用空间的本性不可比构造了一类空间,其上的非本性算子类和严格余奇异算子类是不重合的。     

On Banach lattices of operators

Let Λ₁ and Λ₂ be infinte-dimensional, Banach lattices such thatc _o is not finitely representable in Λ₂. Then the bounded linear operators from Λ₁ to Λ₂ form a lattice if and only if Λ₁ is an abstract AL space.     

Some properties of the class of positive Dunford-Pettis operators

We characterize Banach lattices for which each positive Dunford-Pettis operator is M-weakly compact (resp. L-weakly compact) and we give some consequences.     

Banach空间结构理论的重大进展--关于Gowers-Maurey系列成果

最近,ＧｏｗｅｒｓＷ．Ｔ．和ＭａｕｒｅｙＢ．构造出第一例遗传不可分解空间,否定地解决了无条件基序列问题,由此导致了Ｂａｎａｃｈ空间的结构理论研究中系列问题的解决,本综述介绍这一新动向,反映了Ｇ－Ｍ系列成果,全文分为七个部分,１个历史回顾与问题沿革;２．Ｇ－Ｍ空间ＸＧ１及其遗传不可分解性质,３．关于空间ＸＧ１上的算子构成;４个关于共轭空间ＸＧ１,５．关于Ｇ－Ｍ的系列成果,６．Ｇ－Ｍ型空间构造的