赋范空间的λ——性质

本文讨论赋范空间的λ－性质，得到了ｌ１（ｘｎ）和Ｌ∞（Ｘｎ）具有λ－性质的充分必要条件，解决了Ｒ．Ｍ．Ａｒｏｎ和Ｒ．Ｈ．Ｌｏｈｍａｎ在文（１）中提出的两个问题，此外，还研究了Ｂａｎａｃｈ空间，ｌｐ（Ｘｎ）（１〈ｐ〈∞）具有一致λ－性质的充分必要条件。     

Orlicz-Bochner序列空间的(K)性质

讨论Orlicz-Bochner序列空间的(K)性质,利用生成函数M及Orlicz-Bochner序列空间的性质,并在假设Banach空间X具有Schur性质的前提下,得到Orlicz-Bochner序列空间具有(K)性质的充要条件,进而得到Orlicz-Bochner序列空间具有弱不动点性质的一个充分条件.     

关于K—极凸Banach空间

引进Ｋ－极凸Ｂａｎａｃｈ空间，证明了ＸＫ－极凸当且仅当Ｘ自反、Ｋ－严格凸且有（Ｈ）性质，得到了Ｋ－极凸空间的一些性质，并讨论了Ｋ－极凸与Ｋ－Ｋ－强光滑、Ｋ－一致凸及完全Ｋ－凸的关系。     

Banach空间的渐近赋范性质与（S）性质

本文证明渐近赋范性质与（ｓ）性质具有对偶性，并且给出一个Ｂａｎａｃｈ空间为ＨａｈｎＢａｎａｃｈ光滑的充要条件。     

Toeplitz算子Fredholm性质的局部化描述

利用Banach代数中的局部化定理描述Bergman空间上Toeplitz算子的Fredholm性质,证明了类似于Hardy空间上的一个局部化定理．讨论了L∞（D）的极大理想空间的一些性质．     

对偶ω^＊-可分的Banach空间上存在具有球覆盖性质的1＋ε-等价范数

我们称赋范空间具有球覆盖性质,如果它的单位球球面能被不含原点的一列开球所覆盖．本文证明了每个对偶ω^＊-可分的Banach空间都可赋1＋ε-等价范数,使该空间对于这个新范数具有球覆盖性质．     

关于C0半群的渐近稳定性

研究Banach空间上等距C0群的性质，据此讨论了C0半群的渐近稳定性和C0半群完全轨线的存在性．在文章的最后，回答了黄发伦提出的一个关于渐近稳定性的猜测．     

多元Bernstein—Sikkema算子的逼近性质

本文对定义在单形上的Ｂｅｒｎｓｔｅｉｎ－Ｓｉｋｋｅｍａ算子讨论其导数的收敛性质，同时出给出了高阶渐近展开式及Ｂｅｓｏｖ空间中的几个等价关系。     

Banach空间的p-弱近似性质

给出了Banach空间的p-弱近似性质和p-有界弱近似性质的定义,获得了这些性质的一些刻画.利用这些刻画证明了如果一个Banach空间X的对偶空间X~*有p-弱近似性质(或p-有界弱近似性质),则X有p-弱近似性质(或p-有界弱近似性质),在一般情况下反之不成立.     

对偶ω*-可分的Banach空间上存在具有球覆盖性质的1+ε-等价范数

我们称赋范空间具有球覆盖性质,如果它的单位球球面能被不含原点的一列开球所覆盖.本文证明了每个对偶ω~*-可分的Banach空间都可赋1+ε-等价范数,使该空间对于这个新范数具有球覆盖性质.     

算子级数的无条件收敛

讨论Banach空间中算子级数的无条件收敛.分别在自反和含无条件基的条件下推广了Vcctor和Cary关于Hilbert空间的两个相应结果.即[1]的定理2和命题6.并且肯定回答了他们留下的问题.     

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.     

Sampling in Reproducing Kernel Banach Spaces

It is well-known the close relationship between reproducing kernel Hilbert spaces and sampling theory. The concept of reproducing kernel Hilbert space has been recently generalized to the case of Banach spaces. In this paper, some sampling results are proven in this new setting of reproducing kernel Banach spaces.     

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.

     

Banach空间的弱*序列紧性

本文研究了Ｂａｎａｃｈ空间的弱序列紧性．Ｂａｎａｃｈ空间Ｘ称为有（ｗ）性质，如果Ｘ（Ｘ的共轭空间）的每个有界序列有弱收敛子列．我们证明了，如果Ｂａｎａｃｈ空间Ｘ有（ｗ）性质，那么ｌｐ（Ｘ）（１≤ｐ＜＋∞）与ｃ０（Ｘ）也有（ｗ）性质．     

On best approximation by rational and holomorphic mappings between Banach spaces

After proving a generalized version of Garkavi's theorem, we give as applications proofs of existence results on best approximation by polynomials, and fractional linear and holomorphic operators between Banach spaces. We also obtain theorems on best approximation by some types of rational functions defined in open subsets of Banach spaces. By considering a natural non-normable distance we prove that every mapping bounded on the bounded subsets of a Banach space has best approximation by polynomials of degree less than or equal to a fixed natural number n.     

迭代逼近m-增生映象的零点

设E是具有一致正规结构的实Banach空间,其范数是一致Gateaux可微的.设A是m-增生映象,使得C=■是E的凸子集,数列{α_n)■[0,1],{r_n}■ (0,∞),在适当的条件下,则由(1.2)式定义的迭代序列{x_n}强收敛于A~(-1)(0)中的点.其次证明了:设E是一致凸Banach空间,其范数是Frechet可微的.设数列{α_n},{β_n)■(0,1),{r_n}■(0,∞),满足适当的条件.如果A~(-1)(0)∩B~(-1)(0)≠φ,则由(3.20)式定义的序列{x_n}弱收敛于A~(-1)(0)∩B~(-1)(0)中的点.其结果推广和改进了Kamimura,Takahashi(2000)的定理2及Xu H.K.(2006)的定理4.1,定理4.2和定理4.3:(i)Kamimura,Takahashi(2000)定理2中的假设"自反Banach空间E的每个有界闭凸子集对非扩张自映象有不动点性质"被去掉;(ii)Xu H.K.(2006)的假设"E是具有弱连续对偶映象J_φ的自反Banach空间",被本文的假设"E是具有一致正规结构且其范数是一致Gateaux可微的Banach空间"所取代.从而补充了Xu H.K.(2006)未包含的另外一些Banach空间.同时还证明了逼近两个m-增生映象的公共零点,其结果也推广和改进了Mainge的相应结果.     

Banach–Lie groupoids and generalized inversion

We study a few basic properties of Banach–Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach–Lie groupoid sheds fresh light on earlier research on some infinite-dimensional manifolds associated with Banach algebras.     

On the weak compactness in Banach lattices

The Grothendieck's criterion of weak compactness in spaceC(S) has been extended by C.P.Niculescu for weakly sequentially complete Banach lattices. This paper first provides some sufficient and necessary conditions for weakly sequentially complete Banach lattices, with the result that the criterion C.P.Niculescu obtained of the weak compactness has actually characterized the weakly sequentially Banach lattices. At the same time we extend C.P.Niculescu's important results. Then we solve negatively Problem 1.11 which was posed by C.P.Niculescu, and a sufficient condition for this is given which generalizes Pelczyński's result.