冯·诺依曼代数的可测算子的性质

本文研究了冯·诺依曼代数的可测算子的基本性质,定义了阶梯算子,证明了任意一个正可测算子可以由阶梯算子在定义域内按照强算子拓扑逼近,从而证明了任意一个可测算子可以由投影在定义域内按照强算子拓扑逼近.此外,还讨论了可测算子与有界算子的复合算子的可测性.     

U-标算子的结构分析

类似与标型谱算子,U-标算子是否拟仿射相似于自伴算子是一“公开问题”．尽管对具纯离散谱的U-标算子答案是肯定的,但一般情况下并不成立．本文继续探讨这一问题,证明了U-标算子在一强范数拓扑意义下是Hermite算子,或者说U-标算子拟仿射相似于Hermite算子,并给出U-标算子是标型谱算子的充要条件．     

一类解析算子函数空间上的复合算子

本文用算子函数论的方法，研究了解析算子函数的Banach空间X，X0上的复合算子．给出此复合算子为有界的条件，并刻划了此复合算子在X0上为紧的特征．     

一类无穷维Hamilton算子的Fredholm性及本质谱

讨论了一类无穷维Hamilton算子的Fredholm性,由于无穷维Hamilton算子是分块算子矩阵,将它的Fredholm性用它的元素算子的某种组合来描述,给出了无穷维Hamilton算子是Fredholm算子的充分必要条件.     

关于空间l∞上的算子理想

讨论算子代数Ｂ（ｌ∞）中各算子理想之间的关系，说明严格奇异算子理想Ｓ（ｌ∞）与弱紧算子理想、也与非本性算子理想重合，成为Ｂ（Ｌ∞）中最大的、非平凡的闭算子理想；它们不与紧算子理想重合。     

非游荡算子的性质*

本文研究一类具有混纯性质的线性算子:非游荡算子,该类算子仅在无穷维线性空间中.我们给出非游荡算子紧集上的超循环分解.     

拟紧算子的谱性质和遍历性

对Banach空间X上的一个线性有界算子A,若存在一紧算子Q和一自然数m,使得‖A~m-Q‖<1,则称A是拟紧算子.本文使用算子谱理论的方法,从多个方面刻划了算子的拟紧性.     

Banach格上的b-Dunford-Pettis算子

为了进一步研究Banach格上算子的性质,受b-序有界集和Dunford-Pettis集定义的启发,给出了b-Dunford-Pettis算子的定义,研究了该算子与b-AM-紧算子(Dunford-Pettis全连续算子,弱极限算子,序Dunford-Pettis算子)间的关系;利用b-Dunford-Pettis算子与Dunford-Pettis算子的共轭关系,证明了b-Dunford-Pettis算子满足控制性.     

Hamacher算子的一个新性质及其应用

完善了Hamacher算子的定义;发现了Hamacher算子的一个新性质;分析了Hamacher算子与其它模糊算子之间的关系。最后,根据Hamacher算子的新性质,给出了一个新的模糊评判的方法。     

解析函数的Banach空间上之复合算子

本文研究了一类解析函数的Ｂａｎａｃｈ空间Ｘ上之复合算子，这类空间包含了Ｂｌｏｃｈ空间，并且可看作Ｂｅｒｇｍａｎ空间Ｌ１ａ（Ｄ）中具有原子分解的解析函数的对偶空间．我们刻划了这类空间上紧复合算子及Ｆｒｅｄｈｏｌｍ复合算子的特征，此外，还研究了具有闭值域的复合算子．     

Dirichlet空间上Toeplitz算子的紧性

本文给出了 Ｄｉｒｉｃｈｌｅｔ空间上 Ｔｏｅｌｐｉｔｚ算子为紧算子的充要条件,并证明具有 Ｃ１－符号的 Ｔｏｅｐｌｉｔｚ算子为紧算子当且仅当它为零算子,当且仅当符号的边值为零．     

Positivity of Toeplitz operators on harmonic Bergman space

In this paper, we study positive Toeplitz operators on the harmonic Bergman space via their Berezin transforms. We consider the Toeplitz operators with continuous harmonic symbols on the closed disk and show that the Toeplitz operator is positive if and only if its Berezin transform is nonnegative on the disk. On the other hand, we construct a function such that the Toeplitz operator with this function as the symbol is not positive but its Berezin transform is positive on the disk. We also consider the harmonic Bergman space on the upper half plane and prove that in this case the positive Toeplitz operators with continuous integrable harmonic symbols must be the zero operator.     

Operators on the orthogonal complement of the Dirichlet space

In this paper we prove that a dual Hankel operator is zero if and only if its symbol is orthogonal to the Dirichlet space in the Sobolev space, and characterize the symbols for (semi-)commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space.     

The finite sum of finite products of Toeplitz operators on the polydisk

A limit theorem is established for a finite sum of finite products of Toeplitz operators on the Hardy space of the polydisk. As a consequence we show that the product of six Toeplitz operators with pluriharmonic symbols is compact iff the product equals zero iff one of these Toeplitz operators equals zero.     

解析Toeplitz代数的本质换位及其相关问题

在本文中，我们决定出多复变Ｈａｒｄｙ空间Ｈ２上解析Ｔｏｅｐｌｉｔｚ代数的本质换位．即一个算子与所有解析Ｔｏｅｐｌｉｔｚ算子本质可换，当且仅当它是符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的紧扰动．由此，符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（Ａｃ）在Ｃａｌｋｉｎ代数中的像是极大可换闭代数，这导致了Ｌ．Ｃｏｂｕｒｎ正合列的极大扩充．从这个事实，证明了符号属于Ａｃ的Ｔｏｅｐｌｉｔｚ算子的本质谱是连通的，这大大改进了Ｃ－Ｓ最近的工作．从本文的主要定理，证明了Ｔｏｅｐｌｉｔｚ代数Ｆ（Ｌ∞）的本质换位和本质中心是由符号属于ＱＣ的Ｔｏｅｐｌｉｔｚ算子生成的代数Ｆ（ＱＣ），这些结果又导致了对代数Ｆ（Ｈ∞）＋Ｋ自同构群的确定．     

多圆柱上Dirichlet空间中Toeplitz算子的交换性

本文讨论了多圆柱上Dirichlet空间中的正规Toeplitz算子以及全纯指标和反全纯指标的两个Toeplitz算子的交换性.我们证明具有多圆柱上Dirichlet空间中反全纯指标的两个Toeplitz算子可交换当且仅当两个指标是线性相关的,同时证明全纯指标和反全纯指标的两个Toeplitz算子可交换当且仅当两个指标有一个是常数.     

Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols

We consider Toeplitz operators with symbols that are almost periodic matrix functions of several variables. It is shown that under certain conditions on the group generated by the Fourier support of the symbol, a Toeplitz operator is Fredholm if and only if it is invertible.

     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Algebraic properties of Toeplitz operators on the Dirichlet space

We study some algebraic properties of Toeplitz operators on the Dirichlet space. We first characterize (semi-)commuting Toeplitz operators with harmonic symbols. Next we study the product problem of when product of two Toeplitz operators is another Toeplitz operator. As an application, we show that the zero product of two Toeplitz operators with harmonic symbol has only a trivial solution. Also, the corresponding compact product problem is studied.     

Toeplitz算子在Dirichlet空间上的亚正规性

In this paper,we prove that the necessary and sufficient condition for a Toeplitz operator Tu on the Dirichlet space to be hyponormal is that the symbol u is constant for the case that the projection of u in the Dirichlet space is a polynomial and for the case that u is a class of special symbols,respectively.We also prove that a Toeplitz operator with harmonic polynomial symbol on the harmonic Dirichlet space is hyponormal if and only if its symbol is constant.