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.     

离散交换群上Toeplitz 算子的代数性质

广泛的意义下定义 Toeplitz 算子, 给出了Toeplitz 算子乘积仍为Toeplitz 算子的充分必要条件, Toeplitz算子是正规算子的充分必要条件以及 Toeplitz 算子可交换的一个必要条件,从而推广了经典 Toeplitz 算子的相应结果.     

Hibert空间中广义框架的非交性

引入了Hilbert空间H中广义框架的非交性、强非交性,讨论了它们的一些性质;并且引入了保非交算子、强保非交算子,证明了酉算子、可逆算子是强保非交算子,下有界算子、余等距算子是保非交算子.     

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.     

交互作用Fock空间l2(Γ,{λn})上的梯度算子和散度算子

主要研究基于l2(N)上交互作用Fock空间l2(Γ,{λn})中的梯度算子和散度算子.首先定义交互作用Fock空间l2(Γ,{λn})上的梯度算子和散度算子;然后研究梯度、散度算子所具有的算子性质;最后研究由梯度算子和散度算子构成的复合算子与该空间中其他算子的关系.结果表明:交互作用Fock空间l2(Γ,{λn})中的梯度算子和散度算子是稠定线性闭算子,而且它们互为共轭算子,另外,由于该空间中交互因子的特殊性,可得到在l2(Γ,{λn})的n-粒子空间中,梯度、散度算子所构成的复合算子和计数算子相等.     

具积分边值条件的迁移方程的性质

在L~1空间上研究了一类增生的细菌群体中具积分边界条件的迁移方程.得出迁移算子是预解正算子,微分算子的共轭算子及共轭算子的定义域.证明了迁移算子的共轭算子定义域的正锥在共轭空间的正锥中共尾.最后证明了迁移算子的增长界等于其谱界.     

L~1空间中具有周期边界条件的迁移方程的性质

在L~1空间研究板几何中具有周期边界条件的迁移方程.证明了迁移算子是预解正算子,得到了微分算子的共轭算子及共轭算子的定义域.证明了迁移算子的共轭算子定义域的正锥在共轭空间的正锥中共尾.最后证明了迁移算子的增长界等于其谱界.     

含参量蕴涵算子的构造及其性质

从空间几何的角度给出了几类含参量蕴涵算子，它们可以将常见的三种蕴涵算子Lukasiewicz算子、如算子及Godel算子包含其中，此外，还讨论了蕴涵算子的正则性质。     

无穷维Hamilton算子的辛自伴性

本文运用算子扰动理论研究了无穷维Hamilton算子的共轭算子,进而得到了无穷维Hamilton算子为辛自伴算子的若干充分条件.     

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．     

Krein's Theorem for the Dissipative Operators with Finite Impulsive Effects

In this article, singular dissipative operators with finite impulsive conditions are investigated. In particular, after passing to the inverse operators, it is obtained that the imaginary parts of the inverse operators are nuclear. Finally, using Krein's theorem, it is proved that all root vectors of the singular dissipative operators with finite impulsive conditions are complete in the Hilbert space.     

Spectral analysis of eigenparameter dependent boundary value transmission problems

In this paper, the singular second order differential operators are considered defined on the multi-interval. Some boundary and transmission conditions are imposed on the maximal domain functions with the spectral parameter. After constructing the differential operators associated with the boundary value transmission problems on the suitable Hilbert spaces, it is proved that these operators are the maximal dissipative operators. Finally constructing the model operators which are established with the help of the scattering functions, it is proved that all root vectors of the maximal dissipative operators are complete in the Hilbert spaces.     

Symplectic geometry and dissipative differential operators

Everitt and Markus characterized the domains of self-adjoint differential operators in terms of Lagrangian subspaces of complex symplectic spaces. In this paper we define Dissipative and strictly Dissipative subspaces for complex symplectic spaces and characterize the domains of dissipative and strictly dissipative differential operators in terms of these subspaces.     

P耗散算子的研究

本文利用广义半内积空间及广义不定度规空间,解决Ｂａｎａｃｈ空间中Ｐ耗散算子的极大耗散扩张表示,并得到很有意义的应用．     

Dissipative extension theory for linear relations

This work is devoted to dissipative extension theory for dissipative linear relations. We give a self-consistent theory of extensions by generalizing the theory on symmetric extensions of symmetric operators. Several results on the properties of dissipative relations are proven. Finally, we deal with the spectral properties of dissipative extensions of dissipative relations and provide results concerning particular realizations of this general setting.     

The Birman-Schwinger principle in von Neumann algebras of finite type

We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman-Schwinger principle in this setting. As an application of this result, revisiting the Birman-Krein formula in the abstract scattering theory, we represent the de la Harpe-Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.     

The one‐dimensional Schrödinger operator on bounded time scales

In this paper, we consider the one‐dimensional Schrödinger operator on bounded time scales. We construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self‐adjoint, and other extensions of the dissipative Schrödinger operators in terms of boundary conditions. In particular, using Lidskii's theorem, we prove a theorem on completeness of the system of root vectors of the dissipative Schrödinger operators on bounded time scales. Copyright © 2016 John Wiley & Sons, Ltd.     

Extensions,dilations, and spectral problems of singular Hamiltonian systems

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit‐point case at a(b) and limit‐circle case at b(a)) acting in the Hilbert space In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self‐adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at a” and “dissipative at b.” For 2 cases, a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl‐Titchmarsh function of the corresponding self‐adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.     

Spectral analysis of the Direct Sum Hamiltonian Operators

In this paper we investigate the deficiency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foia¸s characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces.     

Realization theorems for stationary multiscale systems

We construct coisometric and quasi-coisometric realizations for transfer operators of multiscale causal stationary dissipative systems.