关于Krein空间的补空间

在[1]中,L.de Brange教授引入了Krein空间的补空间的概念,这是他的重要思想,有许多应用。本文主要讨论一下补空间的简单性质。 设H、K是Krein空间,HK。记H到K的嵌入算子为i:H→K。如果i是连续压缩,那么称H是连续压缩地嵌入K。我们记为H→K。这时P=ii~*:K→K是一个自共轭算子,而且P~2≤P。反之,若P是Krein空间K中一个自共轭算子,且P~2≤P,在[1]中证明了存在唯一的Krein空间H→K,使P=ii~*,这儿i是H到K中嵌入。     

一类次正常算子的表示

[1]中提出如下的问题:若S是Hilbert空间H上次正常算子,而且S~*S-SS~*是有限秩算子。能否绐出S的一般形式。当S~*S-SS~*是一秩算子时,S=aI bU_ ,这儿U-是单向平移算子。在[2]中对自对偶次正常算子情况,给出了一个表示。这里我们从另一个角度来部分回答上述问题。 Hilbert空间H上算子S称为次正常算子是指存在一个Hilbert空间R(?)H上正常算子N,NH(?)H,而S=N|H。称N为S的正常延拓,相对于R=H(?)H-,正常算子N有表示     

张量空间中的真正锥

本文指出了文献 [1 ]在张量空间中定义的两种锥是一致的 ,证明了它们是张量空间中的最小真正锥 ,并可用来表示有限维实空间中由锥不变算子所组成的锥 ,因而可用来研究锥不变算子 .     

球几何迁移算子的豫解算子的非全连续性

雷勒(J.Lehner)在[1]中说到:在希尔柏特空间H中球几何迁移算子A的豫解算子是全连续算子,这个结论是不正确的,下面给出证明:设希尔柏特空间H是图中的半圆上以P(x,y)=y为权的绝对平方可积函数的空间,内积定义为其中。线性算子A定义如下:的定义域为关于x绝对连续,其中是大于零的常数,     

可列谱的仿正规算子是正规算子

本文中,总设H是复平面C上的Hilbert空间,φ(H)是H上的线性有界算子全体。设T∈(H),称T为仿正规算子。若对所有x∈H,‖Tx‖~2 ‖T~2x‖ ‖x‖。易知半亚正规算子(因而亚正规算子)是仿正规算子。仿正规算子的正规性条件是一个引人注意的问题。1972年,T.Saito在其专著[1]中提出了一个问题:多项式紧的仿正规算子是否正规算子?1982年,文[2]指出多项式紧的仿正规算子必是正规算子的紧摄动。本文中,我们利用超穷     

算子的*交换性

<正> 在[1]中讨论了C代数的推广,如J代数、JC代数等.其中引入了算子的*交换概念,但未对*交换的算子进行讨论.本文利用极分解式讨论算子的*交换性。 下面H为复Hilbert空间,(H)表示H→H的线性有界算子全体.(H)中的运算及范数等均按通常的意义.     

关于Banach空间算子的本性谱

这篇注记把文献[1]中关于希尔伯特空间算子本性谱特征刻划的某些结果推广到一般巴拿赫空间上去。其中之一是说明,零属于算子 T 的左本性谱的充分必要条件是存在一个紧算子 K,使得 T+K 的零空间无限维,或 R(T)不可补。     

一般Wiener-Hopf算子和双侧位移在算子分解中的应用

Wiener Hopf 积分方程不仅在物理、力学等方面有着重要的应用,而且为泛函分析提供了一类算子的模型,因此有关学者对它们进行了多方面的研究([1]、[4]、[6]).本文的主要目的是在可分的 Hilbert 空间 H 上,利用双侧位移算子给出算子类 F_p={A∈L(H)可逆;T_(?)(A)∈Φ(R(P))}中元素的分解表达式:A=A_V~xA_ T_0.从而不但推广了[4]的     

Bergman空间L_a~1(Ω)上的Toeplitz和Hankel算子

本文讨论了 Bergman空间 L1a(Ω )中 Toeplitz和 Hankel算子的 W* 紧性 ,得到与 L2a(Ω )上 T- H算子紧性 [4]类似的某些结果     

Meyer-Knig-Zeller算子在Hlder范数下的逼近性质

首先介绍了Hlder空间中相关范数、连续模的基本概念以及Meyer-KnigZeller算子的定义,然后讨论了Meyer-Knig-Zeller算子在Hlder空间中的逼近性质.利用连续模与K-泛函的等价关系,得到了在Hlder范数下Meyer-Knig-Zeller算子对[0,1]上连续函数逼近的正定理.     

Hypoellipticity in spaces of ultradistributions—Study of a model case

In this work we study C ^??-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying H?rmander??s condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.     

A criterion for Hill operators to be spectral operators of scalar type

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrö dinger operator) H = ?d ² /dx ² + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrödinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion.The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.     

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ $\mathbb{T}^d $ , where $\mathbb{T}^d $ is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ?^d, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ $\mathbb{T}^d $ if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H₊(k), k ∈ $\mathbb{T}^d $ , corresponding to a two-particle system with repulsive interaction.     

Probabilistic Approximation of the Evolution Operator

A method for approximation of the operator e^?itH, where \(H = - \frac{1}{2}\frac{{{d^2}}}{{d{x^2}}} + V(x)\), in the strong operator topology is proposed. The approximating operators have the form of expectations of functionals of a certain random point field.     

Self-adjointness of the dirac operator in a space of vector functions

This paper is devoted to the proof of the self-adjointness of the minimal operator defined on the space L₂(? ∞, ∞; H) (H being a separable Hilbert space) by the expression L=iJ(d/dt)+A+B(t). The coefficients in this expression are self-adjoint operators on H, with A being unbounded, AJ+JA = 0, and the function ∥B(t)∥ H being assumed to lie in L ₂ ^loc (? ∞, ∞). The result obtained is applicable to the Dirac operator.     

Noncommutative Grassmannian <Emphasis Type="Italic">U</Emphasis>(1) sigma model and a Bargmann-Fock space

We consider a Grassmannian version of the noncommutative U(1) sigma model specified by the energy functional E(P) = ‖[a, P]‖ _HS ² , where P is an orthogonal projection operator in a Hilbert space H and a: H → H is the standard annihilation operator. With H realized as a Bargmann-Fock space, we describe all solutions with a one-dimensional range and prove that the operator [a, P] is densely defined in H for a certain class of projection operators P with infinite-dimensional ranges and kernels. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 347–357, December, 2007.     

Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ? ⁿ . The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hörmander’s L ² estimates for the $\bar{\partial}$ operator are key ingredients in the proof.     

Traces of operators with a relatively compact perturbation

This is a continuation of the authors’ series of papers on the theory of regularized traces of abstract discrete operators. We prove a theorem in which the perturbing operator B is subordinate to the operator A ₀ in the sense that BA ₀ ^?δ is a compact operator belonging to some Schatten-von Neumann class of finite order. Apart from covering new classes of operators, and in contrast to our preceding papers, we give a unified statement of the theorem regardless of whether the resolvent of the unperturbed operator belongs to the trace class. Two examples are given in which the result is applied to ordinary differential operators as well as to partial differential operators.     

On composition operators onH p -spaces in several variables

In this paper, we prove that a composition operator onH ^p (B) is Fredholm if and only if it is invertible if and only if its symbol is an automorphism onB, and give the representation of the spectra of a class of composition operators. In addition, using composition operator, we discuss intertwining Toeplitz operators. Supported by NNSF and PDSF     

New Studies on the Fock Space Associated to the Generalized Airy Operator and Applications

In this work, we introduce the Fock space \(F_\nu (\mathbb {C})\) associated to the Airy operator \(L_\nu \), and we establish Heisenberg-type uncertainty principle for this space. Next, we study the Toeplitz operators, the Hankel operators and the translation operators on this space. Furthermore, we give an application of the theory of extremal function and reproducing kernel of Hilbert space, to establish the extremal function associated to a bounded linear operator \(T{:}\,F_\nu (\mathbb {C})\rightarrow H\), where H be a Hilbert space. Finally, we come up with some results regarding the extremal functions, when T is the difference operator and the Dunkl-difference operator, respectively.