基于二次算子族的无界Hamilton算子根向量组的基性质

利用无界Hamilton算子导出的二次算子族,本文研究了一类无界Hamilton算子根向量组的Schauder基性质.首先,建立了无界Hamilton算子的根向量与相应的二次算子族的根向量之间的关系.其次,借助二次算子族谱的相关性质,刻画了无界Hamilton算子的本征值分布以及本征值的代数指标,并得到了无界Hamilton算子的根向量组是某个Hilbert空间的一个块状Schauder基的充要条件.最后,将所得结果应用于矩形薄板弯曲问题.     

二次算子族及非负无穷维Hamilton算子的谱分布

本文讨论了一类在弦和梁的微小振动中出现的二次算子族L(λ)=λ~2MλK-A的谱分布问题,进而将所得结论与无穷维Hamilton算子联系起来,利用无穷维Hamilton算子的特殊结构,得到了一类非负无穷维Hamilton算子的谱分布,这为无穷维Hamilton算子的半群方法提供了理论保证.     

Recovery of differential equations with nonlinear dependence on the spectral parameter

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line is studied. We give a formulation of the inverse problem, prove the uniqueness theorem and provided a procedure for constructing the solution of the inverse problem. We also establishe connections with inverse problems for partial differential equations.     

Bergman Space Structure,Commutative Algebras of Toeplitz Operators,and Hyperbolic Geometry

We exhibit a surprising but natural connection among the Bergman space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. The commutative C*-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil. It turns out that the C*-algebra generated by Toeplitz operators with this class of symbols is commutative. Submitted: January 15, 2001?Revised: February 7, 2002     

Schrödinger operators and associated hyperbolic pencils

For a large class of Schrödinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are significantly easier to study. Then, we give some applications to the original Schrödinger operators including one-dimensional Schrödinger operators with L²-operator-valued potentials, multidimensional Schrödinger operators with slowly decaying potentials.     

The Theory of Operators with Dominant Main Diagonal. I.

In this paper a characterization of the symmetric operators on a finite dimensional Hilbert space which have a matrix representation with a dominant diagonal with respect to any orthonormal basis are obtained. The set of such operators is a solid, reproducing, normal and acute cone in the space of symmetric operators. These results are applied to localizing the spectrum of operators pencils.     

ON THE GENERALIZED RESOLVENT OF LINEAR PENCILS IN BANACH SPACES

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil λ→ T λ S are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.     

Operator pencils on the algebra of densities

We continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role played by the geometry of the extended manifold where the algebra of densities is a special class of functions. Firstly we consider basic examples. We give a projective line of diff(M)-equivariant pencil liftings for first order operators and describe the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO) pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO pencil lifting to describe all regular proj-equivariant pencil liftings. In particular, the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian.     

Sums of characteristic values of compact polynomial operator pencils

We consider polynomial pencils whose coefficients are compact operators. Bounds for the sums of absolute values, real and imaginary parts of characteristic values are derived. Applications to differential and difference equations are discussed.     

On the approximation of convex functions by Bernstein-type operators

As a consequence of Jensen's inequality, centered operators of probabilistic type (also called Bernstein-type operators) approximate convex functions from above. Starting from this fact, we consider several pairs of classical operators and determine, in each case, which one is better to approximate convex functions. In almost all the discussed examples, the conclusion follows from a simple argument concerning composition of operators. However, when comparing Szász-Mirakyan operators with Bernstein operators over the positive semi-axis, the result is derived from the convex ordering of the involved probability distributions. Analogous results for non-centered operators are also considered.     

(NCI)算子与(NFI)算子

引入两类与不变子空间密切相关的算子,即(NCI)算子与(NFI)算子.考虑这两类算子的性质,例如存在性,相似不变性,以及与强不可约算子的关系等.并在可分Banach空间上讨论谱为单点集{0}的算子能否小紧摄动成为(NFI)算子.     

Banach 空间上算子的几种不可约性

本文根据Cowen-Douglas 算子的定义引入两类与强不可约算子有紧密联系的算子类— B_n 算子与B 算子. 说明了在可分Banach 空间上存在B_n 算子与B 算子. 文章详细讨论了几种具有不可约性的算子类之间的关系并得到了一个关系图. 本文还给出这些算子类的一些性质, 包括算子的(拟) 相似不变性等.     

Numerical Approaches to Compute Spectra of Non-Self Adjoint Operators and Quadratic Pencils

In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators. This leads to solve nonlinear eigenvalue problems. We begin with a review of theoretical results for the spectra of quadratic operators, especially for the Schrödinger pencils. Then we present the numerical methods developed to compute the spectra: spectral methods and finite difference discretization, in infinite or in bounded domains. The numerical results obtained are analyzed and compared with the theoretical results. The main difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are very unstable.     

复对称算子的一些等价性质

证明了在复对称算子的前提下,对数-亚正规算子与正规算子是等价的,并且给出了复对称算子的一些等价性质;最后通过给出例子来说明我们的结论.     

Enclosure results for second-order relative spectra by elementary means

Motivated by the general approach due to Shargorodsky we derive enclosure results for the second-order relative spectrum of bounded selfadjoint operators by studying quadratic operator pencils. The quality of the results is discussed by means of a simple example.

     

Homogeneous operators and homogeneous integral operators

We introduce and study in a general setting the concept of homogeneity of an operator and, in particular, the notion of homogeneity of an integral operator. In the latter case, homogeneous kernels of such operators are also studied. The concept of homogeneity is associated with transformations of a measure—measure dilations, which are most natural in the context of our general research scheme. For the study of integral operators, the notions of weak and strong homogeneity of the kernel are introduced. The weak case is proved to generate a homogeneous operator in the sense of our definition, while the stronger condition corresponds to the most relevant specific examples—classes of homogeneous integral operators on various metric spaces—and allows us to obtain an explicit general form for the kernels of such operators. The examples given in the article—various specific cases—illustrate general statements and results given in the paper and at the same time are of interest in their own way.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.     

Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces

The behavior of certain weighted Hardy-type operators on rearrangement-invariant function spaces is thoroughly studied. Emphasis is put on the optimality of the obtained results. First, the optimal rearrangement-invariant function spaces guaranteeing the boundedness of the operators from/to a given rearrangement-invariant function space are described. Second, the optimal rearrangement-invariant function norms being sometimes complicated, the question of whether and how they can be simplified to more manageable expressions is addressed. Next, the relation between optimal rearrangement-invariant function spaces and interpolation spaces is investigated. Last, iterated weighted Hardy-type operators are also studied.     

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.     

Elementary operators on algebras of unbounded operators

Summary Structural results about elementary operators of length one, local elementary operators and injectivity preserving maps are proved. These are generalizations of results concerning algebras of bounded operators on Banach spaces to algebras of unbounded operators on Hilbert spaces.