广义算子半群与广义分布参数系统的适定性

首先,针对广义分布参数系统的求解问题,提出了由Hilbert空间中有界线性算子所引导的广义算子半群和广义积分半群;其次,讨论了广义预解算子的性质、广义算子半群与广义积分半群的性质;最后,研究了广义分布参数系统的适定性问题.     

On the closability of paranormal operators

We show two examples of operators acting on some Hilbert space and having invariant domains: a paranormal operator, which is not closable and a paranormal and closable operator, which closure is not paranormal. We start by establishing some general lemmas and propositions associating the families of operators mentioned above.     

Hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the nonsymmetric case.     

Lebesgue type decompositions for nonnegative forms

A nonnegative form on a complex linear space is decomposed with respect to another nonnegative form : it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by which is almost dominated by . The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions.     

Iterated solutions of linear operator equations with the Tau method

The Tau Method produces polynomial approximations of solutions of differential equations. The purpose of this paper is (i) to extend the recursive formulation of this method to general linear operator equations defined in a separable Hilbert space, and (ii) to develop an iterative refinement procedure which improves on the accuracy of Tau approximations. Applications to Fredholm integral equations demonstrate the effectiveness of this technique.

     

Larson理想中算子的插值

设a为nest代数，R^∞为A的Larson理想，R(A)为A的根，[R(A)]^s为R（A）的强拓扑闭。在文本中，我们给出[R(A)]^s的纯代数构造；且引进了一个新算子集合I，并证明了：若A的不变子空间格为几乎原子的，则R^∞＝[R(A)]^s=I。利用上述结果，我们研究了当A的不变子空间格为几乎原子时的Larson理想中的算子插值问题。我们得到算子方程AX＝Y在R^∞中有解A的充分必要条件。     

Subgrid stabilization of Galerkin approximations of linear monotone operators

This paper presents a stabilized Galerkin technique for approximatingmonotone linear operators in a Hilbert space. The key idea consistsin introducing an approximation space that is broken up intoresolved scales and subgrid scales so that the bilinear formassociated with the problem satisfies a uniform inf-sup conditionwith respect to this decomposition. An optimal Galerkin approximationis obtained by introducing an artificial diffusion on the subgridscales.     

广义分布参数系统的状态反馈稳定性问题(1)

关于系统的状态反馈稳定性问题的研究一直是现代控制理论研究的重要问题之一.广义分布参数系统是比分布参数系统更广的一类系统,在研究复合材料热导体中的温度分布等问题时会出现这样的系统.本文讨论了H ilbert空间中一阶广义分布参数系统的状态反馈稳定性问题.应用泛函分析及线性算子半群理论的方法给出了使闭环广义分布参数系统渐进稳定的充要条件,充分条件及状态反馈的构造性表达式.这对研究广义分布参数系统的状态反馈稳定性问题具有重要的理论价值.     

Connection colligations on Hilbert bundles

The theory of operator colligations in a Hilbert space is extended to connection colligations on Hilbert bundles. Any invariant subbundle of a Hilbert bundle leads to a decomposition of connection colligation into simpler components. The problem of finding all invariant subbundles of a Hilbert bundle is reduced to a search of subspaces of a fixed Hilbert space which are invariant with respect to the holonomy group of the linear connection.     

Applications of polar decompositions of idempotent and 2-nilpotent operators

An operator means a bounded linear operator on a complex Hilbert space. Using two polar decompositions of idempotent and 2-nilpotent operators, we shall study numerical radii of these two operators and finally we shall discuss two operator transformations, one of which is the generalized Aluthge transformation and another is an extension of the operator transformation by Patel-Tanahashi-Uchiyama.     

Spectrum of a linear operator in a family of banach spaces containing Hilbert space

The mutual relation is established between the spectra of a bounded linear operator acting in a family of Banach spaces. It is assumed in addition that one of the spaces is a Hilbert space and that the operator acting on it is self-adjoint. An example is presented illustrating the properties proved.Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 495–498, October, 1977.     

Certain approximate methods for solving linear operator equations

A solution of linear operator equations in the Hilbert space is constructed by using the best polynomial approximation of the inverse operator. This approach gives rise to certain iteration processes. Error estimates manifest that the suggested schemes may be fairly efficient.     

F格上的逼近算子

粗集理论是波兰学者Pawlak提出的知识表示新理论.Pawlak代数是粗集理论中粗集系统的抽象,其公理系统包含了知识粗表示所必须的全部性质.本文深入研究了F格上的逼近算子,建立了F格上弱逼近算子之间的某些代数运算,从而从理论上建立了各种知识粗表示之间的联系.我们还定义了逼近算子的闭包,进而用逼近算子导出拓扑,为信息系统的近似提供必要的数学基础.最后,作为特例,我们研究了粗集理论中由相似关系导出逼近算子的某些性质.     

Linear preservers of isomorphic types of lattices of invariant operator ranges

We describe all linear self-mappings of the space of bounded linear operators in an infinite dimensional separable complex Hilbert space which preserve the isomorphism class of the lattice of invariant operator ranges.     

On the level sets of the resolvent norm of a linear operator

We construct a bounded linear operator on a Banach space anda closed densely defined operator on a Hilbert space with resolventnorms that are constant in a neighbourhood of zero. We alsodiscuss cases where the norm of the resolvent of a bounded linearoperator cannot be constant on an open set.     

Hypercyclicity of special operators on Hilbert function spaces

In this paper we give some sufficient conditions for the adjoint of a weighted composition operator on a Hilbert space of analytic functions to be hypercyclic. This paper is a part of the second author’s Doctoral thesis written at Shiraz University under the direction of the first author.     

Investigation of the spectral properties of a fourth order differential operator with finite transmission conditions

In this paper, the spectral analysis of a singular dissipative fourth order differential operator in lim‐4 case with finite transmission conditions is investigated. For this purpose, the inverse operator with explicit form is used. Finally, with the help of Liv?ic's theorem, it is proved that all root vectors of the fourth order dissipative operator in lim‐4 case with finite transmission conditions are complete in the Hilbert space. Copyright © 2015 John Wiley & Sons, Ltd.     

Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.     

算子框架的稳定性

算子框架是广义的框架,它包括框架序列和子空间框架.本文分别讨论了在Bessel算子列和算子框架的扰动下,其对偶框架的稳定性.得到了一些新结果,这些结果是序列框架稳定性的推广.     

Structure of the point spectrum of a linear operator

In order that a set of complex numbers be the point spectrum of some linear operator in a separable Hilbert space it is necessary and sufficient that it be a set of type F_σ.