Adjoints of linear fractional composition operators on the Dirichlet space

The adjoint of a linear fractional composition operator acting on the classical Dirichlet space is expressed as another linear fractional composition operator plus a two rank operator. The key point is that, in the Dirichlet space modulo constant functions, many linear fractional composition operators are similar to multiplication operators and, thus, normal. As a particular application, we can easily deduce the spectrum of each linear fractional composition operator acting on such spaces. Even the norm of each linear fractional composition operator is computed on the Dirichlet space modulo constant functions. It is also shown that all this work can be carried out in the Hardy space of the upper half plane.This work was partially supported by Plan Nacional I+D Ref. BFM2000-0360 and Junta de Andalucía Ref. FQM-260. The first named author was also supported by Plan Propio de la Universidad de Cádiz.     

非线性Lipschitz算子的Lipschitz对偶算子及其应用

在文山中我们对非线性Ｌｉｐｓｃｈｉｔｚ算子定义了其Ｌｉｐｓｃｈｉｔｚ对偶算子，并证明了任意非线性Ｌｉｐｓｃｈｉｔｚ算子的Ｌｉｐｓｃｈｉｔｚ对偶算子是一个定义在Ｌｉｐｓｃｈｉｔｚ对偶空间上的有界线性算子．本文还进一步证明：设Ｃ为 Ｂａｎａｃｈ空间 Ｘ的闭子集，Ｃ＊Ｌ为Ｃ的 Ｌｉｐｓｃｈｉｔｚ对偶空间，Ｕ为 Ｃ＊Ｌ上的有界线性算子，则当且仅当 Ｕ为 ｗ＊－ｗ＊连续的同态变换时，存在Ｌｉｐｓｃｈｉｔｚ连续算子Ｔ，使Ｕ为Ｔ的Ｌｉｐｓｃｈｉｔｚ对偶算子．这一结论的理论意义在于：它表明一个非线性Ｌｉｐｓｃｈｉｔｚ算子的可逆性问题可转化为有界线性算子的可逆性问题．作为应用，通过引入一个新概念──ＰＸ－对偶算子，在一般框架下给出了非线性算子半群的生成定理．     

A Non-Separable Reflexive Banach Space on Which there are Few Operators

It is shown that there exists a non-separable reflexive Banachspace on which every bounded linear operator is the sum of ascalar multiple of the identity operator and an operator ofseparable range. There is a strong sense that such a Banachspace has as few operators as its linear and topological propertiesallow.     

Cosine of angle and center of mass of an operator

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.     

Random commutation

We investigate the commutation between a continuous linear random operator and a continuous linear deterministic operator on a Banach space. From this we obtain probabilistic versions of theorems by Fuglede and Putnam, both of them dealing with the commutation between continuous linear operators with continuous normal operators on a Hilbert space.     

Perturbed projection methods for split equations of the first kind

The paper investigates the approximate solution of linear operator equations of type (J-K)u=v, with J assumed invertible, obtained by applying a linear projection on both sides of the equation, together with a linear perturbation operator on the left side and a perturbation element on the right side.     

Banach空间的Lipschitz对偶及其应用

本文引进Banach空间E的一个全新对偶空间概念—Lipschitz对偶空间,并证明：任何Banach空间的Lipschitz对偶空间是某个包含E的Banach空间的线性对偶空间,以所引进的新对偶空间为框架,本文定义了非线性Lipschitz算子的Lipshitz对偶算子,证明：任何非线性Lipschitz算子的Lipschitz对偶算子是有界线性算子．所获结果为推广线性算子理论到非线性情形（特别,运用线性算子理论研究非线性算子的特性）开辟了一条新的途径．作为例证,我们应用所建立的理论证明了若干新的非线性一致Lipschitz映象遍历收敛性定理.     

算子概率范数与共鸣定理

提出概率赋范线性空间上集合有界性的简化定义,利用算子概率范数概念。进一步研究概率赋范线性空间上的线性算子理论,并在算子概率赋范空间上,建立了概率有界、概率半有界、非概率无界意义下的共鸣定理。     

Resolvent Tests for Similarity to a Normal Operator

The main result of this paper is the resolvent similarity criterionwhich says that linear growth of the resolvent towards the spectrumis sufficient for a Hilbert space contraction with finite rankdefect operators and spectrum not covering the unit disc tobe similar to a normal operator. Similar results are provedfor operators having a spectral set bounded by a Dini-smoothJordan curve; in particular, a dissipative operator with finiterank imaginary part is similar to a normal operator if and onlyif its resolvent grows linearly towards the spectrum. Relevantresults on the insufficiency of linear resolvent growth notaccompanied by smallness of defect operators are presented.Also it is proved that there is no restriction on the spectrum,other than finiteness, which together with linear resolventgrowth implies similarity to a normal operator. The constructionof corresponding examples depends on a characterization of well-knownAhlfors curves as curves of linear length growth with respectto linear fractional transformations. 1991 Mathematics SubjectClassification: 11D25, 11G05, 14G05.     

一类不可分解Banach空间上线性算子的谱性质

给出一类不可分解的∑_e~1型Banach空间上线性算子(不一定有界)的谱结构,并讨论这种空间上生成C_0群或C_0半群的线性算子的有界性、特殊的谱性质和谱结构,还给出这种空间上闭算子是有界算子的一个充分条件。     

Convergence results of a modified regularized gradient method for nonlinear ill-posed problems

A modified iteratively regularized gradient method and its continuous version are proposed for nonlinear ill-posed problems, in which the Tikhonov regularization term is generated by a linear operator. The linear operator may have some physical meaning. And by employing the linear operator, scaling the problem can be avoided in the case that the nonlinear operator in the problem has larger gradient. Adopting a posteriori and a priori stopping rule respectively, we establish the convergence results by using a modified approximate source condition. The numerical results show that the linear operator effects the performance greatly.     

Elementary rotations of operators on regular Banach spaces

In this paper, we prove the existence of an elementary rotation (a Julia operator) for any continuous, linear, adjointable operator in a regular Banach space with inner product. The proof is based on a more general theorem of the author on the existence of an elementary rotation for any linear operator in a category with quadratic splitting. This result is a generalization of a well-known result about the existence of an elementary rotation for any continuous linear operator in a Krein space. The result can be useful for constructing isometric and unitary dilations as well as characteristic functions of continuous linear operators acting in regular Banach spaces with inner product. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 175–192, 2006.     

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.     

正则余弦算子函数的内插和外插定理

该文研究正则余弦算子函数的内插和外插．证明了线性算子A在Banach空间X中生成一个指数有界的C-正则余弦函数当且仅当存在Banach空间Y和线性算子B使得：［R这里是C在Y中的有界扩张，B在Y中生成一个强连续余弦算子函数且A＝B|x．     

一个双曲-椭圆耦合系统解的存在唯一性

本文研究了一个双曲-椭圆耦合系统.通过能量方法建立了有关微分算子的一些先验估计,构造了一个闭线性算子,证明了该闭线性算子为一个有界收缩线性算子半群的无穷小生成元.在此基础上,利用半群理论具体证明了双曲-椭圆耦合系统解的存在唯一性.     

Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.     

Formulae of partial reduction for linear systems of first order operator equations

This paper deals with the reduction of non-homogeneous linear systems of first order operator equations with constant coefficients. An equivalent reduced system, consisting of higher order linear operator equations having only one variable and first order linear operator equations in two variables, is obtained by using the rational canonical form.     

The regularization of variational inequalities and a general approximation scheme for regularized solutions in Banach spaces

For variational inequalities with monotone operators in Banach spaces a regularized perturbed inequality is constructed by means of a strongly monotone linear operator, a general scheme of approximating regularized solutions is investigated, and an example of construction of such a linear operator based on information on the operator of the problem is given.Translated from Urainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1273–1276, September, 1991.     

A method for finding the fixed vector of a stochastic operator

The author proposes a new method for finding the fixed vector of a stochastic operator based on the representation of the operator of the linear system as the product of an noninvertible operator of simple form and an invertible operator. A complement to Jentsch’s theorem is obtained.     

关于正则Fredholm多值线性算子对

将单值算子的Fredholm对的概念推广到多值线性算子的范畴上,讨论了Fredholm多值线性算子对的一些初等性质,在适当的条件下,获得了正则的Fredholm多值线性算子对的与正则Fredholm算子对的相平行的一些结果.