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．     

Kakeya极大算子及其分数次情形的正则性

研究中心Kakeya(Nikodym)极大算子K_N(N2)及其分数次情形K_(α,N)(0αd)的正则性.特别地,建立了中心分数次Kakeya极大算子K_(α,N)是从W~(1,p)(R~d)到W~(1,q)(R~d)上的有界连续算子,其中1p∞,q=dp/(d-αp)和0≤αd/p.还证明了中心Kakeya极大算子K_N是分数次Sobolev空间W~(s,p)(R~d),非齐次Triebel-Lizorkin空间F_s~(p,q)(R~d)以及非齐次Besov空间B_s~(p,q)(R~d)上的有界连续算子,其中0s1,1p,q∞.此外,也考虑分数次Kakeya极大函数的弱导数的两种点态估计以及其离散情形的正则性.     

离散算子差分法的单元函数

给出了弹性力学离散算子差分法的离散格式,并给出了该方法的几个板弯曲单元和平面四边形单元,通过对它们的考察,分析了离散算子差分方法中的离散格式对单元位移函数的反映能力。在离散算子差分方法中,无论单元位移函数是否协调,其位移函数均能在离散格式中得到十分好的再现,说明了离散算子差分方法的离散格式是一种性能很优良的离散格式。     

具纯离散谱的u-标算子

本文研究一类具纯离散谱的非自伴算子,证明了该类算子在弱拓扑意义下可以特征展开的充分必要条件是该类算子是u-标的(u-scalar),又等价于该类算子拟仿射相似于自伴算子.并给出例子,说明其在弱拓扑意义下可以特征展开,但不属于经典的标型谱算子(Spectral operator of scalar type).     

Asymptotic type for sectorial operators and an integral of fractional powers

There is a standard notion of type for a sectorial linear operator acting in a Banach space. We introduce a notion of asymptotic type for a linear operator V, involving estimates on the resolvent ⁻¹(λI+V) as λ→0. We show, for example, that if V is sectorial and of asymptotic type ω then the fractional power Vα is of asymptotic type αω for a suitable range of positive α. Moreover, we establish various properties of the operator ; in particular, this operator is of asymptotic type 0, for a sectorial operator V. This result has an application to the construction of operators satisfying the well-known Ritt resolvent condition.     

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     

On Tensor Products of l-Operators and bo-Operators

It is proved that the tensor product of two linear operators is a cone summing operator (respectively, order bounded operator) if and only if both operators are cone summing (respectively, order bounded).     

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．     

Computation of eigenvalues of perturbed discrete semibounded operators

To compute the eigenvalues of a perturbed discrete semibounded operator, systems are obtained for the first time in which the number of equations is equal to the multiplicity of the corresponding eigenvalues of the unperturbed operator.     

The deformed exponential functions of two variables in the context of various statistical mechanics

In the recent development in various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In last two decades, the Tsallis and Kaniadakis versions have found a lot of applications. In this paper, we consider the deformations which have two purposes. First, we introduce them like beginning of a more general mathematical approach where the Tsallis and Kaniadakis exponential functions are the special cases. Then, we wish to pay attention to the mathematical community that they have a lot of interesting properties from mathematical point of view and possibilities in applications. Really, we will show the differential and difference properties of our deformations which are important for the formation and explanation of continuous and discrete models of numerous phenomena.     

Infinite-horizon deterministic dynamic programming in discrete time: a monotone convergence principle and a penalty method

We consider infinite-horizon deterministic dynamic programming problems in discrete time. We show that the value function of such a problem is always a fixed point of a modified version of the Bellman operator. We also show that value iteration converges increasingly to the value function if the initial function is dominated by the value function, is mapped upward by the modified Bellman operator and satisfies a transversality-like condition. These results require no assumption except for the general framework of infinite-horizon deterministic dynamic programming. As an application, we show that the value function can be approximated by computing the value function of an unconstrained version of the problem with the constraint replaced by a penalty function.     

一类积—微分算子的离散本征值

本文讨论的积-微分算子是一类以众多应用领域为背景的无界、非自伴线性算子.在较一般的假设下,籍助L~2空间的线性算子理论,我们证明了这类算子存在离散本征值的充分条件,并获得了可供实际工作者参考的估计式.     

On factoring an operator using elements of its kernel

A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation ? acting on some ring of functions, this paper considers the more general situation of an endomorphism 𝔇 acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of 𝔇 followed by left multiplication by elements of the algebra. Assume that the set of such operators is closed under multiplication and that a Wronski-like matrix produced from some finite list of elements of the algebra is invertible (analogous to the linear independence condition). Then, it is shown that the set of operators whose kernels contain all of those elements is the left ideal generated by an explicitly given operator. In other words, an operator has those elements in its kernel if and only if it has that generator as a right factor. Three examples demonstrate the application of this result in different contexts, including one in which 𝔇 is an automorphism of finite order.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

Existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretization. The discretization combines a time stepping on a non-uniform time grid, which generalizes the well-known Störmer-Verlet scheme, with an internal approximation scheme.The linear operator acting on the zero-order term is supposed to induce an inner product, whereas the nonlinear time-dependent operator acting on the first-order time derivative is assumed to be hemicontinuous, monotone and coercive (up to some additive shift), and to fulfill a certain growth condition. The analysis also extends to the case of additional nonlinear perturbations of both the operators, provided the perturbations satisfy a certain growth and a local Hölder-type continuity condition. A priori estimates are then derived in abstract fractional Sobolev spaces.Convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids.     

Discrete gevrey regularity attractors and uppers–semicontinuity for a finite difference approximation to the ginzburg–landau equation

A semi-discrete spatial finite difference approximation to the complex Ginzburg-Landau equation with cubic non-linearity is considered. Using the fractional powers of a sectorial operator, discrete versions of the Sobolev spaces H ⁵, and Gevrey classes of regularityG, are introduced.Discrete versions of some standard Sobolev space norm inequalities are proved.     

一类次线性算子在广义Morrey空间上的加权有界性

本证明了如果次线性算子T在Orlicz空间上有介,则也有Morrey空间上有界,该算子包括极大算子和奇异积分算子等重要算子。     

Calculation of eigenvalues of a discrete self-adjoint operator perturbed by a bounded operator

We generalize the method of regularized traces which calculates eigenvalues of a perturbed discrete operator for the case of an arbitrary multiplicity of eigenvalues of the unperturbed operator. We obtain a system of equations, enabling one to calculate eigenvalues of the perturbed operator with large ordinal numbers. As an example, we calculate eigenvalues of a perturbed Laplace operator in a rectangle.     

Power bounded operators and supercyclic vectors

By the well-known result of Brown, Chevreau and Pearcy, every Hilbert space contraction with spectrum containing the unit circle has a nontrivial closed invariant subspace. Equivalently, there is a nonzero vector which is not cyclic.

We show that each power bounded operator on a Hilbert space with spectral radius equal to one has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant positive cone.

     

Dirichlet空间上Toeplitz算子的紧性

本文给出了 Ｄｉｒｉｃｈｌｅｔ空间上 Ｔｏｅｌｐｉｔｚ算子为紧算子的充要条件,并证明具有 Ｃ１－符号的 Ｔｏｅｐｌｉｔｚ算子为紧算子当且仅当它为零算子,当且仅当符号的边值为零．