On Sets of Measurable Operators Convex and Closed in Topology of Convergence in Measure

For a von Neumann algebra with a faithful normal semifinite trace, the properties of operator “intervals” of three types for operators measurable with respect to the trace are investigated. The first two operator intervals are convex and closed in the topology of convergence in measure, while the third operator interval is convex for all nonnegative operators if and only if the von Neumann algebra is Abelian. A sufficient condition for the operator intervals of the second and third types not to be compact in the topology of convergence in measure is found. For the algebra of all linear bounded operators in a Hilbert space, the operator intervals of the second and third types cannot be compact in the norm topology. A nonnegative operator is compact if and only if its operator interval of the first type is compact in the norm topology. New operator inequalities are proved. Applications to Schatten–von Neumann ideals are obtained. Two examples are considered.     

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

A system of ordinary differential equations of mixed order on an interval (0, r₀) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako , where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt , Mennicken and Naboko . In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.     

Perturbations of Invertible Operators and Stability of g-Frames in Hilbert Spaces

In this paper, we study the perturbations of invertible operators and stability of g-frames in Hilbert spaces. In particular, we obtain some conditions under which the perturbations of an invertible operator are still an invertible operator, the perturbations of a right invertible operator or a surjective operator are still a right invertible operator or surjective operator. Then we apply the perturbations of invertible operators to study the stability of g-frames which is close related with the invertibility (or right invertibility) property of operators.     

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．     

冯·诺依曼代数的可测算子的性质

本文研究了冯·诺依曼代数的可测算子的基本性质,定义了阶梯算子,证明了任意一个正可测算子可以由阶梯算子在定义域内按照强算子拓扑逼近,从而证明了任意一个可测算子可以由投影在定义域内按照强算子拓扑逼近.此外,还讨论了可测算子与有界算子的复合算子的可测性.     

On spectral properties of the discrete Schrödinger operator with pure imaginary finite potential

In this paper, we consider the spectral properties of the discrete Schrödinger operator in the space of square integrable two-sided sequences with a pure imaginary potential of finite rank with zero mean value. We show that if such potentials are small, then the spectrum of the operator under study coincides with the spectrum of the unperturbed operator, and the operator itself is similar to a self-adjoint operator.     

非线性二元算子方程组的迭代求解方法

Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are improved and generalized.     

基于逐点优化模糊推理的模糊控制器及其插值机理

本文从直观优化思想出发,提出一种逐点优化模糊推理方法,称之为POFI方法. 首先分别给出了基于POFI方法的Mamdani蕴涵算子、代数积蕴涵算子和Zadeh蕴涵算子的FMP算法与FMT算法中寻求推理后件(或前件)的计算表达式.然后分别获得了基于POFI方法的Mamdani蕴涵算子模糊控制器、代数积蕴涵算子模糊控制器、Zadeh蕴涵算子模糊控制器及加乘算子模糊控制器的插值表示,由此发现这些模糊控制器具有函数逼近的泛性.其次指出在基于POFI方法的单输入单输出情况下,Mamdani蕴涵算子模糊控制器、代数积蕴涵算子模糊控制器及加乘算子模糊控制器三者相互等效,并且在基于POFI方法的双输入单输出情况下,Mamdani蕴涵算子模糊控制器与代数积蕴涵算子模糊控制器是等效的.     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

Some Geometric Properties of Analytic Functions Involving a New Fractional Operator

In this paper, we introduce and study a new fractional operator and its implications in terms of the Ruscheweyh derivative operator, the Sălăgean operator and a certain fractional differintegral operator. Some geometric properties of the analytic functions involving this operator are derived. We also consider some applications and derive certain corollaries of our main results. Some useful consequences and relationship of certain results with known results are also pointed out.

     

Characterization of Riesz bases of wavelets generated from multiresolution analysis

We investigate Riesz bases of wavelets generated from multiresolution analysis. This investigation leads us to a study of refinement equations with masks being exponentially decaying sequences. In order to study such refinement equations we introduce the cascade operator and the transition operator. It turns out that the transition operator associated with an exponentially decaying mask is a compact operator on a certain Banach space of sequences. With the help of the spectral theory of the compact operator we are able to characterize the convergence of the cascade algorithm associated with an exponentially decaying mask in terms of the spectrum of the corresponding transition operator. As an application of this study we establish the main result of this paper which gives a complete characterization of all possible Riesz bases of compactly supported wavelets generated from multiresolution analysis. Several interesting examples are provided to illustrate the general theory.     

Study of Volterra integro-differential equations arising in viscoelasticity theory

Volterra integrodifferential equations with unbounded operator coefficients in a Hilbert space that are operator models of integrodifferential equations arising in viscoelasticity theory are studied. These equations are shown to be well-posed in Sobolev spaces of vector functions, and spectral analysis is applied to the operator functions that are the symbols of the given equations.     

Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making

Trapezoidal intuitionistic fuzzy numbers (TrIFNs) is a special intuitionistic fuzzy set on a real number set. TrIFNs are useful to deal with ill-known quantities in decision data and decision making problems themselves. The focus of this paper is on multi-attribute group decision making (MAGDM) problems in which the attribute values are expressed with TrIFNs, which are solved by developing a new decision method based on power average operators of TrIFNs. The new operation laws for TrIFNs are given. From a viewpoint of Hausdorff metric, the Hamming and Euclidean distances between TrIFNs are defined. Hereby the power average operator of real numbers is extended to four kinds of power average operators of TrIFNs, involving the power average operator of TrIFNs, the weighted power average operator of TrIFNs, the power ordered weighted average operator of TrIFNs, and the power hybrid average operator of TrIFNs. In the proposed group decision method, the individual overall evaluation values of alternatives are generated by using the power average operator of TrIFNs. Applying the hybrid average operator of TrIFNs, the individual overall evaluation values of alternatives are then integrated into the collective ones, which are used to rank the alternatives. The example analysis shows the practicality and effectiveness of the proposed method.     

Asymptotics of Eigenvalues of Differential Operator With Alternating Weight Function

We study a differential operator of the sixth order with an alternating weight function. The potential of the operator has a first-order discontinuity at some point of the segment, where the operator is being considered. The boundary conditions are separated. We study the asymptotics of solutions to the corresponding differential equations and the asymptotics of eigenvalues of the considered differential operator.     

Operator Frames for Banach Spaces

In this paper, operator Bessel sequences, operator frames, Banach operator frames, Operator Riesz bases for Banach spaces and dual frames of an operator frame are introduced and discussed. The necessary and sufficient condition for a Banach space to have an operator frame, a Banach operator frame or an operator Riesz basis are given. In addition, operator frames and operator Riesz bases are characterized by the analysis operator of operator Bessel sequences.     

Rank one perturbations of not semibounded operators

Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension.     

The analytic model of a hyponormal operator with rank one self-commutator

The analytic model of a general hyponormal operator with rank one self-commutator is obtained. A Cauchy integral representation for the adjoint is given, certain kernels are discussed, and conditions are given under which the hyponormal operator is similar to a subnormal Toeplitz operator.     

On the Fredholm Property of a Class of Convolution-Type Operators

The notions of the L-convolution operator and the ?-Wiener–Hopf operator are introduced by replacing the Fourier transform in the definition of the convolution operator by a spectral transformation of the self-adjoint Sturm–Liouville operator on the axis ?. In the case of the zero potential, the introduced operators coincide with the convolution operator and theWiener–Hopf integral operator, respectively. A connection between the ?-Wiener–Hopf operator and singular integral operators is revealed. In the case of a piecewise continuous symbol, a criterion for the Fredholm property and a formula for the index of the ?-Wiener–Hopf operator in terms of the symbol and the elements of the scattering matrix of the operator ? are obtained.     

Quasi-multipliers of operator spaces

We use the injective envelope to study quasi-multipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasi-multipliers. We obtain generalizations of the Banach-Stone theorem.     

Graded Differential Equations and Dirac Type Operators

Singularities of solutions of graded differential equations as filtrations are studied. Symbols and the transfer operator associated with the h-adic filtration are computed. The condition that the transfer operator is a differential operator over the characteristic manifold of codimension 1 for this filtration is given. The Dirac operator is introduced by using the De Broglie principle: singularities of wave-like solutions move as material points. Use of the transfer operator associated with the h-adic filtration implies an invariant definition of the Pauli operator. A natural parallel translation arises as quantum analog of the Levi-Civita connection.