Boundedness of sublinear operators in weighted grand Morrey spaces

The boundedness of sublinear integral operators in grand Morrey spaces defined by means of measures generated by the Muckenhoupt weights is established. The operators under consideration involve operators of Harmonic Analysis such as Hardy–Littlewood and fractional maximal operators, Calderoń–Zygmund operators, potential operators etc.     

Banach algebra generated by a finite number of bergman polykernel operators,continuous coefficients,and a finite group of shifts

We study the Banach algebra generated by a finite number of Bergman polykernel operators with continuous coefficients that is extended by operators of weighted shift that form a finite group. By using an isometric transformation, we represent the operators of the algebra in the form of a matrix operator formed by a finite number of mutually complementary projectors whose coefficients are Toeplitz matrix functions of finite order. Using properties of Bergman polykernel operators, we obtain an efficient criterion for the operators of the algebra considered to be Fredholm operators.     

Lattice operators underlying dynamic systems

This paper investigates algebraic and continuity properties of increasing set operators underlying dynamic systems. We recall algebraic properties of increasing operators on complete lattices and some topologies used for the study of continuity properties of lattice operators. We apply these notions to several operators induced by a differential equation or differential inclusion. We especially focus on the operators associating with any closed subset its reachable set, its exit tube, its viability kernel or its invariance kernel. Finally, we show that morphological operators used in image processing are particular cases of operators induced by constant differential inclusion.     

关于IVIFS算子与IFS算子关系的讨论

Atanassov教授在给出区间教直觉模糊集算子的定义以后曾给出断语:直觉模糊集算子满足的结论,都可以推广到相应的区问数直党模糊集算子.本文详细分析多条直觉模糊集算子的性质与相应区间数直党模糊集算子的性质之间的关系,证明这些直觉模糊集算子的性质都不能推广到区间数直党模糊集算子,因而对Atanassov的论断进行了澄清.     

用小波限制紧算子逼近奇异积分算子的速度

根据Beylin-Coifman-Rokhlin和Yang的观点，用合适的小波基可以刻画性地研究C.Z算子或拟微分算子，这样就可以用小波来计算算子，比如可以从奇异积分算子的B-C-R算法刻画出发，用小波限制紧算子进行逼近，本文旨在计算作为原算子与逼近算子的差的误差算子的H1到L1的连续性范数的最佳衰减速度和在Lp上的连续性范数的衰减速度的范围。     

Two classes of operators with irreducibility and the small and compact perturbations of them

This paper gives the concepts of finite dimensional irreducible operators((FDI) operators)and infinite dimensional irreducible operators((IDI) operators). Discusses the relationships of(FDI)operators,(IDI) operators and strongly irreducible operators((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an(FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in(Σ FDI)(X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in(Σ FDI)(X) with respect to the norm topology on a Banach space X with a Schauder basis, where(ΣFDI)(X) := {T∈B(X) : T=Σki=1Ti, Ti ∈(FDI), k ∈ N}.     

无穷维Hilbert空间上框架的算子与范数

在框架理论研究中,哪类可逆算子能使得某些框架性质保持不变这个问题是基本和重要的,本文在无穷维Hilbert空间上对下述两个问题进行研究.问题1:哪类可逆算子能使得框架算子保持不变;问题2:哪类可逆算子能使得框架范数只相差一列常数.本文从抽象的算子理论和具体的构造方法两方面对问题1给出解答.利用框架的相容算子的概念,当把问题2中的可逆算子集换成一类较小的算子集时,得到了问题2的回答.     

On Integral Operators of Generalized Wiener-Hopf-Type

A class of singular integral operators on the positive halfaxis, constituted by the one-sided HILBERT transformation, the WIENER -HOPF operators, the multiplicative convolution operators, and some multiplication operators, generated by continuous functions, are studied with BANACH algebra methods. With the help of symbol functions the FREDHOLM operators wil be characterized. Moreover one gets information about an index formula and FREDHOLM inverses.     

某些算子和交换子在非齐型空间上的Morrey-Herz空间中的有界性

引入了非齐型空间上的齐次Morrey-Herz 空间和弱齐次Morrey-Herz空间并建立了Hardy-Littlewood极大算子,Calder\'on-Zygmund算子和分数次积分算子在齐次Morrey-Herz空间中的有界性以及在弱齐次Morrey-Herz空间中的弱型估计. 此外,还证明了$\rb$函数与Calder\'on-Zygmund算子或分数次积分算子生成的多线性交换子以及与Hardy-Littlewood极大算子相关的极大交换子在齐次Morrey-Herz空间中的有界性.     

On infinite dimensional quadratic Volterra operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.     

On uniform approximation by some classical Bernstein-type operators

We investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval. The discussed examples include the Szász operators, the Szász-Durrmeyer operators, the gamma operators, the Baskakov operators, and the Meyer-König and Zeller operators. We show that some results of Totik remain valid for unbounded functions, at the same time that we give simple rates of convergence in terms of the usual modulus of continuity. We also show by a counterexample that the result for Meyer-König and Zeller operators does not extend to Cheney and Sharma operators.     

King type modification of q-Bernstein-Schurer operators

Very recently the q-Bernstein-Schurer operators which reproduce only constant function were introduced and studied by C. V. Muraru (2011). Inspired by J. P. King, Positive linear operators which preserve x ² (2003), in this paper we modify q-Bernstein-Schurer operators to King type modification of q-Bernstein-Schurer operators, so that these operators reproduce constant as well as quadratic test functions x ² and study the approximation properties of these operators. We establish a convergence theorem of Korovkin type. We also get some estimations for the rate of convergence of these operators by using modulus of continuity. Furthermore, we give a Voronovskaja-type asymptotic formula for these operators.     

U-ideals of factorable operators

We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons's construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are u-ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.     

Equivalence after extension and Schur coupling coincide for inessential operators

In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators that can in norm be approximated by invertible operators. In this paper we prove that the implication EAE $?$ SC also holds for inessential Banach space operators. The inessential operators were introduced as a generalization of the compact operators, and include, besides the compact operators, also the strictly singular and strictly co-singular operators; in fact they form the largest ideal such that the invertible elements in the associated quotient algebra coincide with (the equivalence classes of) the Fredholm operators.     

Spectral theory of higher order difference operators

The spectral theory of higher order difference operators is studied by means of asymptotic summation, thereby extending many results of differential operators to the discrete setting. The spectra of degenerate fourth-order operators are also investigated. The results are then compared with those of corresponding differential operators. Even though there are many similarities between both classes of operators, the spectral results may be quite distinct.     

Invertible composition operators on Hp

The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H² is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.     

Monotone Operators Representable by l.s.c. Convex Functions

A theorem due to Fitzpatrick provides a representation of arbitrary maximal monotone operators by convex functions. This paper explores representability of arbitrary (nonnecessarily maximal) monotone operators by convex functions. In the finite-dimensional case, we identify the class of monotone operators that admit a convex representation as the one consisting of intersections of maximal monotone operators and characterize the monotone operators that have a unique maximal monotone extension.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

An algorithmic representation of fractional powers of positive operators

We are concerned with a new representation of fractional powers of operators by a series using exclusively contractive operators or operators with uniformly bounded powers. This representation is suitable for the construction of efficient approximations for which the error estimates are given and the convergence is investigated. The rate of convergence turns out to be exponential for bounded operators and polynomial for unbounded operators     

利用格蕴涵代数找出所有LI-理想的方法

利用格蕴涵代数中理想的定义未找出所有的理想需要花费大量的时间.给出了利用格蕴涵代数的蕴涵运算表找出格蕴涵代数所有理想的方法;再利用析取运算表找出格蕴涵代数中所有sl_理想的方法;最后,利用蕴涵否运算表找出格蕴涵代数中所有LI-理想的方法.