The σ<Emphasis Type="Italic">g</Emphasis>-Drazin Inverse and the Generalized Mbekhta Decomposition

In this paper we define and study an extension of the g-Drazin for elements of a Banach algebra and for bounded linear operators based on an isolated spectral set rather than on an isolated spectral point. We investigate salient properties of the new inverse and its continuity, and illustrate its usefulness with an application to differential equations. Generalized Mbekhta subspaces are introduced and the corresponding extended Mbekhta decomposition gives a characterization of circularly isolated spectral sets.     

Approximation of functions of two variables by certain linear positive operators

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an rth order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.     

Bochner identities for Kählerian gradients

We discuss algebraic properties for the symbols of geometric first order differential operators on Kähler manifolds. Through a study of the universal enveloping algebra and higher Casimir elements, we know a lot of relations for the symbols, which induce Bochner identities for the operators. As applications, we have vanishing theorems, eigenvalue estimates, and so on.     

Eigenvalue Distribution of Positive Definite Kernels on Unbounded Domains

We study eigenvalues of positive definite kernels of L² integral operators on unbounded real intervals. Under the assumptions of integrability and uniform continuity of the kernel on the diagonal the operator is compact and trace class. We establish sharp results which determine the eigenvalue distribution as a function of the smoothness of the kernel and its decay rate at infinity along the diagonal. The main result deals at once with all possible orders of differentiability and all possible rates of decay of the kernel. The known optimal results for eigenvalue distribution of positive definite kernels in compact intervals are particular cases. These results depend critically on a 2-parameter differential family of inequalities for the kernel which is a consequence of positivity and is a differential generalization of diagonal dominance.     

An Implicit Function Theorem for One-sided Lipschitz Mappings

Implicit function theorems are derived for nonlinear set valued equations that satisfy a relaxed one-sided Lipschitz condition. We discuss a local and a global version and study in detail the continuity properties of the implicit set-valued function. Applications are provided to the Crank–Nicolson scheme for differential inclusions and to the analysis of differential algebraic inclusions.     

Classification of Toeplitz operators on hardy spaces of bounded domains in the plane

We construct an orthonormal basis for the class of square integrable functions on bounded domains in the plane in terms of the classical kernel functions in potential theory. Then we generalize the results of Brown and Halmos about algebraic properties of Toeplitz operators and Laurent operators on the unit disc to general bounded domains. This is a complete classification of Laurent operators and Toeplitz operators for bounded domains.     

Two-sided estimate for the modulus of continuity of a convolution

We study the differential properties of the convolution of functions with a generalized Bessel-Macdonald kernel. The integral properties of a function are characterized in terms of its decreasing permutation. The differential properties of the convolution are described in terms of its modulus of continuity of arbitrary order in the uniform norm. We obtain order-sharp estimates for the modulus of continuity of the convolution. By way of application, we present two-sided estimates for the modulus of continuity of the classical Bessel potential.     

Generalized spherical Aluthge transforms and binormality for commuting pairs of operators

In this paper, we introduce the notion of generalized spherical Aluthge transforms for commuting pairs of operators and study nontrivial joint invariant (resp. hyperinvariant) subspaces between the generalized spherical Aluthge transform and the original commuting pair. Next, we study the norm continuity through generalized Aluthge transform maps. We also study how the Taylor spectra and the Fredrolm index of commuting pairs of operators behave under the spherical Duggal transform. Finally, we introduce the notion of Campbell binormality for commuting pairs of operators and investigate some of its basic properties under spherical Aluthge and Duggal transforms. Moreover, we obtain new set inclusion diagrams among normal, quasinormal, centered, and Campbell binormal commuting pairs of operators.     

On Bregman-Type Distances for Convex Functions and Maximally Monotone Operators

Given two point to set operators, one of which is maximally monotone, we introduce a new distance in their graphs. This new concept reduces to the classical Bregman distance when both operators are the gradient of a convex function. We study the properties of this new distance and establish its continuity properties. We derive its formula for some particular cases, including the case in which both operators are linear monotone and continuous. We also characterize all bi-functions D for which there exists a convex function h such that D is the Bregman distance induced by h.     

On the asymptotic stability of discontinuous systems analysed via the averaging method

The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.     

A Filtration of Wick Algebra and Its Application to Quantum SDEs

A family of closed snbalgebras, indexed by R(the set of real numbers), of the Wick algebra is constructed. Fundamental properties of tile family are shown including the increasing property and the right-continuity. The notion of adaptedness to the family is defined for quantum stochastic processes in terms of generalized operators. The existence and uniqueness of solutions adapted to the family is established for quantum stochastic differential equations in terms of generalized operators.     

Simultaneous Approximation for Generalized Baskakov-Durrmeyer-Type Operators

The present paper deals with the study of a Durrmeyer-type integral modification of certain modified Baskakov operators. Here we study simultaneous approximation properties for these operators by using the iterative combinations. We obtain an asymptotic formula and an error estimation in terms of higher order modulus of continuity for these operators.      

Strong inclusion orders between L-subsets and its applications in L-convex spaces

Abstract

In this paper, the concept of strong inclusion orders between L-subsets is introduced. As a tool, it is applied to the following aspects. Firstly, the notion of algebraic L-closure operators is proposed and the resulting category is shown to be isomorphic to the category of L-convex spaces (also called algebraic L-closure spaces). Secondly, restricted L-hull operators, as generalizations of restricted hull operators, are introduced and the resulting category is also proved to be isomorphic to the category of L-convex spaces. Finally, by using the properties of strong inclusion orders, it is shown that the category of convex spaces can be embedded in the category of stratified L-convex spaces as a reflective subcategory and the concrete form of the coreflective functor from the category of L-convex spaces to the category of stratified L-convex spaces is presented.     

Spaces of Ideal Convergent Sequences of Bounded Linear Operators

Abstract

The notation of I–convergence was introduced and studied by Kostyrko, Macaj, Salat, and Wilczynski. Recently, the concept of I–convergent for a sequence of bounded linear operators has been studied by Khan and Shafiq. This has motivated us to introduce and study some new spaces of double sequences of bounded linear operators and their basic topological and algebraic properties of these spaces. And we study some of their basic topological and algebraic properties of these spaces. We prove some inclusion relations on these spaces.     

An existence result for non-smooth vibro-impact problems

We are interested in mechanical systems with a finite number of degrees of freedom submitted to frictionless unilateral constraints. We consider the case of a convex, non-smooth set of admissible positions given by , ν?1, and we assume inelastic shocks at impacts. We propose a time-discretization of the measure differential inclusion which describes the dynamics and we prove the convergence of the approximate solutions to a limit motion which satisfies the constraints. Moreover, if the geometric properties ensuring continuity on data hold at the limit, we show that the transmission of velocities at impacts follows the inelastic shocks rule.     

One-sided Perron Differential Inclusions

The main qualitative properties of the solution set of almost lower (upper) semicontinuous one-sided Perron differential inclusion with state constraints in finite dimensional spaces are studied. Using the technique introduced by Veliov (Nonlinear Anal 23:1027–1038, 1994) we give sufficient conditions for the solution map of the above state constrained differential inclusion to be continuous in the sense of Hausdorff metric. An application on the propagation of the continuity of the state constrained minimum time function associated with the nonautonomous differential inclusion and the target zero is given. Some relaxation theorems are proved, which are used afterward to derive necessary and sufficient conditions for invariance.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

Equivariant heat invariants of the laplacian and nonmininmal operators on differential forms

In this paper,we compute the first two equivariant heat kernel coeffcients of the Bochner Laplacian on differential forms.The first two equivariant heat kernel coeffcients of the Bochner Laplacian with torsion are also given.We also study the equivariant heat kernel coeffcients of nonminimal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula.     

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

Anamorphoses and Flat Morphological Operators on Power Lattices

Flat morphological operators are operators on grey-level images derived from increasing set operators by a combination of thresholding and stacking. For analog grey-levels, they commute with anamorphoses or contrast mappings, that is, continuous increasing grey-level transformations; when the underlying set operator is upper semi-continuous, they also commute with thresholding. For bounded discrete grey-levels, commutation with increasing grey-level transformations and with thresholding is guaranteed, without any continuity conditions. In this paper we consider flat operators for images defined on an arbitrary space of points and taking their values in an arbitrary complete lattice. We study their commutation with increasing transformations of values. This requires some continuity requirements on the transformations of values or on the underlying set operator, which are expressed in terms of the lattice of values. We obtain as particular cases the known conditions for analog and discrete grey-levels, and also new conditions for other examples of values: multivalued vectors or any finite set of values.