Relative Boundedness Research Articles

Characterizations of infinitesimal relative boundedness for higher order Schrödinger operators

Let m∈N and H=(−Δ)m/2+V be a higher order Schrödinger operators in the Euclidean space Rn with V∈Lloc1(Rn). In this paper, the authors characterize the infinitesimal relative boundedness and Trudinger's subordination for H on Lp(Rn) with p∈[1,∞), via the limit behavior of the family of operators {V(λ2−Δ)−m/2}λ∈(0,∞) and a generalized Kato-type class condition. The latter is weaker than the classical Kato class condition corresponding to the case p=1. All these characterizations are new even when H=−Δ+V is a second order Schrödinger operator.

Relative boundedness in non-Archimedean Banach spaces

This paper treats general unbounded operators, closed operators, and relative boundedness in non-Archimedean Banach spaces.

Integro-differential equations with bounded operators in Banach spaces

The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.

A Brief Exposition of the Space of Relatively Bounded Nonlinear Operators

This note aims to present some rudimentary aspects of relatively bounded nonlinear (not necessarily linear) operators. To be precise, we prove that the set of all nonlinear operators between two normed linear spaces that are relatively bounded with respect to a fixed operator between these spaces itself is a normed linear space under a suitable norm. Furthermore, it is shown that if the codomain is a Banach space, then the space of all relatively bounded nonlinear operators is also complete. Some elementary results on the algebraic structure of the space of all nonlinear operators that are bounded with respect to a fixed operator are also recorded. The results reported in this note illustrate that with some predictable minor modifications, the fundamental classical results on the space of all bounded linear operators can be carried over to a more general setting. Here the word `general' is used in a double sense that the boundedness is generalized to relative boundedness and the restriction of linearity is dropped to include nonlinear maps.

Relative Continuity, Proximal Boundedness and Best Proximity Point Theorems

This articles discusses some properties of relatively continuous mappings, a natural generalization of continuous mappings. Also, we introduce the notion of proximal boundedness and dilate upon a relationship among proximal boundedness, proximal completeness and proximal compactness. Finally, we utilize such results to elicit an extension of a Schauder’s fixed point theorem, which states that every continuous self-mapping on a non-void compact convex subset of a normed linear space has a fixed point, to the case of non-self relatively continuous mappings. In fact, such an extension is proved in the form of a best proximity point theorem for relatively continuous mappings in the framework of a proximally compact space that has semi-sharp proximinality, thereby ascertaining the existence of an optimal approximate solution to some equations. Such an optimal approximate solution is known as a best proximity point and elicited as a result of approximation with the minimization of the error due to approximation. Further, an application of such a result is explored to elicit a common best proximity point theorem for a family of commuting affine mappings. Also, we furnish another application of our main result to find best proximity solution to an ordinary differential equation.

«Точки» и «запятые» в структуре ситуации: абсолютный и относительный предел и средства их выражения (аспектуальные характеристики и падеж прямого дополнения)

The paper deals with some features of the event structure that influence its formal expression and determine the wider context of a clause. The notion of relative/absolute boundedness is used to indicate the exhaustiveness of the event. The relative boundedness is a feature that makes it possible to use the genitive partitive case as the direct object. Within such perfective verbs as vypit’ ‘drink.PF’ and sjest’ ‘eat.PF’, the nominal case determines the interpretation of a verbal phrase: the accusative indicates the absolute boundedness while use of the genitive results in the relative boundedness. As for degree achievements, delimitative Aktionsart and semelfactive Aktionsart, it is the verb that determines the interpretation of a verbal phrase. What all the examined cases have in common is that the use of the genitive as the direct object marker indicates the relative boundedness (partial resultativeness): the object of the event is a part of the cumulative object of a wider (actual or ideal) situation.

Identification Problem for Strongly Degenerate Evolution Equations with the Gerasimov–Caputo Derivative

For a linear differential equation in a Banach space with a degenerate operator multiplying the fractional Gerasimov–Caputo derivative, we prove a theorem on the existence of a unique solution of the inverse problem with an unknown time-dependent coefficient. It is assumed that the relative boundedness condition is satisfied for the pair of operators occurring in the equation and an overdetermation condition with an operator whose kernel contains the degeneracy subspace of the equation under study is specified. The result obtained is illustrated with an example of a model system of partial differential equations unsolved for the fractional time derivative, containing an unknown coefficient, and equipped with initial and boundary conditions and an overdetermination condition.

Essential spectra of self-adjoint relations under relatively compact perturbations

ABSTRACTThis paper is concerned with the stability of essential spectra of self-adjoint relations under relatively compact perturbation in Hilbert spaces. Relationships between relative boundedness and relative compactness of linear relations are established, and some necessary and sufficient conditions of relative compactness and relative boundedness of linear relations are given. It is shown that the essential spectra of self-adjoint relations are invariant under either relatively compact perturbation or a more general perturbation. The results obtained in the present paper generalize the corresponding results for operators to relations, and some of which weaken certain assumptions of the related existing results.

Perturbation of unbounded linear operators by $$\gamma $$ γ -relative boundedness

Diagana (Handbook on operator theory. Springer, Basel, pp 875–880, 2015) studied some sufficient conditions such that if S, T and K are three unbounded linear operators with S being a closed operator, then their algebraic sum \(S+T+K\) is also a closed operator. The main focus of this paper is to extend these results to the closable operator by adding a new concept of the gap and the \(\gamma \)-relative boundedness inspired by the work of Jeribi et al. (Linear Multilinear Algebra 64:1654–1668, 2015). After that, we apply the obtained results to study the specific properties of some block operator matrices.

Unbounded operators in Hilbert space, duality rules, characteristic projections, and their applications

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces whose intersection contains a fixed vector space $$\mathscr {D}$$ . In the case when $$\mathscr {D}$$ is dense in one of the Hilbert spaces (but not necessarily in the other), we make precise an operator-theoretic linking between the two Hilbert spaces. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and the operator theory of reflection positivity.

Boundedness and closedness of linear relations

This paper focuses on boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and stability of closedness of linear relations under bounded and relatively bounded perturbations is studied. The results obtained in the present paper generalize the corresponding results for single-valued linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results.

Relatively Bounded and Relatively Compact Perturbations for Limit Circle Hamiltonian Systems

This paper is concerned with relatively bounded and relatively compact perturbations of limit circle Hamiltonian systems of arbitrary order which may be formally non-symmetric. In this paper, a relationship between the relative boundedness and relative compactness of an operator is obtained, and a sufficient and necessity condition is given for a relatively compact operator V with respect to an operator T to be relatively compact with respect to a finite-dimensional closed extension of T. Furthermore, properties of the limit circle Hamiltonian systems are derived, the regularity field of the minimal operator corresponding to the limit circle Hamiltonian system is given, and then it is proved that the relative boundedness and relative compactness of a class of multiplication operators with respect to the maximal operator corresponding to the limit circle Hamiltonian system are equivalent.

Essential spectra of some matrix operators involving γ-relatively bounded entries and an application

In this paper, we introduce the concept of relative boundedness with respect to an axiomatic measure of noncompactness , as a generalization of the notion of relative boundedness. Then, some spectral properties of a unbounded block operator matrix involving -relatively bounded inputs are given. The obtained results are applied to investigate the essential spectra of a two-dimensional transport operator in with abstract boundary conditions relating the incoming and the outgoing fluxes.

Solvability of Initial Problems for One Class of Dynamical Equations in Quasi-Sobolev Spaces

The equations, which are not solved with respect to the highest derivative, are now actively studied. Such equations are also called the Sobolev type equations. Note that these equations in Banach spaces are studied quite well. Quasi-Sobolev spaces are quasi normalized complete spaces of sequences. Recently these spaces began to be studied. The interest to such spaces and its equations is connected with a desire to fill up the theory more than with practical applications. The paper is devoted to the study of solvability of the Cauchy problem and the Showalter -- Sidorov problem for a class of equations considered in the quasi-Sobolev spases. To this end we use properties of the equation operators, namely the relative boundedness of the operators. To illustrate abstract results we consider an analogue of the Hoff equation in the quasi-Sobolev spaces.

Relatively Bounded Extensions of Generator Perturbations

The Miyadera perturbation theorem provides as a by-product that operators defined on a core for the generator of a C0-semigroup and satisfying the Miyadera condition have a relatively bounded extension to the domain of the generator. We show that a weakening of the Miyadera condition characterizes relative boundedness with respect to the generator. We also investigate extensions of these results to Hille-Yosida operators. The various conditions we use in the abstract part are illustrated by several examples. 2000 MSC: 47D06, 47D62, 47A55, 47B60

Implementing innovation in construction: contexts, relative boundedness and actor‐network theory

Theoretical understanding of the implementation and use of innovations within construction contexts is discussed and developed. It is argued that both the rhetoric of the ‘improvement agenda’ within construction and theories of innovation fail to account for the complex contexts and disparate perspectives which characterize construction work. To address this, the concept of relative boundedness is offered. Relatively unbounded innovation is characterized by a lack of a coherent central driving force or mediator with the ability to reconcile potential conflicts and overcome resistance to implementation. This is a situation not exclusive to, but certainly indicative of, much construction project work. Drawing on empirical material from the implementation of new design and coordination technologies on a large construction project, the concept is developed, concentrating on the negotiations and translations implementation mobilized. An actor‐network theory (ANT) approach is adopted, which emphasizes the roles that both human actors and non‐human agents play in the performance and outcomes of these interactions. Three aspects of how relative boundedness is constituted and affected are described; through the robustness of existing practices and expectations, through the delegation of interests on to technological artefacts and through the mobilization of actors and artefacts to constrain and limit the scope of negotiations over new technology implementation.

Form boundedness of the general second‐order differential Operator

AbstractWe give explicit necessary and sufficient conditions for the boundedness of the general second‐order differential operatorwith real‐ or complex‐valued distributional coefficientsaij,bj, andc, acting from the Sobolev spaceW1, 2(ℝn) to its dualW−1, 2(ℝn). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of ℒ︁ with respect to the Laplacian onL2(ℝn).In particular, we establish a complete characterization of the form boundedness of the Schrödinger operator$(i \nabla + \vec{a})^2 + q$with magnetic vector potential$\vec{a} \in L^2_{{\rm loc}} (R^{n})^{n}$andq∈D′(ℝn). © 2005 Wiley Periodicals, Inc.

PIERRE BOURDIEU AND THE PRACTICES OF LANGUAGE

This paper synthesizes research on linguistic practice and critically examines the legacy of Pierre Bourdieu from the perspective of linguistic anthropology. Bourdieu wrote widely about language and linguistics, but his most far reaching engagement with the topic is in his use of linguistic reasoning to elaborate broader sociological concepts including habitus, field, standardization, legitimacy, censorship, and symbolic power. The paper examines and relates habitus and field in detail, tracing the former to the work of Erwin Panofsky and the latter to structuralist discourse semantics. The principles of relative autonomy, boundedness, homology, and embedding apply to fields and their linkage to habitus. Authority, censorship, and euphemism are traced to the field, and symbolic power is related to misrecognition. And last, this chapter relates recent work in linguistic anthropology to practice and indicates lines for future research.

Pseudorelativistic operator for a two-electron ion

By means of a unitary transformation scheme, the Coulomb-Dirac operator for two electrons in a central potential is transformed into a pseudorelativistic operator which allows for the decoupling of the electron and positron degrees of freedom to arbitrary order $n$ in the potential strength. In case of $n=2$, relative boundedness properties and positivity of the resulting operator are shown for subcritical potential strength.

Relative form boundedness and compactness for a second-order differential operator

If A and L are self-adjoint operators on a Hilbert space H such that A is nonnegative and L⩾ εI for some ε>0 we study conditions under which A is form bounded or form compact with respect to L and contrast these concepts with the stronger properties that A be relatively operator bounded or compact with respect to L. In particular several definitions of form compactness are shown to be equivalent. The principal application of the theory is to self-adjoint second order operators. In this case conditions that A be form bounded or form compact with respect to L are shown in some cases to be necessary and sufficient. Examples include the energy operator of the hydrogen atom.