scholarly journals Singularly perturbed rank one linear operators

2021 ◽  
Vol 56 (2) ◽  
pp. 162-175
Author(s):  
M.E. Dudkin ◽  
O. Yu. Dyuzhenkova
Keyword(s):  
Adjoint Operator ◽  
Linear Operators ◽  
Singularly Perturbed ◽  
Basic Principles ◽  
Dirac Delta ◽  
Symmetric Perturbation ◽  
Rank One ◽  
Definition Of ◽  
Adjoint Operators ◽  
Symmetric Operators

The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.

scholarly journals Self adjoint operators and matrix measures

1971 ◽  
Vol 4 (3) ◽  
pp. 289-305 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Spectral Representation ◽  
Adjoint Operator ◽  
Multiplication Operator ◽  
Linear Operators ◽  
Positive Matrix ◽  
Adjoint Operators ◽  
Matrix Measures

Given a self adjoint operator, T, on a Hilbert space H, and given an integer n ≥ 1, we produce a collection , N ∈ L, of n × n positive matrix measures and a unitary map U: such that UTU−1, restricted to the co-ordinate space , is the multiplication operator F(t) → tF(t) in that space. This is a generalization of the spectral representation theory of Dunford and Schwartz, Linear operators, II (1966).

ON THE PROBLEM OF THE RIGHT HAMILTONIAN UNDER SINGULAR FORM-SUM PERTURBATIONS

2000 ◽  
Vol 12 (01) ◽  
pp. 1-24 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
VOLODYMYR KOSHMANENKO
Keyword(s):  
Quadratic Forms ◽  
Symmetric Operator ◽  
Adjoint Operator ◽  
Deficiency Indices ◽  
Singular Form ◽  
Approximating Sequence ◽  
Sum Formula ◽  
Definition Of ◽  
The Right ◽  
Adjoint Operators

Let a perturbation of the self-adjoint operator H0>0 in the Hilbert space ℋ be given by an operator V (or by a quadratic form ν) which is possibly singular and in general nonpositive, so H0+V on [Formula: see text] is only a symmetric operator with nontrivial deficiency indices. The definition of the sum [Formula: see text] in the sense of quadratic forms is extended to cases which are not covered by the well-known KLMN-theorem and conditions are found which ensure the unique self-adjoint realization of H in ℋ. It is also shown that ℋ coincides with the strong resolvent limit of the approximating sequence Hn = H0+Vn, where Vn are bounded self-adjoint operators such that Vn → V in a suitable sense. Essentially that operator V might be strongly singular and acts in the H0-scale of spaces, V:ℋ+→ℋ-.

scholarly journals Direct and Inverse Spectral Problems for Rank-One Perturbations of Self-adjoint Operators

2021 ◽  
Vol 93 (2) ◽  
Author(s):  
Oles Dobosevych ◽  
Rostyslav Hryniv
Keyword(s):  
Inverse Problem ◽  
Discrete Spectrum ◽  
Adjoint Operator ◽  
Inverse Spectral Problems ◽  
Spectral Problems ◽  
Rank One ◽  
Inverse Spectral ◽  
Adjoint Operators

AbstractFor a given self-adjoint operator A with discrete spectrum, we completely characterise possible eigenvalues of its rank-one perturbations B and discuss the inverse problem of reconstructing B from its spectrum.

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals International law protecting the environment during armed conflict: gaps and opportunities

2010 ◽  
Vol 92 (879) ◽  
pp. 569-592 ◽  
Author(s):  
Michael Bothe ◽  
Carl Bruch ◽  
Jordan Diamond ◽  
David Jensen
Keyword(s):  
International Law ◽  
Armed Conflict ◽  
Environmental Law ◽  
International Humanitarian Law ◽  
Environmental Damage ◽  
Humanitarian Law ◽  
International Environmental Law ◽  
Basic Principles ◽  
Principle Of Proportionality ◽  
Definition Of

AbstractThere are three key deficiencies in the existing body of international humanitarian law (IHL) relating to protection of the environment during armed conflict. First, the definition of impermissible environmental damage is both too restrictive and unclear; second, there are legal uncertainties regarding the protection of elements of the environment as civilian objects; and third, the application of the principle of proportionality where harm to the environment constitutes ‘collateral damage’ is also problematic. These gaps present specific opportunities for clarifying and developing the existing framework. One approach to addressing some of the inadequacies of IHL could be application of international environmental law during armed conflict. The detailed norms, standards, approaches, and mechanisms found in international environmental law might also help to clarify and extend basic principles of IHL to prevent, address, or assess liability for environmental damage incurred during armed conflict.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

EFFECTIVE STEPS ON THE WAY TO MODERNIZATION OF CIVIL SERVICE IN THE SYSTEM OF JUDICIARY BODIES: INTRODUCTION OF ELEMENTS OF GAME THEORY

2018 ◽  
Author(s):  
Olena Hladunova ◽  
Keyword(s):  
Game Theory ◽  
Justice System ◽  
Civil Service ◽  
Civil Servants ◽  
International Standards ◽  
Scientific Article ◽  
Basic Principles ◽  
Conflict Situations ◽  
Definition Of ◽  
The Right

In this scientific article the main elements of game theory are analyzed, the achievements of domestic and foreign scientists devoted to the consideration of such theory are investigated. The expediency of involving in the practical activity of the civil service in the system of judicial authorities effective methods used in the field of business and consisting in the use of game technologies, which have proven their effectiveness in terms of providing quality services. It is focused on the fact that game theory can play a key role in the decision-making process, however, it is important to strictly adhere to the limits of its application. Possible conflict situations in the work of civil servants of the justice system are formulated and it is investigated that in conflict conditions each so-called participant of the game makes his course, i.e. chooses his strategy, as a result of which the relevant conflict situation is outlined and a set of strategies of all players. Some examples of the use of elements of game theory are given and the content of certain types of strategies is revealed. In particular, a strategy is described, which is denoted by the term "screening". Taking into account the definition of ways to modernize the civil service, the need to include in standardized training programs for civil servants of the justice system category "B" training course, which will include the basic principles of game theory for their active use in conflict, skills to compromise in relationships with visitors to the court - recipients of court services, selection of the right strategy, consideration of theoretical and game modeling of personnel management tasks, focusing on the ability to obtain and timely provide the necessary information to create a new civil service in the judiciary that meets international standards.

scholarly journals Deep foundations of happiness. Principal fundamentals of different types of understanding happiness

2021 ◽  
Vol 76 (1) ◽  
pp. 23-29
Author(s):  
L. Samarska ◽  
◽  
N.M. Sas ◽  
Keyword(s):  
Time Management ◽  
Well Being ◽  
Free Play ◽  
Deep Foundations ◽  
Subjective Well Being ◽  
Physiological Basis ◽  
Basic Principles ◽  
Theoretical Approaches ◽  
Different Types ◽  
Definition Of

The choice of the article’s topic is conditioned by the necessity to develop happiness management (such as knowledge management, time management, etc.). With considerable attention to the definition of “happiness”, the analysis of recent publications reveals that it is crucial to understand the deep foundations of happiness, create a typology, reveal the basic principles of different types of understanding of happiness, which was chosen as the topic research. Theoretical approaches to the definition of “happiness” are chosen sociology of imagination of G. Durand, the theory of archetypes of C. Jung, and the theory of images and dreams of G. Bachelard. In the context of this system of views, the idea of happiness is the result of a free play of the imagination, which, while being on the path from past to future, is transformed, revealed, comes accurate as a result of previous collective and individual intermediate ideas, and is enriched and concretised by individual people, social groups, individuals. The mythos of happiness across nations and people differs in the way, method, and tools of individuation, the discovery of the Self. The anthropological tract of happiness has an end to its existence. It is determined by the cessation of existence, the life of nations and individuals. The desire to experience pleasure (according to Freud), the desire to rise (according to Durand) are reflexive, which determines the physiological basis of happiness. Representations of happiness determine priorities, coordinate the direction of thinking, actions, reactions to external circumstances, and choose ways to achieve happiness. This is done through the transcendental function (according to Jung) – a psychological function that arises from the connection of the content of the unconscious with the content of consciousness. Achieving happiness allows one to strengthen the subjectivity and reveal their uniqueness, which allows them to identify typological features (archetypes) of behavioural reactions of people based on individual and group ideas about happiness. The authors reveal the basic foundations of such archetypes of happiness as hedonism, eudemonia, “rat racing”, nihilism, subjective well-being.

scholarly journals Improving the efficiency of planning training hours for cadets’ weapon practice

2020 ◽  
Vol 26 ◽  
pp. 00088
Author(s):  
Zinaida Kuznetsova ◽  
Aleksey Chaban ◽  
Larisa Kotkova ◽  
Ruslan Miftakhov ◽  
Stanislav Kuznetsov
Keyword(s):  
Mental Stress ◽  
Professional Training ◽  
Military Training ◽  
First Year ◽  
Sports Training ◽  
Basic Principles ◽  
School Year ◽  
Adaptation Mechanisms ◽  
Definition Of ◽  
Methodological Aspects

Cadets’ weapons practice is one of the important components of professional training. Its success is in the training hours’ distribution and mastering the main tasks at the early stages of military training. The correct distribution of weapon practice means and methods is very important. Planning of all weapon practice means begins with the definition of conditions and logistics. It is important to implement the principles of sports training. The implementation of the sports training basic principles during the school year allows expanding the cadets ’ body adaptation mechanisms to daily physical and mental stress. The development of innovative technologies for planning hours and the implementation of weapon practice tools allows to optimize and improve the effectiveness of cadets’ military training from the first year. Methodological aspects of planning and calculating hours by types of cadets’ weapon practice are presented, taking into account the tasks and direction of the training tools implementation. Difficulties are analyzed and ways to eliminate them in the process of improving the cadets weapon practice are found. The key tasks in the structure of weapon practice and the direction of their implementation are defined. The author’s version of cadets’ weapon practice planning allows to control and adjust both physical and mental loads.

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