THE MOST IMPORTANT KNOWLEDGE ABOUT THE LINEAR OPERATOR

​САМОЕ ВАЖНОЕ ЗНАНИЕ О ЛИНЕЙНОМ ОПЕРАТОРЕ

Дуневич Ирина Семеновна

магистрант, кафедра педагогического образования математического образование, Белгородский государственный национальный исследовательский университет,

РФ, г. Белгорода

Богачев Роман Евгеньевич

научный руководитель, доц., Белгородский государственный национальный исследовательский университет,

РФ, г. Белгорода

ABSTRACT

This paper is devoted to the spectral theory of operators. In a separate chapter, the spectral theory of compact operators is considered in more detail. The most important problems of this theory are statements about the reduction of the studied operators to the so-called diagonal form - spectral theorems, statements about the properties of the spectrum and eigenvalues.

The purpose of this course work is to acquaint those who are interested in mathematics with the spectral theory of operators, in particular, with the spectral theory for compact operators.

The object of research is linear operators, and the subject is operations with linear operators.

In this work, the following tasks were set:

1. Describe the concept of linear operators.

2. Specify the main operations and theorems.

3. Consider and describe the practice of finding the eigenvalue of a linear operator.

This work consists of three chapters:

1) Linear operator;

2) Spectral theory of operators;

3) Methods for finding eigenvalues of linear operators.

The first chapter deals with the concepts of a linear operator, linear transformations, ad joint and self-adjoins operators.

The second chapter deals with the concept of the spectrum of an operator, a theorem for a closed linear operator, the spectral radius, the concept of a bounded operator and compact operators, and a theorem that is an important characteristic property of compact operators.

АННОТАЦИЯ

Данная работа посвящена спектральной теории операторов. В отдельной главе более подробно рассматривается спектральная теория компактных операторов. Важнейшими задачами этой теории являются утверждения о приведении изучаемых операторов к так называемому диагональному виду – спектральные теоремы, утверждения о свойствах спектра и собственных значениях.

Цель данной курсовой работы – познакомить тех, кто интересуется математикой со спектральной теорией операторов, в частности, со спектральной теорией для компактных операторов.

Объект исследования - линейные операторы, а предмет - операции с линейными операторами.

В данной работе были поставлены следующие задачи:

1. Описать понятие линейных операторов.

2. Указать основные операции и теоремы.

3. Рассмотреть и описать практику нахождения собственного значения линейного оператора.

Данная работа состоит из трёх глав:

1) Линейный оператор;

2) Спектральная теория операторов;

3) Методы отыскания собственных значений линейных операторов.

В первой главе рассматривается понятия линейного оператора, линейные преобразования, сопряжённый и самосопряжённый оператор.

Во второй главе рассматривается понятие спектра оператора, теорема для замкнутого линейного оператора, спектральный радиус понятие об ограниченном операторе и компактных операторах, а также теорема, являющаяся важным характеристическим свойством компактных операторов.

Keywords: linear operator, differential operators, limited transformation, continuous, bounded linear.

Ключевые слова: линейный оператор, дифференциальные операторы, ограниченные преобразования, непрерывный, ограниченный линейный.

1. Linear operator

1.1. The concept of a linear operator

An ordinary function whose values belong to a field of scalars is called a functional.

In general, a function with a set of values is not always defined on a general Hilbert space, but only on some of its subsets. This subset is meant to be the domain of the function. The set of values of a function is the set into which this function maps its domain of definition. For convenience, we agree to denote the domain of definition by D, the Hilbert space that contains it, by H_1, the set of values ​​by R, and the space containing it by H_2.

The operator (transformation) L is also called linear when its domain of definition D is a linear subspace (exactly dense, not necessarily) and then it is also linear on D.

Here the set of a linear operator is also a linear subspace.

1.2. Linear transformations

This graph G(T) of our transformation of linear T is a subspace in the product of subspaces H_1 × H_2 , which is also formed according to the rule

But the linear transformation T is called (closed) only when its function graph is (closed) in the subset H_3. Otherwise, the closedness of this operator T is defined as follows:

let

  Then                                  (3a)

  и 

Note that, as a rule, all differential operators are closed. Therefore, it is necessary to consider closed operators, namely their class.

From here it follows that a linear transformation T is a limited transformation if D= H_1 and

                                                                                    (4a)

And the norm of the restricted linear transformation T is the number

                                                                                        (5a)

A linear transformation is bounded when it is continuous at the origin. And that means it is also continuous at its every point. The linear transformation is bounded, that is, continuous.

Let T_1, T_2 be bounded linear operators, on H_1 to H_2 . It follows from here that the sum T_1+T_2 is a bounded linear operator.

                                                                             (6a)

And reading the definitions (αT) x=α T x, where α is an element of a scalar field, then the operator αT is within the constraint only when T is constrained. Consequently, all linearly bounded set of operators create a linear space, and their operator norm is the norm on this entire space.

The nor med linear space of operators obtained by us is defined as follows: L(H_1,H_2). It follows from here that the linear space L(H_1,H_2) is (complete). Indeed, if {TM} is the Cauchy sequence of the space, then for any element x of the space H_1 we have

                                                                           (7a)

Therefore, {Tm} is the spatial Cauchy sequence H2 , and we denote its limit by Tm. From here it is clear that the operator T is linear and bounded.

 If a

  и  , then

Also, it uses ad joint and self-adjoins operator. Then, let T be a bounded linear operator from the set H_1 to H_2 - the ad joint operator T^* (defined on H_2 and taking values ​​in H) is defined by the condition y=T*x only if there is a vector (y) such that [ y, z]=[x, T z] for any zϵH_1.

We assume that H_1=H_2=H. An operator L with a dense domain of definition is called self-ad joint if L=L^*.

Список литературы:

1. Ахнезер, М.Н., Глазман И.Н. Теория линейных операторов в гильбертовом пространстве М.Н. Ахнезер, И.Н. Глазман. – Киев: Ваша школа, 1977. – 336 с.
2. Балакришнан, A. Slochaslic Differential system. A.V. Balakrishnan. – N.-Y.: Springer-Velar, 1960. – 451 с.
3. Балакришнан, A. Communication theory. A.V. Balakrishnan. – N.-Y.: McGraw-Hill, 1968. – 432 с.
4. Блюм, Е. Numerical Analysis and Computation. Theory and Placlice. Е. Блюм. N.-Y.:Addison Wesley. 1972.–274 c.