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2019/2020

Функциональный анализ (Теория операторов)

Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский

Программа дисциплины

Аннотация

Functional analysis studies infinite-dimensional vector spaces equipped with a norm (or, more generally, with a topology), operators between such spaces, and representations of algebraic structures on such spaces. The classical areas of Functional Analysis are the spectral theory of linear operators, the geometry of Banach spaces, distribution theory, operator algebra theory, etc. Among relatively new areas are noncommutative geometry à la Connes, operator space theory (a.k.a. «quantum functional analysis»), and locally compact quantum groups. Functional analysis has numerous applications in differential equations, harmonic analysis, representation theory, geometry, topology, calculus of variations, optimization, quantum physics, etc. This course is a continuation of the course «Introduction to Functional Analysis» (fall 2019). We plan to discuss those aspects of functional analysis which deal with rather general classes of linear operators on Banach and Hilbert spaces. This means that we will not consider, for example, differential operators at all, because their theory can be well presented in a separate course only. Instead, we concentrate on those topics which emphasize the role of algebraic methods in functional analysis.
Цель освоения дисциплины

Цель освоения дисциплины

  • Students will be introduced to some topics of operator theory (with an emphasis on spectral theory) and to the fundamentals of Banach algebra theory.
Результаты освоения дисциплины

Результаты освоения дисциплины

  • Prove the continuity of concrete linear operators between topological vector spaces.
  • Given a linear operator, understand whether or not it is compact.
  • Find the essential spectra of linear operators.
  • Find the maximal spectra of concrete commutative Banach algebras.
  • Describe the functional calculi and the spectral decompositions of concrete selfadjoint operators.
Содержание учебной дисциплины

Содержание учебной дисциплины

  • Topological vector spaces and duality.
  • Compact and Fredholm operators. The Riesz–Schauder theory. The general index theory.
  • Commutative Banach algebras. The Gelfand transform. The commutative Gelfand–Naimark theorem.
  • Spectral theory of normal operators on a Hilbert space. The spectral theorem.
  • Distributions (if time permits).
Элементы контроля

Элементы контроля

  • неблокирующий Created with Sketch. midterm grade
    The midterm exam (oral) will be at the end of March and will include only the material of the 3rd module
  • неблокирующий Created with Sketch. exercise sheets grade
    To get the maximum grade for the exercise sheets, you should solve 75% of all the exercises. If you solve more, you will earn bonus points. You can also earn bonus points for working actively at the exercise classes and for solving «bonus exercises» (marked as «B» in the sheets).
  • неблокирующий Created with Sketch. final exam
    The oral exam will be at the end of May and will include only the material of the 4th module.
Промежуточная аттестация

Промежуточная аттестация

  • Промежуточная аттестация (4 модуль)
    0.35 * exercise sheets grade + 0.3 * final exam + 0.35 * midterm grade
Список литературы

Список литературы

Рекомендуемая основная литература

  • Aydın Aytuna, Reinhold Meise, Tosun Terzioğlu, & Dietmar Vogt. (2011). Functional Analysis and Complex Analysis. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=974875

Рекомендуемая дополнительная литература

  • Francois Treves. (2013). Topological Vector Spaces, Distributions and Kernels. [N.p.]: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1151250
  • Reed, M. (1972). Methods of Modern Mathematical Physics : Functional Analysis. Oxford: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=567963
  • Treves, F. (1967). Topological Vector Spaces, Distributions and Kernels. San Diego, Calif: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=239470
  • Treves, F., Smith, P. A., & Eilenberg, S. (1967). Topological Vector Spaces, Distributions and Kernels : Pure and Applied Mathematics, Vol. 25. New York: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1261097