2020/2021

# Functional Analysis 2 (Operator Theory)

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Category 'Best Course for New Knowledge and Skills'

Type:
Optional course (faculty)

Delivered by:
Faculty of Mathematics

Where:
Faculty of Mathematics

When:
3, 4 module

Language:
English

ECTS credits:
6

Contact hours:
72

### Course Syllabus

#### Abstract

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.

#### Learning Objectives

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

#### Expected Learning Outcomes

- 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.
- Given a linear operator, understand whether or not it is compact.
- Prove the continuity of concrete linear operators between topological vector spaces.

#### Course Contents

- 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).

#### Interim Assessment

- Interim assessment (4 module)0.35 * exercise sheets grade + 0.3 * final exam + 0.35 * midterm grade

#### Bibliography

#### Recommended Core Bibliography

- 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

#### Recommended Additional Bibliography

- 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