Functional Analysis 3

Course guarantor:

Pavel Šťovíček

Lecturer:

Pavel Šťovíček

Tutor:

Pavel Šťovíček

Supervisor:

Department of Mathematics

Synopsis:

Advanced parts of functional analysis needed for theory of representations of Lie groups and quantum theory. Compact

operators, their ideals, unbounded selfadjoint operators, theory of selfadjoint extension of symmetric operators, Stones

theorem, quadratic forms and Bochner integral. The basics of Banach algebras and C*-algebras.

Requirements:

Basic calculus and linear algebra courses (01MANA, 01MAA2-4, 01LALA, 01LAA2), the first two parts of Functional analysis course (01FAN1, 01FA2).

Syllabus of lectures:

1. Tensor product of vector spaces.

2. Compact operators in Banach and Hilbert spaces.

3. Ideals of compact operators.

4. Symmetric operators, selfadjoint operators, selfadjoint extensions of symmetric operators.

5. Spectral theorem for unbounded selfadjoint operators.

Syllabus of tutorials:

1. Summarization of properties of bounded operators on Hilbert spaces.

2. Compact operators.

3. Symmetric, selfadjoint, closed operators. Essential spectrum.

Study Objective:

The goal of study is to finish the basic functional analysis course oriented mainly on modern quantum theory and solving of problems which arise in physical and technical applications.

Study materials:

Key references:

[1] J. Blank, P. Exner, M. Havlíček: Hilbert Space Operators in Quantum Physics, Springer, 2008.

[2] B. Simon: Operator Theory: A Comprehensive Course in Analysis, Part 4, AMS, Rhode Island, 2015.

[3] K. Yoshida, Functional Analysis, Springer Science & Business Media, New York 2013

[4] J. B. Conway, A Course in Functional Analysis, Springer Science & Business Media, New York 2013

Recommended references:

[5] W. Rudin: Real and Complex Analysis, (McGrew-Hill, Inc., New York, 1974)

[6] A. N. Kolmogorov, S. V. Fomin: Elements of the Theory of Functions and Functional Analysis, (Dover Publications,

1999)

[7] A. E. Taylor: Introduction to Functional Analysis, (John Wiley and Sons, Inc., New York, 1976)

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans: