Functional Analysis 3

Vztahy:

In order to register for the course 01FAN3, the student must have successfully completed or received credit for and not exhausted all examination dates for the course 01FAN2. The course 01FAN3 can be graded only after the course 01FAN2 has been successfully completed.

Garant předmětu:

Lecturer:

Tutor:

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, Stone’s

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. Basics of theory of Banach and Hilbert spaces, bounded linear operators, closed linear operators, Hilbert-Schmidt

operators, bounded selfadjoint operators.

2. Tensor product of Hilbert spaces.

3. Compact operators in Banach and Hilbert spaces.

4. Ideals of compact operators.

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

6. Spectral theorem for unbounded selfadjoint operators.

7. Oneparametric groups of unitary operators, Stone’s theorem.

8. Quadratic forms, representation theorem, Friedrich’s extension.

9. Bochner integral.

10. Banach algebras, C*-algebras.

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:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematická fyzika (elective course)
• Matematické inženýrství (compulsory course in the program)