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Course, academic year 2023/2024
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Functional Analysis 2 - NMMA402
Title: Funkcionální analýza 2
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: prof. RNDr. Ondřej Kalenda, Ph.D., DSc.
Class: M Mgr. MA
M Mgr. MA > Povinné
Classification: Mathematics > Functional Analysis
Incompatibility : NRFA054
Interchangeability : NRFA054
Is interchangeable with: NRFA054
Annotation -
Last update: doc. RNDr. Pavel Pyrih, CSc. (12.05.2022)
Mandatory course for the master study programme Mathematical analysis. Recommended for the first year of master studies. Continuation of the course NMMA401. Devoted to advanced topics in functional analysis - spectral theory in Banach algebras, Gelfand transform, spectral theory of bounded and unbounded operators.
Course completion requirements -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (05.02.2024)

The rules for 2023/2024:

The course is finished by a credit and an exam. Before passing the exam it is necessary to gain the credit.

The credit will be awarded after complete and correct solution of two homeworks and presenting a correct solution of one problem during the classes.

If the submitted solution of a homework is not complete and correct, a correction should be provided. The number of iterations is not a priori limited.

Detailed rules will be specified at the webpage of the lecturer.

Literature -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (01.02.2024)

Rudin, W.: Functional analysis. Second edition, McGraw-Hill, Inc., New York, 1991

Meise R. and Vogt D. : Introduction to functional analysis, Oxford University Press, New York, 1997

Requirements to the exam -
Last update: doc. Mgr. Marek Cúth, Ph.D. (03.02.2023)

The exam is oral with the possibility of a written preparation. Mainly knowledge and understanding of the notions and theorems explained during the semester will be tested. In addition, solving selected problems using the methods explained during the course will be a part of the exam. The lectures are the main source of materials for the exam.

Syllabus -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (09.05.2022)
1. Banach algebras

Definition, examples, adding a unit, renorming

Invertible elements, Neumann series

Spectrum and its properties, spektral radius

C*-algebra, self-adjoint and normal elements

Holomorphic calkulus

2. Gelfand transform

Complex homomorphisms and maximal ideals in commutative Banach algebras

Gelfand transform and its properties

Applications for commutative C*-algebras - Gelfand-Neimark theorem

Applications for non-commutative C*-algebras - continuous funkction calkulus

3. Operators on a Hilbert space

Self-adjoint operators, normal operators, positive operators, unitary operators, projections

Continuous and measurable calkulus, spectral measure and integral with respect to it, spectral decomposition of a normal operator

Polar decomposition, positive and negative part

4. Unbounded operators

Unbounded operators on Banach spaces, closed operators, densely defined operators, spectrum

Unbounded operators on Hilbert spaces, adjoint operator, symmetric and self-adjoint operators

Cayley transform, deficiency indices

Integral of an unbounded function with respect to a spectral measure

Spectral decomposition of a self-adjoint operator

Entry requirements -
Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (05.02.2024)

Continuation of the course NMMA401. Devoted to advanced topics in functional analysis. Expected knowledge includes elements of functional analysis (the content of courses NMMA331 and NMMA401), complex analysis (Cauchy theorem, Cauchy formula) and measure and integration.

 
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