(in Polish) Spectral theory and functional calculus 290-MA3-1STFC

The student expands their competence to include the fundamentals of the theory of Banach algebras and C*-algebras—the modern language of functional analysis, noncommutative geometry, and quantum theories.
The concepts mastered include the spectrum and its properties, the functional calculus for operators, the Gelfand transform, and the characterization of commutative C*-algebras—the Gelfand-Naimark theorem.

As the course is conducted as a seminar, students and the instructor divide the topics to be covered among themselves, then take turns presenting and discussing them.The topics discussed will cover the following points but may be expanded to include additional elements, depending on the participants' research interests:

1) Banach algebras, examples and basic facts

2) Invertible elements in Banach algebras

3) Spectrum and the spectral radius

4) The Gelfand representation

5) Analytic functional calculus

6) C*-algebras, examples and basic facts

7) Charaterisation of commutative C*-algebras - Gelfand-Naimark Theorem

8) Continuous functional calculus and its applications.