General information
| Course type | AMUPIE |
| Module title | Spectral Theory |
| Language | English |
| Module lecturer | Andrzej Sołtysiak |
| Lecturer's email | asoltys@amu.edu.pl |
| Lecturer position | profesor UAM |
| Faculty | Faculty of Mathematics and Computer Science |
| Semester | 2022/2023 (winter) |
| Duration | 60 |
| ECTS | 6 |
| USOS code | 06-DTSPUM0-E |
Timetable
Module aim (aims)
The aim of the course is to present very important and classical result from operator theory such as the spectral theorem for self-adjoint operators on a Hilbert space.
Pre-requisites in terms of knowledge, skills and social competences (where relevant)
Background knowledge in mathematical analysis and functional analysis.
Syllabus
Week 1: Banach spaces and bounded linear operators on a Banach space– short reminder of main results.
Week 2: Hilbert spaces, the orthogonal projection theorem.
Week 3: Bounded linear operators on a Hilbert space, the adjoint of a bounded linear operator, normal, self-adjoint, and unitary operators.
Week 4: Invariant and reducing subspaces.
Week 5: The spectrum of a bounded linear operator on a Banach space.
Week 6: Main theorems concerning spectra.
Week 7: The spectrum of a bounded linear operator on a Hilbert space.
Week 8: The spectrum of a compact operator on a Banach space. The Riesz theorem.
Week 9: Integral operators. The Fredholm theorems.
Week 10: Orthogonal projection operators.
Week 11: The spectral theorem in a finite-dimensional space. The spectral theorem for normal compact operators.
Week 12: Positive operators.
Week 13: The functional calculus for self-adjoint operators.
Week 14: The spectral theorem for self-adjoint operators.
Week 15: The spectral theorem for normal and unitary operators.
Reading list
- J. B. Conway, Course in Functional Analysis, Springer-Verlag, New York 1997.
- N. Dunford, J. T. Schwartz, Linear Operators, Part II, Interscience Publishers, New York 1963.
- P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Chelsea Publ. Comp., New York 1951.
- P. R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, New York 1982.
- W. Mlak, Hilbert Spaces and Operator Theory, PWN and Kluver, Warszawa and Dordrecht 1991.
- F. Riesz, B. SZ.-Nagy, Functional Analysis, Frederick Ungar Publ. Co., New York 1955.
- W. Rudin, Functional Analysis, McGraw-Hill Inc., New York 1991.