General information

Course type AMUPIE
Module title Spectral Theory
Language English
Module lecturer Andrzej Sołtysiak
Lecturer's email
Lecturer position profesor UAM
Faculty Faculty of Mathematics and Computer Science
Semester 2022/2023 (winter)
Duration 60
USOS code 06-DTSPUM0-E


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.



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