Spectral Theory

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.