Advanced Course: Operator Ideals and Their Applications
Location and date:
Room 3.06, Jadranska 21, 3rd Floor,
The central result in finite-dimensional operator theory (theory of matrices) is the Jordan normal form.
Our main topics will be special classes of linear operators in Hilbert spaces, namely, compact operators and (unbounded) self-adjoint operators.
The purpose of the course is to get acquainted with basic results of the theory of compact operators and operator ideals in Hilbert spaces. We will also focus on numerous applications of this theory (bound state problems, scattering theory, nonlinear wave equations etc.). Required knowledge are the basic results of linear algebra. Some knowledge of Hilbert spaces would be desirable and helpful.
The following topics are expected to be treated:
- Compact operators, Hilbert--Schmidt operators, Hilbert--Schmidt Theorem, Fredholm alternative, Spectral Theorem (for compact operators), Canonical expansion.
- Singular values, Trace Class, Neumann--Schatten ideals.
- Fredholm determinant, Mercer's Theorem, Lidskii's Theorem.
- Applications: Bound state problems, 1d scattering, KdV and all that.
My lecture notes
I. C. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators,
Amer. Math. Soc., Providence, 1969.
B. Simon, Trace Ideals and Their Applications, 2nd edn., Amer. Math. Soc., Providence, RI, 2005.
B. Simon, A Comprehensive Course in Analysis, Part 4: Operator Theory, Amer. Math. Soc., Providence, RI, 2015.