Introduction to spectral theory

Part I
1. Resolvent and spectrum of bounded operators
    • Resolvent set and resolvent of an operator.
    • Generalities on the spectrum: non empty compact set, spectral radius, Gelfand formula.
    • The case of normal and self-adjoint operators on Hilbert spaces
2. Compact operators on Hilbert spaces
    • Definition and general properties
    • Spectrum of compact operators
    • Diagonalization of self-adjoint compact operators
3. Spectral theorem for bounded self-adjoint operators
    • Continuous functional calculus
    • Spectral measures
    • The spectral theorem: multiplicative form
    • Bounded functional calculus
4. (Time permitting) Essential and discrete spectrum: definition, Weyl criterion, stability of essential spectrum by compact perturbation
    

Part II
1. Introduction
    • Historical perspective
    • Spectral theorem in finite dimension
    • Notation and preliminaries
    • Formulation(s) of the spectral theorem
    • Spectral theorem in quantum mechanics
2. Harmonic analysis
    • Preliminaries
    • The Poisson transform and Radon-Nikodym derivatives
    • Poisson representation of harmonic functions
    • The Borel transform
    • Poltoratskii's theorem
3. Spectral theorem revisited
    • Cyclic case
    • General case
    • Harmonic analysis and spectral theorem
4. (Time permitting) Spectral theory of rank one perturbations

The basic reference for Part I is the book by M. Reed and B. Simon Methods of Modern Mathematical Physics, vol 1.
The basic reference for Part II are the lecture notes
Additional references will be provided during the lectures (and will be available via dedicated Dropbox file).
 
 

Jakob Björnberg and Daniel Ueltschi