Selected topics of spectral theory in Hilbert spaces 1000-1M24TSH

1. Review: Spectral properties of compact operators on Banach spaces.

2. Spectral measures and resolution of the identity; spectral theorem for normal operators on Hilbert spaces; functional calculus; positive and unitary operators; polar decomposition.

3. Theory of unbounded operators: closability, adjoint, spectrum. Spectral theorem for unbounded operators.

4. Applications I: Laplace operator. Connections between the spectrum and the geometry of the underlying space.

5. Applications II: Quantum mechanics. Uncertainty principle; harmonic oscillator.

6. Applications III: Non-commutative probability. Operator algebras as spaces of random variables; operator spectrum as a generalization of a distribution; laws of large numbers and central limit theorem in the non-commutative setting.