# MA5012 Operator Theory - Sommersemester 2016

**Lecturer:**Simone Warzel

**Exercises:**Frank Hofmaier

## Time and Location

**Lecture:**Tue 14:15-15:45 in MI 03.10.011, and Th 12:30-14:00 in MI 03.08.011

**Supplements:**Th 14:00-15:00

**Exercises:**

## News

Exercise classes in the week 16.-20.5. are cancelled due to Pentacost holidays. The class on July 7 is cancelled. Oral exams will be taken place during the period July 21-28. Exact schedules were distributed by mail.## Prerequisites

Functional analysis [MA3001], complex analysis [MA2006/MA2008]## Contents

**Topics:**Banach algebras and their spectral theory; spectral theorem in Hilbert spaces; unbounded operators; semigroups

**Summary:**The central topic of the lecture will be spectral theory and spectral calculus. This is an indispensable technique for the solution of linear evolution equations which arise in many applications ranging from mathematical physics and biology to engineering. The course will follow a top-down approach to the topic starting from the spectral theory in Banach algebras down to the theory of self-adjoint (possibly unbounded) operators in a Hilbert space. In the course of the lecture, applications of these techniques with an emphasis on the area of mathematical physics will be covered.

- Week 1 (Banach and C^* algebras: notion of spectrum and resolvent, von Neumann series theorem, spectral radius formula; outlook into functional calculus)
- Week 2 (Gelfand theory of commutative Banach algebras: (complex) homomorhisms, ideals, Gelfand representation theory)
- Week 3 (Spectral theorem for normal operators in a C^* algebra)
- Week 4 (Spectral theorem for bounded normal operators in a Hilbert space: spectral measures, measurable functional calculus)
- Week 5 (States and the GNS representation)
- Week 6/7 (Unbounded operators on Hilbert spaces: closed and closable operators, the adjoint)
- Week 8 ((Essential) self-adjointness, inverses and the notion of spectra)
- Week 9 (Self-adjointness, a glance at quantum mechanics and Stone's theorem)
- Week 10 (Proof of Stone's theorem; spectral theorem for unbounded self-adjoint operators)
- Week 11 (Spectral theory for unbounded self-adjoint operators; Semigroups of operators: Basics)
- Week 12 (Semigroups of operators: contractions; Hille-Yosida's and Phillip's theorem)
- Week 13 (Approximating semigroups; Trotter product formula)
- Week 14 (Spectra of semigroups; Outlook: Quantum Dynamical Semigroups and Lindblad generators)

## Exercise Sheets

- Sheet 1 (now including solutions)
- Sheet 2 (now including solutions)
- Sheet 3 (now including solutions)
- Sheet 4 (now including solutions)
- Sheet 5 (now including solutions)
- Sheet 6 (now including solutions)
- Sheet 7 (now including solutions)
- Sheet 8 (now including solutions)
- Sheet 9 (now including solutions)
- Sheet 10 (now including solutions)
- Sheet 11 (now including solutions)
- Sheet 12 (now including solutions)

## Literature

- Barry Simon: Operator Theory (AMS 2015) [very readable; we will cover only a selection]
- Peter D. Lax: Functional Analysis (Wiley, 2002) [very readable; selected chapters from the second half]
- Kehe Zhu: An Introduction to Operator Algebras (CRC Press 1993) [very readable reference for the algebraic part of the course]
- Walter Rudin: Functional Analysis (Mc Graw Hill 1991) [good reference for the algebraic part of the course]
- John B. Conway: A Course in Functional Analysis (Springer, 1990) [selected chapters from the second half]
- Gert K. Pedersen: Analysis Now (Springer, 1989) [advanced text]
- Dirk Werner: Funktionalanalysis (Springer, 1995) [very readable; in German]