### Spectral Theory (Summer Semester 2023)

- Lecturer: Prof. Dr. Roland Schnaubelt
- Classes: Lecture (0163700), Problem class (0163800)
- Weekly hours: 4+2
- Audience: Mathematics, Physics (from 6. semester)

**All information and materials for this lecture are provided in the course "Spectral theory" within Ilias, including communication via email and forum. Please join this course if you want to participate.**

Schedule | |||
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Lecture: | Tuesday 8:00-9:30 | SR 2.66 | Begin: 18.4.2023 |

Thursday 11:30-13:00 | SR 2.58 | ||

Problem class: | Wednesday 14:00-15:30 | SR 2.66 | Begin: 19.4.2015 |

Lecturers | ||
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Lecturer | Prof. Dr. Roland Schnaubelt | |

Office hours: Tuesday at 12:00 - 13:00, and by appointment. | ||

Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30) | ||

Email: schnaubelt@kit.edu | ||

Problem classes | Richard Nutt M.Sc. | |

Office hours: Just drop by whenever I am in the office. | ||

Room 2.043 Kollegiengebäude Mathematik (20.30) | ||

Email: richard.nutt@kit.edu |

The spectrum of a linear operator on a Banach space generalizes the concept of an eigenvalue of a matrix. In Banach spaces spectral theoretic methods play an equally important role as the eigenvalue theory in finite dimensions. These methods are used everywhere in analysis and its applications.

At the beginning we treat the basic properties of the spectrum. In view of the applications on differential operators this is not only done for bounded operators, but also for a certain class of unbounded linear operators, the so-called closed operators. To handle differential operators on L^p spaces, we discuss weak derivatives in the L^p setting and Sobolev spaces. Here we also study the Fourier transform. One can develop a detailed spectral theory for two main classes of operators. We first deal with compact operators, where the spectrum determined by the eigenvalues to a large extent. In this context we also prove the so-called Fredholm alternative, which has important applications e.g. to integral equations. Then we investigate (possibly only closed) self adjoint operators on Hilbert spaces. For such operators the spectral theorem is a far reaching extension of the diagonalisation of hermitian matrices. Finally, we treat the functional calculi for self adjoint, bounded and sectorial operators.

We strongly recommend knowledge of the lecture Functional Analysis.

# Examination

There will be an oral exam (of about 30 min) on **16.8.** or **14.9.** It talkes place in room 2.070. Please register online at CAS Campus Management. Afterwards come to the secretariat (Katz/Schaaf) to select one oft the dates (until 11.8. (12:00) for first, until 11.9. (12:00) for second). If you can deregister from the exam (without giving a reason) up to 3 working days before the exam. If have obtained a time slot from the secretariat, please inform it about your withdrawal. The exam can be taken in Englisch or German.

You will not be asked about Sections 3.2, 3.3, 5.2, and also not about the additional contents of the lecture notes (appendices, proofs omitted in the lecture (marked by a footnote), and references to the literature and other lecture notes). However, you should understand why a cited result can be used in the arguments.

# References

On my webpage and in Ilias, parallel to the lectures I will provide lecture notes. Here are several relevant monographs, including classics:

- H.W. Alt: Lineare Funktionalanalysis. Springer, 2006.
- H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2011.
- J.B. Conway: A Course in Functional Analysis. Springer, 2007
- N. Dunford, J.T. Schwartz: Linear Operators. Part I: General Theory. Wiley, 1988.
- N. Dunford, J.T. Schwartz: Linear Operators. Part II: Spectral Theory. Wiley, 1988.
- T Kato: Perturbation Theory of Linear Operators. Springer, 1995.
- M. Reed, B. Simon: Methods of Modern Mathematical Analysis. Volume I: Functional Analysis. Academic Press, 1980.
- M. Reed, B. Simon: Methods of Modern Mathematical Analysis. Volume II: Fourier Analysis, Self-adjointness. Academic Press, 1975.
- B. Simon: Operator Theory. A Comprehensive Course in Analysis, Part 4. American Math. Society, 2015.
- A.E. Taylor, D.C. Lay: Introduction to Functional Analysis. Wiley, 1980.
- J. Weidmann: Lineare Operatoren in Hilberträumen. Teil I Grundlagen. Teubner, 2000.
- D. Werner: Funktionalanalysis. Springer, 2005.