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Spectral Theory (Summer Semester 2010)

  • Lecturer: Prof. Dr. Roland Schnaubelt
  • Classes: Lecture (1564), Problem class (1565)
  • Weekly hours: 4+2
  • Audience: Mathematics, Physics (from 6. semester)
Schedule
Lecture: Monday 8:00-9:30 Neuer Hörsaal Begin: 12.4.2010
Wednesday 9:45-11:15 Nusselt-Hörsaal
Problem class: Wednesday 15:45-17:15 SR 1 Begin: 14.4.2010
Lecturers
Lecturer Prof. Dr. Roland Schnaubelt
Office hours: Wednesday, 11:30 - 13:00, and by appointment
Room 2-047 (Englerstr. 2) Kollegiengebäude Mathematik (20.30)
Email: schnaubelt@kit.edu
Problem classes Dr. Esther Bleich
Office hours: Nach Vereinbarung
Room 3A-28 Allianz-Gebäude (05.20)
Email: esther.bleich@t-online.de

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 im finite dimensions. These methods are used everywhere in analysis and its applications.

At the beginning we discuss 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 treat differential operators on L^p spaces, we introduce weak derivative in the L^p setting and Sobolev spaces. 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 study (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.

Prerequisites: Functional analysis.

Further informations concerning this lecture you find in the Studierendenportal of the KIT at the URL
https://studium.kit.edu/sites/vab/67648/Start/homepage.aspx

References

  • H.W. Alt: Lineare Funktionalanalysis. Springer.
  • J.B. Conway: A Course in Functional Analysis. Springer.
  • N. Dunford, J.T. Schwartz: Linear Operators. Part I: General Theory. Wiley.
  • T Kato: Perturbation Theory of Linear Operators. Springer.
  • A.E. Taylor, D.C. Lay: Introduction to Functional Analysis. Wiley.
  • D. Werner: Funktionalanalysis. Springer.

(More literature can be found in the "Vorlesungspräsenz" in the department's library.)