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Spectral Theory (Summer Semester 2015)

  • Lecturer: Prof. Dr. Roland Schnaubelt
  • Classes: Lecture (0163700), Problem class (0163800)
  • Weekly hours: 4+2
  • Audience: Mathematics, Physics (from 6. semester)
Schedule
Lecture: Monday 9:45-11:15 SR 2.66 Begin: 13.4.2015
Wednesday 11:30-13:00 SR 2.66
Problem class: Thursday 15:45-17:15 SR 2.66 Begin: 16.4.2015
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. Heiko Hoffmann
Office hours: Monday, 16:00 - 17:00, and by appointment
Room 2.048 Kollegiengebäude Mathematik (20.30)
Email: heiko.hoffmann@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 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 derivatives 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: Knowledge of the lecture Functional Analysis or of the lecture Hilbert Spaces and Differential Equations.

Examination

There are oral exams on 12 August 2015 and 21 September 2015.

References

On my webpage one can find the PDF file of the manuscript of my lecture Spectral Theory from summer semester 2010. Presumably, an updated version will be delivered during lecture time. A few relevant monographs:

  • H.W. Alt: Lineare Funktionalanalysis. Springer.
  • H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations. 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.