### Spectral Theory (Summer Semester 2011)

- Lecturer: apl. Prof. Dr. Peer Christian Kunstmann
- Classes: Lecture (0156400), Problem class (0156500)
- Weekly hours: 4+2

Schedule | ||
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Lecture: | Friday 11:30-13:00 | 1C-04 |

Tuesday 11:30-13:00 | 1C-04 | |

Problem class: | Wednesday 15:45-17:15 | Z 1 |

Lecturers | ||
---|---|---|

Lecturer | apl. Prof. Dr. Peer Christian Kunstmann | |

Office hours: Thursday, 13 - 14 Uhr | ||

Room 2.027 Kollegiengebäude Mathematik (20.30) | ||

Email: peer.kunstmann@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 |

Spectral theory is concerned with properties of linear operators in Banach spaces.

Given a linear operator in a complex Banach space with domain the *spectrum* is the set of all complex such that is **not** an isomorphism (here is equipped with the graph norm). Most prominent and already known from finite dimensions (linear algebra) are *eigenvalues* , but in infinite dimensions new phenomena arise.

The complement of in is called the *resolvent set of *.

Central topics in spectral theory are properties of the spectrum including investigation of eigenvalues and eigenvectors, properties of the *resolvent map* , decomposition of the space in invariant subspaces and existence of functional calculi for .

In the lecture we shall study in particular

- spectrum and resolvent for bounded and unbounded operators,
- spectral properties of compact operators in Banach spaces,
- the spectral theorem for self adjoint operators in Hilbert space,
- applications to differential operators and boundary value problems.

# Summary

Summary of the lectures |

# References

J.B. Conway: A Course in Functional Analysis, Springer.

E.B. Davies: Spectral Theory and Differential Operators, Cambridge University Press.

N. Dunford, J.T. Schwartz: Linear Operators, Part I: General Theory, Wiley.

D.E. Edmunds, W.D. Evans: Spectral Theory and Differential Operators, Oxford University Press.

T. Kato: Perturbation Theory of Linear Operators, Springer.

S. Lang: Real and Functional Analysis, Springer.

R. Meise, D. Vogt: Introduction to Functional Analysis, Oxford, Clarendon Press.

W. Rudin: Functional Analysis, McGraw-Hill.

D. Werner: Funktionalanalysis, Springer.