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Institutes

Institute for Analysis

Workgroup Functional Analysis

Spectral Theory

Spectral Theory

Spectral Theory (Summer Semester 2011)

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

Schedule
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

Spectral theory is concerned with properties of linear operators in Banach spaces.
Given a linear operator $A$ in a complex Banach space $X$ with domain $D(A)$ the spectrum $\sigma(A)$ is the set of all complex $\lambda$ such that $\lambda I-A:D(A)\to X$ is not an isomorphism (here $D(A)$ is equipped with the graph norm). Most prominent and already known from finite dimensions (linear algebra) are eigenvalues $\lambda$ , but in infinite dimensions new phenomena arise.
The complement $\rho(A)$ of $\sigma(A)$ in $\mathbb{C}$ is called the resolvent set of $A$ .

Central topics in spectral theory are properties of the spectrum $\sigma(A)$ including investigation of eigenvalues and eigenvectors, properties of the resolvent map $\lambda\mapsto(\lambda I-A)^{-1}$ , decomposition of the space $X$ in invariant subspaces and existence of functional calculi for $A$ .

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.

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.