Modul

Oral examination of approx. 30 minutes.

After participation, students

• understand the concepts of spectrum and resolvent of closed operators on Banach spaces.
• know their basic properties and are able to explain them in simple examples.
• can explain and justify the special features of compact operators and the Fredholm Alternative.
• can deduce algebraic identities and norm bounds for operators by means of the Dunford functional calculus and the spectral calculus for self-adjoint operators. This in particular includes spectral projections and spectral mapping theorems.
• are able to apply this general theory to integral and differential equations, and recognize the importance of spectral theoretic methods in Analysis.

none

• Closed operators on Banach spaces,
• Spectrum and resolvent,
• Compact operators and Fredholm alternative,
• Dunford functional calculus, spectral projections,
• Fourier transform,
• Unbounded self-adjoint operators on Hilbert spaces,
• Spectral theorem,
• Sesquilinear forms and sectorial operators,
• Applications to partial differential equations.

The module „Functional Analysis“ is strongly recommended.

Total workload: 240 hours

Attendance: 90 h

• lectures, problem classes and examination

Self studies: 150 h

• follow-up and deepening of the course content,
• work on problem sheets,
• literature study and internet research on the course content,
• preparation for the module examination