Spectral Theory of Linear Operators

Single operator theory and functional calculi

• Holomorphic functional calculus.

• Gelfand-Naimark theorem for commutative unital C*-algebras.

• Spectral theorem for bounded operators.

• Unbounded operators and notions as: symmetric, self-adjoint, closed, domain, dense domain, deficiency indices, self-adjoint extensions.

• Spectral theorem for unbounded self-adjoint operators.

• Stone's theorem on 1-parameter groups of unitairy transformations.

• The harmonic oscillator.

Operator algebras

• C*-algebras.

• Positive operators in a C*-algebra.

• States on a C*-algebra, and the GNS-construction.

The final grade of the course consists of the following components:

• Take-home mid-term exam

• Oral exam

Final grade calculation: Max(E, 0.15*M+0.85*E).

If the estimated number of students that takes the exam is above 15 then the exam may be written. This will be decided and announced during the first quarter of the course.

Disclaimer: information may change depending on unforeseen circumstances or measures (see: TER Art 29, sub 4).

In case of insufficient results a repair option may exist in accordance with Article 2, Examination requirements, Clause 4, of the Implementation Regulations 2024-2025.

Oral exam: oral resit