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.