Syllabus
*1. Banach algebras Definition, examples, adding a unit, renorming Invertible elements, Neumann series Spectrum and its properties, spektral radius C*-algebra, self-adjoint and normal elements Holomorphic calkulus *
2. Gelfand transform Complex homomorphisms and maximal ideals in commutative Banach algebras Gelfand transform and its properties Applications for commutative C*-algebras - Gelfand-Neimark theorem Applications for non-commutative C*-algebras - continuous funkction calkulus *
3. Operators on a Hilbert space Self-adjoint operators, normal operators, positive operators, unitary operators, projections Continuous and measurable calkulus, spectral measure and integral with respect to it, spectral decomposition of a normal operator Polar decomposition, positive and negative part *
4. Unbounded operators Unbounded operators on Banach spaces, closed operators, densely defined operators, spectrum Unbounded operators on Hilbert spaces, adjoint operator, symmetric and self-adjoint operators Cayley transform, deficiency indices Integral of an unbounded function with respect to a spectral measure Spectral decomposition of a self-adjoint operator
Annotation
Mandatory course for the master study programme Mathematical analysis. Recommended for the first year of master studies.
Continuation of the course NMMA401. Devoted to advanced topics in functional analysis - spectral theory in Banach algebras, Gelfand transform, spectral theory of bounded and unbounded operators.