Operatoranalyse

Operatoranalyse, or operator theory, is a branch of functional analysis that studies linear operators on function spaces, typically Banach or Hilbert spaces. It investigates how operators behave under algebraic operations, how their spectra influence dynamics, and how they can be extended or approximated. The central objects are linear operators, including bounded operators and unbounded densely defined closed operators, along with their adjoints and domains.

Key topics include the spectral theory: spectrum, point spectrum, continuous spectrum, and resolvent; and the spectral theorem for self-adjoint and normal operators, which provides a functional calculus allowing functions of operators. Unbounded operators require careful domain considerations. Semigroups of operators describe evolution processes; the Hille–Yosida theorem characterizes generators of strongly continuous semigroups.

Another major pillar is operator algebras, notably C*-algebras and von Neumann algebras, which abstract and unify operator theory with algebraic methods and yield results in representation theory and quantum mechanics.

Methods include perturbation theory, dilation theory, numerical ranges, and various forms of functional calculus. Applications span quantum mechanics, partial differential equations, signal processing, control theory, and numerical analysis.

History: operator analysis grew from spectral theory of differential operators in the 19th and early 20th centuries, with foundational contributions by Hilbert, Weyl, and von Neumann, and later developments by Gelfand, Naimark, Kadison, and Krein shaping the theory of operator algebras.