Functional Calculi

A functional calculus is a construction which associates with an operator or a family of operators a homomorphism from a function space into a subspace of continuous linear operators, i.e. a method for defining “functions of an operator”. Perhaps the most familiar example is based on the spectral theorem for bounded self-adjoint operators on a complex Hilbert space.

This book contains an exposition of several such functional calculi. In particular, there is an exposition based on the spectral theorem for bounded, self-adjoint operators, an extension to the case of several commuting self-adjoint operators and an extension to normal operators. The Riesz operational calculus based on the Cauchy integral theorem from complex analysis is also described. Finally, an exposition of a functional calculus due to H. Weyl is given.
Contents: Vector and Operator Valued MeasuresFunctions of a Self Adjoint OperatorFunctions of Several Commuting Self Adjoint OperatorsThe Spectral Theorem for Normal OperatorsIntegrating Vector Valued FunctionsAn Abstract Functional CalculusThe Riesz Operational CalculusWeyl's Functional CalculusAppendices:The Orlicz–Pettis TheoremThe Spectrum of an OperatorSelf Adjoint, Normal and Unitary OperatorsSesquilinear FunctionalsTempered Distributions and the Fourier Transform
Readership: Graduate students, mathematicians, physicists or engineers interested in functions of operators.