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.
Vector and Operator Valued Measures; Functions of a Self Adjoint Operator; Functions of Several Commuting Self Adjoint Operators; The Spectral Theorem for Normal Operators; Integrating Vector Valued Functions; An Abstract Functional Calculus; The Riesz Operational Calculus; Weyl's Functional Calculus; Appendices: The Orlicz - Pettis Theorem; The Spectrum of an Operator; Self Adjoint, Normal and Unitary Operators; Sesquilinear Functionals; Tempered Distributions and the Fourier Transform.