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 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

Readership: Graduate students, mathematicians, physicists or engineers interested in functions of operators.