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.
