In this book, which is basically self-contained, we concentrate on partial differential equations in mathematical physics and on operator semigroups with their generators.

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Content

In this book, which is basically self-contained, we concentrate on partial differential equations in mathematical physics and on operator semigroups with their generators. A central theme is a thorough treatment of distribution theory. This is done via convolution products, Fourier transforms, and fundamental solutions of partial differential operators with constant coefficients. Linear initial value problems are treated via operator semigroups. A relationship between so-called Feller-Dynkin semigroups and Markov processes is described. Finally, Feynman-Kac semigroups are introduced.

- Chapter 5. Operators in Hilbert space

- Some results in Banach algebras
- Symbolic calculus
- On square roots in Banach algebras
- On C˚-algebras
- On Gelfand transforms
- Resolution of the identity

- Closed linear operators

- Chapter 6. Operator semigroups and Markov processes

- Generalities on semigroups
- Examples
- Uniformly continuous semigroups
- Self-adjoint semigroups
- Translation group
- Gaussian semigroup
- Wave operator
- Adjoint semigroups
- Dyson-Phillips expansion
- Stone’s theorem
- Convolution semigroups of measures
- Semigroups acting on operators
- Quantum dynamical semigroups
- Semigroups for system theory
- Semigroups and pseudo-differential operators
- Quadratic forms and semigroups
- Ornstein-Uhlenbeck semigroup
- Evolutions and semigroups

- Markov processes
- Feynman-Kac semigroups
- KMS formula

- Harmonic functions on a strip

- Chapter 7. Holomorphic semigroups

- Introduction
- Exponentially bounded analytic semigroups
- Bounded analytic semigroups
- Bounded analytic semigroups and the Crank-Nicolson iteration scheme
- Stability of the Crank-Nicolson iteration scheme

- Chapter 8. Elements of functional analysis

- Theorem of Hahn-Banach
- Baire category

- Banach-Steinhaus theorems: barreled spaces
- Inductive limits
- The open mapping theorem
- Krein-Smulian and the Eberlein-Smulian theorem

- Subjects for further research and presentations
- Bibliography
- Index