|
" Semigroups of linear operators : "
David Applebaum (University of Sheffield).
| Document Type
|
:
|
BL
|
| Record Number
|
:
|
839683
|
| Main Entry
|
:
|
Applebaum, David,1956-
|
| Title & Author
|
:
|
Semigroups of linear operators : : with applications to analysis, probability and physics /\ David Applebaum (University of Sheffield).
|
| Publication Statement
|
:
|
Cambridge, United Kingdom ;New York, NY :: Cambridge University Press,, 2019.
|
|
|
:
|
, ©2019
|
| Series Statement
|
:
|
London Mathematical Society student texts ;; 93
|
| Page. NO
|
:
|
x, 223 pages ;; 23 cm.
|
| ISBN
|
:
|
1108483097
|
|
|
:
|
: 1108716377
|
|
|
:
|
: 9781108483094
|
|
|
:
|
: 9781108716376
|
|
|
:
|
9781108669658
|
|
|
:
|
9781108672641
|
| Bibliographies/Indexes
|
:
|
Includes bibliographical references and index.
|
| Contents
|
:
|
Cover; Series page; Title page; Copyright page; Dedication; Epigraph; Contents; Introduction; 1 Semigroups and Generators; 1.1 Motivation from Partial Differential Equations; 1.2 Definition of a Semigroup and Examples; 1.3 Unbounded Operators and Generators; 1.3.1 Unbounded Operators and Density of Generators; 1.3.2 Differential Equations in Banach Space; 1.3.3 Generators as Closed Operators; 1.3.4 Closures and Cores; 1.4 Norm-Continuous Semigroups; 1.5 The Resolvent of a Semigroup; 1.5.1 The Resolvent of a Closed Operator; 1.5.2 Properties of the Resolvent of a Semigroup
|
|
|
:
|
1.6 Exercises for Chapter 12 The Generation of Semigroups; 2.1 Yosida Approximants; 2.2 Classifying Generators; 2.3 Applications to Parabolic PDEs; 2.3.1 Bilinear Forms, Weak Solutions and the Lax-Milgram Theorem; 2.3.2 Energy Estimates and Weak Solutions to the Elliptic Problem; 2.3.3 Semigroup Solution of the Parabolic Problem; 2.4 Exercises for Chapter 2; 3 Convolution Semigroups of Measures; 3.1 Heat Kernels, Poisson Kernels, Processes and Fourier Transforms; 3.1.1 The Gauss-Weierstrass Function and the Heat Equation; 3.1.2 Brownian Motion and Itô's Formula
|
|
|
:
|
3.1.3 The Cauchy Distribution, the Poisson Kernel and Laplace's Equation3.2 Convolution of Measures and Weak Convergence; 3.2.1 Convolution of Measures; 3.2.2 Weak Convergence; 3.3 Convolution Semigroups of Probability Measures; 3.4 The Lévy-Khintchine Formula; 3.4.1 Stable Semigroups; 3.4.2 Lévy Processes; 3.5 Generators of Convolution Semigroups; 3.5.1 Lévy Generators as Pseudo-Differential Operators; 3.6 Extension to L[sup(p)]; 3.7 Exercises for Chapter 3; 4 Self-Adjoint Semigroups and Unitary Groups; 4.1 Adjoint Semigroups and Self-Adjointness; 4.1.1 Positive Self-Adjoint Operators
|
|
|
:
|
4.1.2 Adjoints of Semigroups on Banach Spaces4.2 Self-Adjointness and Convolution Semigroups; 4.3 Unitary Groups, Stone's Theorem; 4.4 Quantum Dynamical Semigroups; 4.5 Exercises for Chapter 4; 5 Compact and Trace Class Semigroups; 5.1 Compact Semigroups; 5.2 Trace Class Semigroups; 5.2.1 Hilbert-Schmidt and Trace Class Operators; 5.2.2 Trace Class Semigroups; 5.2.3 Convolution Semigroups on the Circle; 5.2.4 Quantum Theory Revisited; 5.3 Exercises for Chapter 5; 6 Perturbation Theory; 6.1 Relatively Bounded and Bounded Perturbations; 6.1.1 Contraction Semigroups; 6.1.2 Analytic Semigroups
|
|
|
:
|
6.2 The Lie-Kato-Trotter Product Formula6.3 The Feynman-Kac Formula; 6.3.1 The Feynman-Kac Formula via the Lie-Kato-Trotter Product Formula; 6.3.2 The Feynman-Kac Formula via Itô's Formula; 6.4 Exercises for Chapter 6; 7 Markov and Feller Semigroups; 7.1 Definitions of Markov and Feller Semigroups; 7.2 The Positive Maximum Principle; 7.2.1 The Positive Maximum Principle and the Hille-Yosida-Ray Theorem; 7.2.2 Crash Course on Distributions; 7.2.3 The Courrège Theorem; 7.3 The Martingale Problem; 7.3.1 Sub-Feller Semigroups; 8 Semigroups and Dynamics; 8.1 Invariant Measures and Entropy
|
| Abstract
|
:
|
"The theory of semigroups of operators is one of the most important themes in modern analysis. Not only does it have great intellectual beauty, but also wide-ranging applications. In this book the author first presents the essential elements of the theory, introducing the notions of semigroup, generator and resolvent, and establishes the key theorems of Hille-Yosida and Lumer-Phillips that give conditions for a linear operator to generate a semigroup. He then presents a mixture of applications and further developments of the theory. This includes a description of how semigroups are used to solve parabolic partial differential equations, applications to Lévy and Feller-Markov processes, Koopmanism in relation to dynamical systems, quantum dynamical semigroups, and applications to generalisations of the Riemann-Liouville fractional integral. Along the way the reader encounters several important ideas in modern analysis including Sobolev spaces, pseudo-differential operators and the Nash inequality."
|
| Subject
|
:
|
Group theory, Textbooks.
|
| Subject
|
:
|
Linear operators, Textbooks.
|
| Subject
|
:
|
Operator theory, Textbooks.
|
| Subject
|
:
|
Semigroups, Textbooks.
|
| Subject
|
:
|
Group theory.
|
| Subject
|
:
|
Linear operators.
|
| Subject
|
:
|
Operator theory.
|
| Subject
|
:
|
Semigroups.
|
| Dewey Classification
|
:
|
515/.7246
|
| LC Classification
|
:
|
QA329.2.A687 2019
|
| |