Title Semigroups of linear operators : with applications to analysis, probability and physics / David Applebaum

Published Cambridge : Cambridge University Press, 2019

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Description 1 online resource (x, 223 pages) Series London Mathematical Society student texts ; 93 London Mathematical Society student texts ; 93. 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 Summary 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 Levy 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 Notes Vendor-supplied metadata Subject Linear operators -- Textbooks Operator theory -- Textbooks Semigroups -- Textbooks Group theory -- Textbooks Teoría de grupos Teoría de operadores Group theory Linear operators Operator theory Semigroups Genre/Form Textbooks Form Electronic book ISBN 9781108672641 1108672647

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