Topics in operator semigroups / Shmuel Kantorovitz.

Format:

Book

Author/Creator:

Kantorovitz, Shmuel, 1935-

Series:

Progress in mathematics (Boston, Mass.) ; v. 281.

Progress in mathematics ; 281

Language:

English

Subjects (All):

Semigroups of operators.

Spectral theory (Mathematics).

Physical Description:

xiii, 266 pages ; 25 cm.

Place of Publication:

Boston, Mass. : Birkhäuser ; London : Springer [distributor], [2010]

Summary:

The theory of operator semigroups was essentially discovered in the early 1930s. Since then, the theory has developed into a rich and exciting area of functional analysis and has been applied to various mathematical topics such as Markov processes, the abstract Cauchy problem, evolution equations, and mathematical physics.

This self-contained monograph focuses primarily on the theoretical connection between the theory of operator semigroups and spectral theory. Divided into three parts with a total of twelve distinct chapters, this book gives an in-depth account of the subject with numerous examples, detailed proofs, and a brief look at a few applications.

Topics include:

The Hille-Yosida and Lumer-Phillips characterizations of semigroup generators

The Trotter-Kato approximation theorem

Kato's unified treatment of the exponential formula and the Trotter product formula

The Hille-Phillips perturbation theorem, and Stone's representation of unitary semigroups

Generalizations of spectral theory's connection to operator semigroups

A natural generalization of Stone's spectral integral representation to a Banach space setting

With a collection of miscellaneous exercises at the end of the book and an introductory chapter examining the basic theory involved, this monograph is suitable for second-year graduate students interested in operator semigroups.

Contents:

Part I General Theory

A Basic Theory 3

A.1 Overview 3

A.2 The Generator 5

A.3 Type and Spectrum 9

A.4 Uniform Continuity 10

A.5 Core for the Generator 11

A.6 The Resolvent 13

A.7 Pseudo-Resolvents 15

A.8 The Laplace Transform 17

A.9 Abstract Potentials 18

A.1O The Hille-Yosida Theorem 20

A.11 The Hille-Yosida Space 22

A.12 Dissipative Operators 25

A.13 The Trotter-Kato Convergence Theorem 28

A.14 Exponential Formulas 32

A.15 Perturbation of Generators 36

A.16 Groups of Operators 42

A.17 Bounded Groups of Operators 43

A.18 Stone's Theorem 44

A.19 Bochner's Theorem 47

B The Semi-Simplicity Space for Groups 49

B.1 The Bochner Norm 49

B.2 The Semi-Simplicity Space 53

B.3 Scalar-Type Spectral Operators 59

C Analyticity 63

C.1 Analytic Semigroups 63

C.2 The Generator of an Analytic Semigroup 65

D The Semigroup as a Function of its Generator 71

D.1 Noncommutative Taylor Formula 71

D.2 Analytic Families of Semigroups 79

E Large Parameter 87

E.1 Analytic Semigroups 87

E.2 Resolvent Iterates 90

E.3 Mean Stability 94

E.4 The Asymptotic Space 103

E.5 Semigroups of Isometries 107

E.6 The ABLV Stability Theorem 109

F Boundary Values 113

F.1 Regular Semigroups and Boundary Values 113

F.2 The Generator of a Regular Semigroup 118

F.3 Examples of Regular Semigroups 121

G Pre-Semigroups 131

G.1 The Abstract Cauchy Problem 132

G.2 The Exponentially Tamed Case 136

Part II Integral Representations

A The Semi-Simplicity Space 141

A.1 The Real Spectrum Case 141

A.2 The Case R + ρ(-A) 154

B The Laplace-Stieltjes Space 161

B.1 The Laplace-Stieltjes Space 161

B.2 Semigroups of Closed Operators 166

B.3 The Integrated Laplace Space -169

B.4 Integrated Semigroups 173

C Families of Unbounded Symmetric Operators 177

C.1 Local Symmetric Semigroups 177

C.2 Nelson's Analytic Vectors Theorem 181

C.3 Local Bounded Below Cosine Families 183

C.4 Local Symmetric Cosine Families 187

Part III A Taste of Applications

A Analytic Families of Evolution Systems 195

A.1 Coefficients Analyticity and Solutions Analyticity 195

A.2 Kato's Conditions 196

A.3 Tanabe's Conditions 198

B Similarity 203

B.1 Overview 203

B.2 Similarity Within the Family S + ὲV 203

B.3 Similarity of Certain Perturbations 217.

Notes:

Includes bibliographical references (pages [253]-261) and index.

ISBN:

9780817649319

081764931X

OCLC:

436264806