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Ⅰ．Linear Dynamical Systems 1

1．Cauchy's Functional Equation 2

2．Finite-Dimensional Systems：Matrix Semigroups 6

3．Uniformly Continuous Operator Semigroups 14

4．More Semigroups 24

a．Multiplication Semigroups on Co(Ω) 24

b．Multiplication Semigroups on Lp(Ω,μ) 30

c．Translation Semigroups 33

5．Strongly Continuous Semigroups 36

a．Basic Properties 37

b．Standard Constructions 42

Notes 46

Ⅱ．ISemigroups,Generators,and Resolvents 47

1．Generators of Semigroups and Their Resolvents 48

2．Examples Revisited 59

a．Standard Constructions 59

b．Standard Examples 65

3．Hille-Yosida Generation Theorems 70

a．Generation of Groups and Semigroups 71

b．Dissipative Operators and Contraction Semigroups 82

c．More Examples 89

4．Special Classes of Semigroups 96

a．Analytic Semigroups 96

b．Differentiable Semigroups 109

c．Eventually Norm-Continuous Semigroups 112

d．Eventually Compact Semigroups 117

e．Examples 120

5．Interpolation and Extrapolation Spaces for Semigroups 123

Simon Brendle 124

a．Sobolev Towers 124

b．Favard and Abstract H？lder Spaces 129

c．Fractional Powers 137

6．Well-Posedness for Evolution Equations 145

Notes 154

Ⅲ．Perturbation and Approximation of Semigroups 157

1．Bounded Perturbations 157

2．Perturbations of Contractive and Analytic Semigroups 169

3．More Perturbations 182

a．The Perturbation Theorem of Desch-Schappacher 182

b．Comparison of Semigroups 192

c．The Perturbation Theorem of Miyadera-Voigt 195

d．Additive Versus Multiplicative Perturbations 201

4．Trotter-Kato Approximation Theorems 205

a．A Technical Tool：Pseudoresolvents 206

b．The Approximation Theorems 209

c．Examples 214

5．Approximation Formulas 219

a．Chernoff Product Formula 219

b．Inversion Formulas 231

Notes 236

Ⅳ．Spectral Theory for Semigroups and Generators 238

1．Spectral Theory for Closed Operators 239

2．Spectrum of Semigroups and Generators 250

a．Basic Theory 250

b．Spectrum of Induced Semigroups 259

c．Spectrum of Periodic Semigroups 266

3．Spectral Mapping Theorems 270

a．Examples and Counterexamples 270

b．Spectral Mapping Theorems for Semigroups 275

c．Weak Spectral Mapping Theorem for Bounded Groups 283

4．Spectral Theory and Perturbation 289

Notes 293

Ⅴ．Asymptotics of Semigroups 295

1．Stability and Hyperbolicity for Semigroups 296

a．Stability Concepts 296

b．Characterization of Uniform Exponential Stability 299

c．Hyperbolic Decompositions 305

2．Compact Semigroups 308

a．General Semigroups 308

b．Weakly Compact Semigroups 312

c．Strongly Compact Semigroups 317

3．Eventually Compact and Quasi-compact Semigroups 329

4．Mean Ergodic Semigroups 337

Notes 345

Ⅵ．Semigroups Everywhere 347

1．Semigroups for Population Equations 348

a．Semigroup Method for the Cell Equation 349

b．Intermezzo on Positive Semigroups 353

c．Asymptotics for the Cell Equation 358

Notes 361

2．Semigroups for the Transport Equation 361

a．Solution Semigroup for the Reactor Problem 361

b．Spectral and Asymptotic Behavior 364

Notes 367

3．Semigroups for Second-Order Cauchy Problems 367

a．The State Space x=XB 1×X 369

b．The State Space x=X×X 372

c．The State Space x=XC 1×X 374

Notes 382

4．Semigroups for Ordinary Differential Operators 383

M.Campiti,G.Metafune,D.Pallara,and S.Romanelli 384

a．Nondegenerate Operators on R and R+ 384

b．Nondegenerate Operators on Bounded Intervals 388

c．Degenerate Operators 390

d．Analyticity of Degenerate Semigroups 400

Notes 403

5．Semigroups for Partial Differential Operators 404

Abdelaziz Rhandi 405

a．Notation and Preliminary Results 405

b．Elliptic Differential Operators with Constant Coefficients 408

c．Elliptic Differential Operators with Variable Coefficients 411

Notes 419

6．Semigroups for Delay Differential Equations 419

a．Well-Posedness of Abstract Delay Differential Equations 420

b．Regularity and Asymptotics 424

c．Positivity for Delay Differential Equations 428

Notes 435

7．Semigroups for Volterra Equations 435

a．Mlid and Classical Solutions 436

b．Optimal Regularity 442

c．Integro-Differential Equations 447

Notes 452

8．Semigroups for Control Theory 452

a．Controllability 456

b．Observability 466

c．Stabilizability and Detectability 468

d．Transfer Functions and Stability 473

Notes 476

9．Semigroups for Nonautonomous Cauchy Problems 477

Roland Schnaubelt 477

a．Cauchy Problems and Evolution Families 477

b．Evolution Semigroups 481

c．Perturbation Theory 487

d．Hyperbolic Evolution Families in the Parabolic Case 492

Notes 496

Ⅶ．A Brief History of the Exponential Function 497

Tanja Hahn and Carla Perazzoli 497

1．A Bird's-Eye View 497

2．The Functional Equation 500

3．The Differential Equation 502

4．The Birth ofSemigroup Theory 506

Appendix 509

A．A Reminder of Some Functional Analysis 509

B．A Reminder of Some Operator Theory 515

C．Vector-Valued Integration 522

a．The Bochner Integral 522

b．The Fourier Transform 526

c．The Laplace Transform 530

Epilogue 531

Determinism：Scenes from the Interplay Between Metaphysics and Mathematics 531

Gregor Nickel 533

1．The Mathematical Structure 533

2．Are Relativity,Quantum Mechanics,and Chaos Deterministic? 536

3．Determinism in Mathematical Science from Newton to Einstein 538

4．Developments in the Concept of Object from Leibniz to Kant 546

5．Back to Some Roots of Our Problem：Motion in History 549

6．Bibliography and Further Reading 553

References 555

List of Symbols and Abbreviations 577

Index 580