##
**One-parameter semigroups for linear evolution equations.**
*(English)*
Zbl 0952.47036

Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000).

In this book the authors (together with 10 coauthors all connected to the Tübinger Functional Analysis Group) give an excellent presentation of the theory of semigroups with applications to various evolution problems. The reader (student or lecturer) can use the text as an introduction to this field following the short cuts indicated by the authors. One can also use it to understand a special application presented in Chapter VI by following back the text. At the same time this book contains many new and very recent results of the theory and its application, thus it is also of great value for experts.

In Chapter I the authors start with a discussion of Cauchy’s functional equation, pass to finite dimensional linear systems and to uniformly continuous semigroups. After a careful discussion of multiplication and translation semigroups they end up with the definition of a strongly continuous semigroup of operators.

Chapter II contains the fundamental results of generators and generation. Semigroups with extra regularity properties are considered extensively. Abstract Sobolev, Favard and Hölder spaces and the fractional powers of generators are defined. These concepts play an important role in the following chapters. In the last section of this chapter Sobolev spaces are used, to discuss uniqueness and well-posedness for evolution equations.

Perturbation and approximation are the topics of Chapter III. First classical results for bounded and relatively bounded perturbations are presented, later the authors give a detailed presentation of the Desch-Schappacher and the Miyadera-Voigt perturbation theorems. The chapter closes with the Trotter-Kato approximation and approximation formulas. Very recent variants and applications are included.

Chapter IV contains the spectral theory for semigroups and its generators. The central topic is the spectral mapping theorem for various types of spectra and its weak versions.

Chapter V treats asymptotic of semigroups, using extensively the results of Chapter IV. The authors give an up to date discussion of different stability concepts. They apply abstract harmonic analysis to analyse stability of weakly and strongly compact semigroups. A short paragraph on ergodic theory concludes this chapter.

“Semigroups Everywhere” is the title of Chapter VI. On 150 pages the authors demonstrate for 9 examples of different evolution problems how semigroups are involved and that they yield a flexible and unifying tool for the study of such problems. This chapter is really a highlight of the book.

The list of references is a detailed survey on the publications on semigroups; it contains many recent (or not yet published) articles. Notes at the end of each chapter contain many hints to additional material. Each section ends with a collection of (not always simple) exercises.

There is another aspect of this book worth to mention. The book starts with a long excerpt from “The Man Without Qualities” by Robert Musil. It ends with a survey of the history of the exponential function and an epilogue on the relationship between evolution problems and determinism. In a convincing way the reader’s attention is invited to the fact that mathematical concepts and theories are always woven into many parts of human culture.

In summary: The book turned out very well.

In Chapter I the authors start with a discussion of Cauchy’s functional equation, pass to finite dimensional linear systems and to uniformly continuous semigroups. After a careful discussion of multiplication and translation semigroups they end up with the definition of a strongly continuous semigroup of operators.

Chapter II contains the fundamental results of generators and generation. Semigroups with extra regularity properties are considered extensively. Abstract Sobolev, Favard and Hölder spaces and the fractional powers of generators are defined. These concepts play an important role in the following chapters. In the last section of this chapter Sobolev spaces are used, to discuss uniqueness and well-posedness for evolution equations.

Perturbation and approximation are the topics of Chapter III. First classical results for bounded and relatively bounded perturbations are presented, later the authors give a detailed presentation of the Desch-Schappacher and the Miyadera-Voigt perturbation theorems. The chapter closes with the Trotter-Kato approximation and approximation formulas. Very recent variants and applications are included.

Chapter IV contains the spectral theory for semigroups and its generators. The central topic is the spectral mapping theorem for various types of spectra and its weak versions.

Chapter V treats asymptotic of semigroups, using extensively the results of Chapter IV. The authors give an up to date discussion of different stability concepts. They apply abstract harmonic analysis to analyse stability of weakly and strongly compact semigroups. A short paragraph on ergodic theory concludes this chapter.

“Semigroups Everywhere” is the title of Chapter VI. On 150 pages the authors demonstrate for 9 examples of different evolution problems how semigroups are involved and that they yield a flexible and unifying tool for the study of such problems. This chapter is really a highlight of the book.

The list of references is a detailed survey on the publications on semigroups; it contains many recent (or not yet published) articles. Notes at the end of each chapter contain many hints to additional material. Each section ends with a collection of (not always simple) exercises.

There is another aspect of this book worth to mention. The book starts with a long excerpt from “The Man Without Qualities” by Robert Musil. It ends with a survey of the history of the exponential function and an epilogue on the relationship between evolution problems and determinism. In a convincing way the reader’s attention is invited to the fact that mathematical concepts and theories are always woven into many parts of human culture.

In summary: The book turned out very well.

Reviewer: Karl-Heinz Förster (Berlin)

### MathOverflow Questions:

A question about uniformly bounded semigroupsapproximation of a Feller semi-group with the infinitesimal generator

Convergence of \(\exp(tQ)\) in operator norm as \(t\rightarrow\infty\)

Generation of strict contraction semigroups

Laplacian dissipative?

Left and right eigenvectors are not orthogonal

Lumer-Phillips-type theorem for non-autonomous evolutions

Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard

Pointwise convergence in functional calculus

Strongly continuous semigroups that cannot be contractions

The Bochner integral about a semigroup of bounded linear operators on a Banach space

### MSC:

47D06 | One-parameter semigroups and linear evolution equations |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

34K30 | Functional-differential equations in abstract spaces |

34K35 | Control problems for functional-differential equations |

35K30 | Initial value problems for higher-order parabolic equations |