Throughout this book, semigroups, semiflows, semidynamical systems have the same meaning: that of a one-parameter family of transformations T so that for some space X,
T(0)=I, the identity transformation on X;
T(t)T(s)=T(t+s), for all t, s ≥0;
T(t):X→X, t ≥0.
As the title indicates, this is a book of problems of widely different difficulty, grouped in 24 chapters, followed by a Chapter 25 of “Notes,” where the author includes some comments, bibliographical references, even some results. The book has its origins in the notes the author has prepared and published in Spanish for the XIII Escuela Venezolana de Matemáticas at the Universidad de los Andes in Merida, Venezuela, 6–15 September 2000. The notes have been expanded considerably since, so that the book includes about four times more material. There are even some chapters in the book dealing with subjects which didn’t even exist in 2000.
While some of the problems in the book are not particularly difficult, most of them are not easy, so they can be used more appropriately in a course similar to the one in Venezuela in 2000: the “students” being faculty and graduate students (maybe a few very motivated undergraduates). Solving the problems in the book is a very interesting enterprise, but the most beneficial thing is that these problems lead to very enlightening discussions and research problems. Just one example is one of the last problems included in the book:
Problem 437 Find corresponding complex field results for the (majority) of problems in this book which are stated (usually implicitly) for the real field.
There is also the possibility of using parts of the book in a (graduate) course or in studying some particular topics: for instance students or researchers interested in Sobolev Spaces can pay very close attention to chapter 16, “Semigroups and Families of Sobolev Spaces.”
In summary, this is a very interesting book which can prove useful to graduate students and mathematicians alike. However, some of them might wish for hints for (some of) the problems, and even full solutions sometimes. This being said, the Notes and the extended list of References are great invitations for more study and research.
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.
-Preface.-1. Introduction.-2. The idea of a semigroup.-3. Translation semigroups.-4. Linear continuous semigroups.-5.Strongly continuous linear semigroups.-6. An Application to the Heat Equation.-7. Some Problems in Analysis.-8.Semigroups of steepest descent.-9. Numerics of semigroups of steepest descent.-10. Nonlinear semigroups studied by linear methods.-11. Measures and linear extension of nonlinear semigroups.-12. Local semigroups and Lie generators.-13. Quasi-analyticity of semigroups.-14. Continuous Newton's method and semigroups-15. Generalized semigroups without forward uniqueness.-16. Semigroups of nonlinear contractions and monotone operators.-17. Notes.-18. References.