This book is aimed at advanced undergraduates or beginning graduate students, who have already had exposure to basic functional analysis and measure theory. The book is different from others on the classical theory of semigroups. In particular, there is more of a focus (as the title suggests) on applications. The probability and stochastic analysis applications are quite nice, and not usual for a basic semigroup theory

book.

The first two chapters, perhaps out of necessity, are devoted to the main thrust of the theory of semigroups. This includes the usual classical results, including the resolvent of a semigroup; Yosida approximants; and applications to parabolic PDEs via the Lax-Milgram Theorem.

In the third chapter, there is already deviation from the “usual” treatment of semigroups. Indeed, in the first section, Brownian motion and Itô’s formula are introduced in the context of the heat equation. Later in this chapter the Lévy-Khintchine formula is discussed. This is a useful result on the characterization of convolution semigroups of measures via their characteristic functions. This leads nicely into a brief analysis of stable semigroups, with applications via so-called \( \alpha \)-stable laws, in theoretical and applied probability. This chapter ends with a nice introduction to pseudo-differential operators.

Chapter 4 returns to more standard semigroup results related to self-adjoint operators. The last section in this chapter is a nice application to quantum dynamical semigroups. The author then in Chapter 5 continues with compact semigroups with a revisit to the previous quantum problems.

Chapter 6 provides a short introduction to perturbation theory with two proofs of one large result: the Feynman-Kac formula. The stochastic approach via Itô’s formula is particularly interesting, and is quite consistent with the author’s affinity for probabilistic applications.

A large jump in generality is given in Chapter 7, where the context is now Markov and Feller semigroups. A full proof of the Courrège Theorem is given. This big result provides the characteristic form of operators which obey the positive maximum theorem. Chapter 8 then returns to a more in-depth treatment of the relationship between semigroups and dynamical systems. Finally, Chapter 9 explores so-called Varopolous semigroups (related to ultracontractive semigroups). It is shown that the Riesz potential operators which occur as Mellin transforms of such semigroups satisfy the Hardy-Littlewood-Sobolev inequality.

Overall, this book is an interesting contribution to the semigroup literature which does not follow a standard route. The focus on applications, with references to full proofs when not given in the text, would serve as a nice introduction to students on this vast subject. In particular, this would allow students to not become overly encumbered by technical details, and be able to focus on interesting applications and significant related results.

Eric Stachura is currently an Assistant Professor of Mathematics at Kennesaw State University. He is generally interested in analysis and partial differential equations.