One-Parameter Semigroups for Linear Evolution Equations (Graduate Texts in Mathematics): Klaus-Jochen Engel, Rainer Nagel

Description

This book gives an up-to-date account of the theory of strongly continuous one-parameter semigroups of linear operators. It includes a systematic discussion of the spectral theory and the long-term behavior of such semigroups. A special feature of the text is an unusually wide range of applications, e.g., to ordinary and partial differential operators, delay and Volterra equations and to control theory, and an emphasis on philosophical motivation and the historical background. The book is written for students, but should also be of value for researchers interested in this field.
Summary: Outstanding Title
Rating: 5
I guess that, besides the classic references from Pazy and Jerome Goldstein, this title outstands in its category. I've heard several words of criticism over Pazy's work due to the lack of applications and several laments over the fact that Goldstein's book is out of print. This book introduces the semigroup theory like no other: it doesn't forget the historic and philosophical aspect of subject and it's full filled with applications and deep beauty. Congrats to the authors and also to the customers that decide to buy this title, may it be PDE or Dinamic Systems, you'll be home free...
Summary: Engel et al.'s One Parameter Semigroups Book
Rating: 5
Semigroups and Clifford algebras have become two of the main trends in mathematics and mathematical physics in the last 5 years. A researcher in either area cannot afford to be without major books and journal articles in these areas, and Engel et al.'s is the best that I have seen of recent semigroup books. Their book is a Graduate Text, but it actually presents many open questions and research ideas, as is usual for Springer and Springer-Verlag publications (and also Kluwer-Plenum and Cambridge University Press). Semigroups were familiar to those of us in probability for many years because of their Markov chain/process relationships, but the astonishing fact is that Engel et al. show that non-Markov applications and non-Hilbert space (in fact, general Banach space) applications and theorems abound. For those of us who are not too fond of Markov processes (like me), and who are also not too fond of Hilbert spaces (the usual space of Quantum Mechanics which went sour and had to be repaired by A. Bohm and still seems in bad need of repair), this is really good news. Another fascinating trend is in the direction of Positive Semigroups, which ties in with ordering results and inequalities (see my review of Clarke et al.). Positive semigroups have applications to transport and delay (differential) equations and semigroups have applications to control theory which means what it sounds like: physics-engineering control of satellites, rockets, robots, and God knows what else. I repeat my usual caution: if your background is mathematically deficient in these areas, hire a reputable consultant or tutor to translate the book into plain English more or less. If the person cannot translate in one session, look for another person, or quiz them on the phone first.

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