© 2010

Topics in Operator Semigroups


  • Concerned with the interplay between the theory of operator semigroups and spectral theory

  • The basics on operator semigroups are concisely covered in this self-contained text

  • Part I deals with the Hille--Yosida and Lumer--Phillips characterizations of semigroup generators, the Trotter--Kato approximation theorem, Kato’s unified treatment of the exponential formula and the Trotter product formula, the Hille--Phillips perturbation theorem, and Stone’s representation of unitary semigroups

  • Part II explores generalizations of spectral theory’s connection to operator semigroups

  • Suitable for a graduate seminar on operator semigroups or for self study


Part of the Progress in Mathematics book series (PM, volume 281)

Table of contents

  1. Front Matter
    Pages 1-11
  2. General Theory

    1. Front Matter
      Pages 1-1
    2. Shmuel Kantorovitz
      Pages 3-48
    3. Shmuel Kantorovitz
      Pages 49-61
    4. Shmuel Kantorovitz
      Pages 63-69
    5. Shmuel Kantorovitz
      Pages 71-86
    6. Shmuel Kantorovitz
      Pages 87-112
    7. Shmuel Kantorovitz
      Pages 113-130
    8. Shmuel Kantorovitz
      Pages 131-138
  3. Integral Representations

    1. Front Matter
      Pages 140-140
    2. Shmuel Kantorovitz
      Pages 141-160
    3. Shmuel Kantorovitz
      Pages 161-175
    4. Shmuel Kantorovitz
      Pages 177-190
  4. A Taste of Applications

    1. Front Matter
      Pages 194-195
    2. Shmuel Kantorovitz
      Pages 195-201
    3. Shmuel Kantorovitz
      Pages 203-218
  5. Back Matter
    Pages 1-47

About this book


The theory of operator semigroups was essentially discovered in the early 1930s.  Since then, the theory has developed into a rich and exciting area of functional analysis and has been applied to various mathematical topics such as Markov processes, the abstract Cauchy problem, evolution equations, and mathematical physics.

This self-contained monograph focuses primarily on the theoretical connection between the theory of operator semigroups and spectral theory. Divided into three parts with a total of twelve distinct chapters, this book gives an in-depth account of the subject with numerous examples, detailed proofs, and a brief look at a few applications.

Topics include:

* The Hille–Yosida and Lumer–Phillips characterizations of semigroup generators

* The Trotter–Kato approximation theorem

* Kato’s unified treatment of the exponential formula and the Trotter product formula

* The Hille–Phillips perturbation theorem, and Stone’s representation of unitary semigroups

*  Generalizations of spectral theory’s connection to operator semigroups

* A natural generalization of Stone’s spectral integral representation to a Banach space setting

With a collection of miscellaneous exercises at the end of the book and an introductory chapter examining the basic theory involved, this monograph is suitable for second-year graduate students interested in operator semigroups.


Nussbaum’s theorem Stone’s representation Trotter product formula Trotter--Kato approximation theorem functional analysis operator semigroups spectral theory unitary semigroups

Authors and affiliations

  1. 1.Dept. Mathematics & Computer ScienceBar-Ilan UniversityRamat GanIsrael

Bibliographic information


From the book reviews:

“This monograph is suitable for second-year graduate students, but it can be recommended also to any researcher interested in operator semigroups.” (László Kérchy, Acta Scientiarum Mathematicarum (Szeged), Vol. 78 (1-2), 2012)

“The present graduate level text expands the previous lecture notes from the same author, Semigroups of operators and spectral theory … . It begins with a succinct introduction to operator semigroups covering classical topics such as generators, the Hille-Yosida theorem, dissipative operators and the Lumer–Phillips theorem, the Trotter convergence theorem, exponential formulas, perturbation theory, Stone’s theorem, and analytic semigroups. … The text is also intended for second-year graduate students … . it will be a valuable source for researchers working in this area.” (G. Teschl, Monatshefte für Mathematik, Vol. 162 (4), April, 2011)

“This book is based on the author’s lecture notes … in which the more advanced parts concentrated on spectral representations. … There is also a presentation of a well-known stability theorem for semigroups under countable spectral conditions. … The increased variety of topics covered will make the book more useful … . Other advantages are the inclusion of an index and some exercises, considerable extensions of the bibliography and the list of contents, and more attractive typesetting.”­­­ (C. J. K. Batty, Mathematical Reviews, Issue 2010 k)