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Topics in Operator Semigroups

  • Book
  • © 2010

Overview

Authors:
  1. Shmuel Kantorovitz

    1. Dept. Mathematics & Computer Science, Bar-Ilan University, Ramat Gan, Israel

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  • 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
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (12 chapters)

  1. Front Matter

    Pages 1-11
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  2. General Theory

    1. Front Matter

      Pages 1-1
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    2. Basic Theory

      • Shmuel Kantorovitz
      Pages 3-48

    3. The Semi-Simplicity Space for Groups

      • Shmuel Kantorovitz
      Pages 49-61

    4. Analyticity

      • Shmuel Kantorovitz
      Pages 63-69

    5. The Semigroup as a Function of its Generator

      • Shmuel Kantorovitz
      Pages 71-86

    6. Large Parameter

      • Shmuel Kantorovitz
      Pages 87-112

    7. Boundary Values

      • Shmuel Kantorovitz
      Pages 113-130

    8. Pre-Semigroups

      • Shmuel Kantorovitz
      Pages 131-138

  3. Integral Representations

    1. Front Matter

      Pages 140-140
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    2. The Semi-Simplicity Space

      • Shmuel Kantorovitz
      Pages 141-160

    3. The Laplace–Stieltjes Space

      • Shmuel Kantorovitz
      Pages 161-175

    4. Families of Unbounded Symmetric Operators

      • Shmuel Kantorovitz
      Pages 177-190

  4. A Taste of Applications

    1. Front Matter

      Pages 194-195
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    2. Analytic Families of Evolution Systems

      • Shmuel Kantorovitz
      Pages 195-201

    3. Similarity

      • Shmuel Kantorovitz
      Pages 203-218

  5. Back Matter

    Pages 1-47
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Keywords

About this book

This book is based on lecture notes from a second-year graduate course, and is a greatly expanded version of our previous monograph [K8]. We expose some aspects of the theory of semigroups of linear operators, mostly (but not only) from the point of view of its meeting with that part of spectral theory which is concerned with the integral representation of families of operators. This approach and selection of topics di?erentiate this book from others in the general area, and re?ect the author’s own research directions. There is no attempt therefore to cover thoroughly the theory of semigroups of operators. This theory and its applications are extensively exposed in many books, from theclassicHille–Phillipsmonograph[HP]tothemostrecenttextbookofEngel and Nagel [EN2] (see [A], [BB], [Cl], [D3], [EN1], [EN2], [Fat], [G], [HP], [P], [Vr], and others), as well as in chapters in more general texts on Functional Analysis and the theory of linear operators (cf. [D5], [DS I–III], [Kat1], [RS], [Y], and many others).

Reviews

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)

Authors and Affiliations

  • Dept. Mathematics & Computer Science, Bar-Ilan University, Ramat Gan, Israel

    Shmuel Kantorovitz

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