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One-Parameter Groups and Semigroups of Operators

  • Konrad Schmüdgen
Chapter
  • 4.4k Downloads
Part of the Graduate Texts in Mathematics book series (GTM, volume 265)

Abstract

Chapter 6 gives a concise introduction into the theory of one-parameter groups or semigroups of operators with an emphasis on the interplay between groups and semigroups and their generators. In the first section, one-parameter unitary groups are investigated, and two fundamental theorems, Stone’s theorem and Trotter’s formula, are proved. Semigroups of operators are applied to Cauchy problems for abstract differential equations on Hilbert space. Then generators of semigroups of contractions on Banach spaces are studied, and the Hille–Yosida theorem is proved. Finally, generators of contraction semigroups on Hilbert space are characterized as m-dissipative operators.

Keywords

Hilbert Space Banach Space Cauchy Problem Unitary Group Contraction Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

Classical Articles

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    Hille, E.: Functional Analysis and Semigroups. Am. Math. Soc. Coll. Publ., vol. 38. Am. Math. Soc., New York (1948) Google Scholar
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    Yosida, K.: On the differentiability and the representation of one-parameter semigroups of linear operators. J. Math. Soc. Jpn. 1, 15–21 (1949) CrossRefGoogle Scholar

Books

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    Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966) CrossRefzbMATHGoogle Scholar
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    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, New York (1972) Google Scholar
  3. [RS2]
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis and Self-Adjointness. Academic Press, New York (1975) Google Scholar
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    Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser-Verlag, Basel (1990) CrossRefGoogle Scholar
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    Warner, G.: Harmonic Analysis on Semi-Simple Lie Groups. Springer-Verlag, Berlin (1972) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Konrad Schmüdgen
    • 1
  1. 1.Dept. of MathematicsUniversity of LeipzigLeipzigGermany

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