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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Classical Articles
Hille, E.: Functional Analysis and Semigroups. Am. Math. Soc. Coll. Publ., vol. 38. Am. Math. Soc., New York (1948)
Yosida, K.: On the differentiability and the representation of one-parameter semigroups of linear operators. J. Math. Soc. Jpn. 1, 15–21 (1949)
Books
Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, New York (1972)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis and Self-Adjointness. Academic Press, New York (1975)
Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser-Verlag, Basel (1990)
Warner, G.: Harmonic Analysis on Semi-Simple Lie Groups. Springer-Verlag, Berlin (1972)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Schmüdgen, K. (2012). One-Parameter Groups and Semigroups of Operators. In: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol 265. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4753-1_6
Download citation
DOI: https://doi.org/10.1007/978-94-007-4753-1_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4752-4
Online ISBN: 978-94-007-4753-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)