Unbounded Self-adjoint Operators on Hilbert Space pp 117-135 | Cite as
One-Parameter Groups and Semigroups of Operators
- 4.4k Downloads
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 SemigroupReferences
Classical Articles
- [Hl]Hille, E.: Functional Analysis and Semigroups. Am. Math. Soc. Coll. Publ., vol. 38. Am. Math. Soc., New York (1948) Google Scholar
- [Yo]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
- [K2]Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966) CrossRefzbMATHGoogle Scholar
- [RS1]Reed, M., Simon, B.: Methods of Modern Mathematical Physics I. Functional Analysis. Academic Press, New York (1972) Google Scholar
- [RS2]Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis and Self-Adjointness. Academic Press, New York (1975) Google Scholar
- [Sch1]Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. Birkhäuser-Verlag, Basel (1990) CrossRefGoogle Scholar
- [Wa]Warner, G.: Harmonic Analysis on Semi-Simple Lie Groups. Springer-Verlag, Berlin (1972) CrossRefGoogle Scholar