Continuous Semigroups in Banach Algebras

In these notes the abstract theory of analytic one-parameter semigroups in Banach algebras is discussed.

Allan M. Sinclair (Author)

9780521285988, Cambridge University Press

Paperback, published 17 June 1982

152 pages
22.8 x 15.2 x 0.9 cm, 0.25 kg

In these notes the abstract theory of analytic one-parameter semigroups in Banach algebras is discussed, with the Gaussian, Poisson and fractional integral semigroups in convolution Banach algebras serving as motivating examples. Such semigroups are constructed in a Banach algebra with a bounded approximate identity. Growth restrictions on the semigroup are linked to the structure of the underlying Banach algebra. The Hille-Yosida Theorem and a result of J. Esterle's on the nilpotency of semigroups are proved in detail. The lecture notes are an expanded version of lectures given by the author at the University of Edinburgh in 1980 and can be used as a text for a graduate course in functional analysis.

1. Introduction and preliminaries
2. Analytic semigroups in particular Banach algebras
3. Existence of analytic semigroups - an extension of Cohen's factorization method
4. Proof of the existence of analytic semigroups
5. Restrictions on the growth of at
6. Nilpotent semigroups and proper closed ideals
Appendix 1. The Ahlfors-Heins theorem
Appendix 2. Allan's theorem - closed ideals in L1( R+,w)
Appendix 3. Quasicentral bounded approximate identities
References
Index.

Subject Areas: Algebra [PBF]