Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

THE HILLE–YOSIDA THEOREM FOR LOCAL CONVOLUTED SEMIGROUPS

Valentin Keyantuoa1, Claus Müllera2 and Peter Vietena2

a1 Department of Mathematics, University of Puerto Rico, Rio Piedras 00931, Puerto Rico (keyantuo@upracd.upr.clu.edu)

a2 Fachbereich Mathematik der Universität Kaiserslautern, Erwin-Schrödinger Strasse, 67663 Kaiserslautern, Germany (claus_mueller@mathematik.uni-kl.de)

Abstract

The characterization theorem for the Banach-space-valued local Laplace transform established by Keyantuo, Müller and Vieten is used to obtain a real variable characterization of generators of local convoluted semigroups. The concept of local convoluted semigroups extends that of distribution as well as ultradistribution semigroups. Complete characterizations existed only for exponentially bounded semigroups integrated $\alpha$ times, whereas for the non-exponential case generation results had been obtained in terms of complex conditions only.

AMS 2000 Mathematics subject classification: Primary 47D03; 47D06; 44A10

(Received March 16 2000)

Keywords

  • local Laplace transform;
  • local convoluted semigroup;
  • integrated semigroup;
  • distribution semigroup;
  • ultradistribution semigroup.