Maximal dominated operator semigroups

Maximal operator semigroups, bounded in a certain sense, on real or complex vector spaces are studied. For any maximal semigroup \MM dominated by a certain pair of homogeneous functions there is an operator quasinorm for which \MM is exactly the semigroup of contractions in this quasinorm. Applications to Riesz spaces are given. In particular, maximal semigroups of matrices dominated by a given positive matrix are characterized. We thus answer the question implicitly posed in 2]. November 20, 1998