IJPAM: Volume 22, No. 1 (2005)

ON REDUCIBILITY OF ADJOINT SEMIGROUPS OF
POSITIVE OPERATORS ON A NORMED RIESZ SPACE

Ömer Gök
Department of Mathematics
Faculty of Arts and Sciences
Yildiz Technical University
Davutpasa Campus
Istanbul, 34210, TURKEY
e-mail: gok@yildiz.edu.tr


Abstract.We prove several results on the existence of invariant closed ideals for adjoint semigroups of positive operators on a normed Riesz space (of dimension greater than 1). If $S$ is a multiplicative semigroup of positive operators on a such space that are quasinilpotent at an atom, then adjoint semigroup $S^{\odot}$ has a non-trivial invariant closed ideal. Furthermore, if $T$ is a non-zero positive operator such that adjoint of $T$, $T'$, is quasinilpotent at an atom and if $S$ is a multiplicative semigroup of positive operators such that $RT \leq TR$ for all $R \in S$, then adjoint semigroup $S^{\odot}$ and adjoint $T'$ have a common non-trivial invariant closed ideal.

Received: May 5, 2005

AMS Subject Classification: 47B65, 47A15, 46B42

Key Words and Phrases: normed Riesz spaces, positive operators, invariant ideals, adjoint semigroups

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 22
Issue: 1