Positivity 7: 3–22, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Positive Operators on Banach Spaces Ordered by
Strongly Normal Cones
EDUARD YU. EMEL’YANOV and MANFRED P. H. WOLFF
Universität Tübingen, Wilhelmstrasse 7, D-72074 Tübingen, Germany
Mathematics Subject Classiﬁcation 2000: Primary 47B60, Secondary: 47A35, 47C15, 46B40
Key words: positive operators, mean ergodic operators, asymptotic domination
In  we have introduced the class of ideally ordered Banach spaces. This class
includes all Banach lattices with order continuous norm as well as the predual of
any von Neumann algebra. We have shown in  that semigroups of positive
operators on ideally ordered Banach spaces possess speciﬁc asymptotic properties.
In this paper we continue these investigations and apply our technique to positive
semigroups with small attractors thus generalizing the main results of [11, 12],
which were stated there for Markov semigroups only. We do not restrict ourselves
to ideally ordered Banach spaces but study in more details Banach spaces ordered
by a strongly normal cone, which includes all Banach lattices as well as all C
algebras. Many results presented here were known before only for Banach lattices.
For simplicity, we deal with single operators, however all results (except Theor-
ems 14 and 16) are true also for strongly continuous semigroups. Since this is not
obvious in the case of the main theorem in Section 3, we prove it in full generality.
Let us introduce some preliminary deﬁnitions and notations, that will be used
through the paper. Let X be a Banach space over R ordered by a closed, normal, and
generating cone X
. We will investigate the asymptotic behavior of the powers of
a positive operator on X. Given an operator T in L(X), denote the Cezàro averages
of T by
(T ) :=