Positivity 4: 245–251, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Asymptotic Behavior of Positive Operators on
, F. RÄBIGER
and M.P.H. WOLFF
Sobolev Institute of Mathematics at NBovosibirsk, Universitetskii pr. 4, 630090 Novosibirsk,
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen,
In the present paper we present some results dealing with the asymptotic behavior
of positive operators on a Banach lattice. Some of the results are known. The others
were recently obtained by the authors. We will consider mean ergodicity, strong
stability, and almost periodicity of positive operators directly and in connection
with some properties of the Banach lattice on which they act. Several results require
countable order completeness or the existence of a topological orthogonal system
in the Banach lattice which is satisﬁed for all classical Banach lattices. This leads
to some open questions which will be posed in the paper. Our terminology and
notations are standard and follow to books of [5, 7] and . We consider bounded
linear operators on complex Banach spaces and Banach lattices.
1. Let X be a Banach space. An operator T : X → X is called power-bounded if
:n ∈ N} < ∞. An operator T is mean ergodic whenever the sequence
converges in E for every x ∈ E. We recall a well known result
from ergodic theory, , Theorem 1.2.
THEOREM 1. Every power-bounded operator on a reﬂexive Banach space is
It is open problem whether the converse of this theorem holds. More precisely, if
every power-bounded operator on a Banach space X is mean ergodic, is X reﬂex-
ive? For Banach lattices, this problem has an afﬁrmative answer (see [14, 8, 2]). On
the other hand on Banach lattices it is natural to consider not all power-bounded op-
erators but only positive power-bounded operators. The following question arises.
QUESTION 2. If every power-bounded positive operator on a Banach lattice E
is mean ergodic, is E reﬂexive?