ATTRACTORS AND ASYMPTOTIC PERIODICITY OF POSITIVE OPERATORS ON BANACH-LATTICES

Several authors used the concept of an attractor to characterize asymp totic periodicity of operators. In our main result we show that a posi tive contraction T on a KB-space (i.e. a weakly sequentally complete B anach lattice) E is asymptotically periodic if and only if T has a qua si order bounded atractor. This generalizes previous results of Komorn ik and Lasota ([Ko1], [KoL]) on Markov operators on L(1)-spaces. As a consequence we can characterize stability of (one-parameter) semigroup s of positive operators by means of attractors.