On the Asymptotic Behaviour of some Positive Semigroups

Abstract

Similar to the theory of finite Markov chains it is shown that in a Banach space $X$ ordered by a closed cone $K$ with nonempty interior int($K$) a power bounded positive operator $A$ with compact power such that its trajectories for positive vectors eventually flow into int($K$), defines a "limit distribution", i.e. its adjoint operator has a unique fixed point in the dual cone. Moreover, the sequence (A^n) converges with respect to the strong operator topology and for each functional $f\in X'$ the sequence $((A^*)^n(f))$ converges with respect to the weak*-topology (Theorem 5). If a positive bounded $C_0$-semigroup of linear continuous operators $(S_t)_{t\geq 0}$ on a Banach space contains a compact operator and the trajectories of the non-zero vectors $x\in K$ have the property from above then, in particular, $(S_t)$ and $(S^*_t)$ converge to their limit operator with repsect to the operator norm, respectively (Theorem 4). For weakly compact Markov operators in the space of real continuous functions on a compact topological space a corresponding result can be derived, that characterizes the long-term behaviour of regular Markov chains.