Weak convergence of operator means

Abstract

For a linear operator with on a Banach space we discuss conditions for the convergence of ergodic operator nets corresponding to the adjoint operator of in the -topology of the space . The accumulation points of all possible nets of this kind form a compact convex set in , which is the kernel of the operator semigroup , where . It is proved that all ergodic nets weakly converge if and only if the kernel consists of a single element. In the case of and the shift operator generated by a continuous transformation of a metrizable compactum we trace the relationships among the ergodic properties of , the structure of the operator semigroups , and , and the dynamical characteristics of the semi-cascade . In particular, if , then a) for any the closure of the trajectory contains precisely one minimal set , and b) the restriction is strictly ergodic. Condition a) implies the -convergence of any ergodic sequence of operators under the additional assumption that the kernel of the enveloping semigroup contains elements obtained from the `basis' family of transformations of the compact set by using some transfinite sequence of sequential passages to the limit.