Weak* convergence of operator means
For a linear contraction U in a Banach space X we discuss conditions for the convergence of ergodic operator nets corresponding to the adjoint operator U* in the W*O-topology of the space End X*. The accumulation points of all possible nets of this kind form a compact convex set L = Ker G in End X*, when G is the operator semigroup - close convex hull of the set D iterations of U*. It is proved that all ergodic nets weakly* converge if and only if the kernel L consists of a single element. For the shift operator U in X = C(Ω) generated by a continuous transformation φ of a metrizable compactum Ω we trace the relationships among the ergodic properties of U, the dynamical characteristics of the semi-cascade (φ,Ω), and the structure of the operator semigroups L, G, and Γ = cl(D). In particular, if card L = 1, then a) the closure of any trajectory in Ω contains precisely one minimal set m, and b) the restriction (φ,m) is strictly ergodic. Condition a) implies the W*O-convergence of any ergodic sequence of operators in End X* under the additional assumption that the kernel of the enveloping semigroup E(φ,Ω) contains elements obtained from iterations of φ by using some transfinite sequence of sequential passages to the limit.