Weak${}^*$ convergence of operator means

For a linear operator $U$ with $\|U^n\| \leqslant \operatorname{const}$ on a Banach space $X$ we discuss conditions for the convergence of ergodic operator nets $T_\alpha$ corresponding to the adjoint operator $U^*$ of $U$ in the $\mathrm{W^*O}$-topology of the space $\operatorname{End} X^*$. The accumulation points of all possible nets of this kind form a compact convex set $L$ in $\operatorname{End} X^*$, which is the kernel of the operator semigroup $G=\overline{\operatorname{co}}\,\Gamma_0$, where $\Gamma_0=\{U_n^*, n \geqslant 0\}$. It is proved that all ergodic nets $T_\alpha$ weakly${}^*$ converge if and only if the kernel $L$ consists of a single element. In the case of $X=C(\Omega)$ and the shift operator $U$ generated by a continuous transformation $\varphi$ of a metrizable compactum $\Omega$ we trace the relationships among the ergodic properties of $U$, the structure of the operator semigroups $L$, $G$ and $\Gamma=\overline{\Gamma}_0$, and the dynamical characteristics of the semi-cascade $(\varphi,\Omega)$. In particular, if $\operatorname{card}L=1$, then a) for any $\omega \in\Omega$ the closure of the trajectory $\{\varphi^n\omega, n \geqslant 0\}$ contains precisely one minimal set $m$, and b) the restriction $(\varphi,m)$ is strictly ergodic. Condition a) implies the $\mathrm{W^*O}$-convergence of any ergodic sequence of operators $T_n \in \operatorname{End} X^*$ under the additional assumption that the kernel of the enveloping semigroup $E(\varphi,\Omega)$ contains elements obtained from the ‘basis’ family of transformations $\{\varphi^n, n \geqslant 0\}$ of the compact set $\Omega$ by using some transfinite sequence of sequential passages to the limit.