Степенная ограниченность и эргодичность в среднем дифференциального оператора в весовых пространствах целых функций

Abstract

During last two decades investigations of many known specialists are devoted to the topological and dynamical properties of the classical operators on weighted Banach spaces of holomorphic functions having uniform estimates with respect to a given radial weight. At present complete results are established for only the problem of a characterization of those spaces in which these operators are bounded or compact. At the same time, their dynamical behavior is studied only for the spaces of a very special type given by power-exponential weights. In this paper we consider some of the dynamical properties of the differentiation operator on weighted spaces of a general kind. By using Young conjugate with the function constructed from a weight by a certain rule, we obtain low estimates for the norms of powers of this operator on every such a space and show that under some general additional conditions these estimates turn into asymptotic equalities.We apply these results to establish some conditions on weights under which the differentiation operator is power bounded or uniformly mean ergodic on the corresponding spaces. It is shown that the obtained results contain previous ones as particular cases.