Ergodicity for a family of operators

The aim of this paper is to introduce the notions of power boundedness, Cesàro boundedness, mean ergodicity, and uniform ergodicity for a family of bounded linear operators on a Banach space. The authors present some elementary results in this setting and show that some main results about power bounded, Cesàro bounded, mean ergodic, and the uniform ergodic operator can be extended from the case of a linear bounded operator to the case of a family of bounded linear operators acting on a Banach space. Also, we show that the Yosida theorem can be extended from the case of a bounded linear operator to the case of a family of bounded linear operators acting on a Banach space.