Transference of ergodic properties of operators via matrix actions

For operators defined on locally convex spaces we define the notions of boundedness and ergodicity associated to an infinite matrix. Given two matrices $A$ and $B$, we study when $A$-bounded operators are $B$-ergodic. Using this approach we obtain equivalent formulations for the classical notions of power boundedness, Cesàro boundedness and mean ergodicity.