Multiplier and averaging operators in the Banach spaces $ces(p)$, $1

The Banach sequence spaces ces(p) are generated in a specified way via the classical spaces ℓp,1<∞. For each pair 1<∞ the (p, q)-multiplier operators from ces(p) into ces(q) are known. We determine precisely which of these multipliers is a compact operator. Moreover, for the case of p=q a complete description is presented of those (p, p)-multiplier operators which are mean (resp. uniform mean) ergodic. A study is also made of the linear operator C which maps a numerical sequence to the sequence of its averages. All pairs 1<∞ are identified for which C maps ces(p) into ces(q) and, amongst this collection, those which are compact. For p=q, the mean ergodic properties of C are also treated.