The Banach sequence spaces ces(p) are generated in a specified way via the classical spaces \( \ell _p, 1< p < \infty .\) For each pair \( 1< p,q < \infty \) 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 \( 1< p,q < \infty \) 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.
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The research of the first two authors was partially supported by the Project MTM2016-76647-P (Spain). The second author thanks the Mathematics Department of the Katholische Universität Eichstätt-Ingolstadt (Germany) for its support and hospitality during his research visit in the period September 2016–July 2017.