Linear operators on the (LB)-sequence spaces $ces(p-)$, 1 < p≤∞

We determine various properties of the regular (LB)-spaces (formula presented), generated by the family of Banach sequence spaces (formula presented). For instance, $ces(p-)$ is a (DFS)-space which coincides with a countable inductive limit of weighted $ell_1$-spaces; it is also Montel but not nuclear. Moreover, $ces(-p)$ and $ces(-q)$ are isomorphic as locally convex Hausdorff spaces for all choices of (formula presented). In addition, with respect to the coordinatewise order, $ces(p-)$ is also a Dedekind complete, reflexive, locally solid, lc-Riesz space with a Lebesgue topology. A detailed study is also made of various aspects (e.g., the spectrum, continuity, compactness, mean ergodicity, supercyclicity) of the Cesàro operator, multiplication operators and inclusion operators acting on such spaces (and between the spaces $ell_r$ and $ces(p-)$).