Linear Operators on the (LB)-Sequence Spaces $$\mathbf{ces(p-), 1< p \le \infty } $$

Abstract

We determine various properties of the regular (LB)-spaces \(ces(p-)\), \(1<p\le \infty \), generated by the family of Banach sequence spaces \(\{ces(q):1<q<p\}\). 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 \(p, q\in (1,\infty ]\). 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 Cesaro operator, multiplication operators and inclusion operators acting on such spaces (and between the spaces \(\ell _{r-}\) and \(ces(p-)\)).