Operators acting in sequence spaces generated by Dual Banach spaces of discrete Cesàro spaces

Abstract

The dual spaces $d(p), 1 \lt p \lt \infty ,$ of the discrete Cesàro (Banach) spaces $\mathrm{ces} (q)$, $1 \lt q \lt \infty ,$ were studied by G. Bennett, A. Jagers and others. These (reflexive) dual Banach spaces induce the non-normable Fréchet spaces $d (p+) := \bigcap_{r \gt p} d (r), $ for $1 \leq p \lt \infty ,$ and the (LB)-spaces $d (p-) := \bigcup_{ 1 \lt r \lt p } d (r), $ for $ 1 \lt p \leq \infty ,$ recently introduced and investigated in [11]. Here a detailed study is made of various aspects, such as the spectrum, continuity, compactness, mean ergodicity and supercyclicity of the Cesàro operator, multiplication operators and inclusion operators when they act on (and between) such spaces.