Spectra and dynamics of generalized Cesàro operators in (LF) and (PLB) sequence spaces

In this paper, we introduce inductive limits of the Fréchet spaces ℓ(p+), ces(p+), and d(p+) (1 ≤p<∞) and projective limits of the (LB)-spaces ℓ(p-), ces(p-), and d(p-) (1 < p ≤ ∞). After having established some topological properties of such spaces as acyclicity and ultrabornologicity, we prove that the generalized Cesàro operators Ct (0 ≤ t ≤ 1) act continuously in these sequence spaces, and we determine the spectra. Finally, we study the ergodic properties, that is, power boundedness, (uniform) mean ergodicity, and supercyclicity, of the operators Ct acting in the (LF)-spaces and in the (PLB)-spaces mentioned above.