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

In this paper, we introduce inductive limits of the Fréchet spaces (p+) ℓ ( p + ) , ces(p+) ces ( p + ) , and $$d(p+)$$ d ( p + ) ( 1 ≤ p < ∞ 1 ≤ p < ∞ ) and projective limits of the (LB)-spaces (p-) ℓ ( p - ) , ces(p-) ces ( p - ) , and $$d(p-)$$ d ( p - ) ( 1 < p ≤ ∞ 1 < p ≤ ∞ ). After having established some topological properties of such spaces as acyclicity and ultrabornologicity, we prove that the generalized Cesàro operators Cₜ C t ( 0 ≤ t ≤ 1 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 C t acting in the (LF)-spaces and in the (PLB)-spaces mentioned above.