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

Abstract

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