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

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