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

Abstract

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