Operators on the Fréchet sequence spaces $${\varvec{ces(p+)}}$$ c e s ( p + ) , $$1\le p<\infty $$ 1 ≤ p < ∞

Abstract

The Frechet sequence spaces $$ ces(p+) $$ are very different to the Frechet sequence spaces $$ \ell _{p+}, 1 \le p < \infty ,$$ that generate them, (Albanese et al. in J Math Anal Appl 458:1314–1323, 2018). The aim of this paper is to investigate various properties (eg. continuity, compactness, mean ergodicity) of certain linear operators acting in and between the spaces $$ ces (p+),$$ such as the Cesaro operator, inclusion operators and multiplier operators. Determination of the spectra of such classical operators is an important feature. It turns out that both the space of multiplier operators $$ {{\mathcal {M}}}(ces (p+)) $$ and its subspace $$ {{\mathcal {M}}}_c (ces (p+)) $$ consisting of the compact multiplier operators are independent of p. Moreover, $$ {{\mathcal {M}}}_c (ces (p+)) $$ can be topologized so that it is the strong dual of the Frechet–Schwartz space $$ ces ( 1+)$$ and $$ ( {{\mathcal {M}}}_c (ces (p+))'_\beta \simeq ces (1+) $$ is a proper subspace of the Kothe echelon Frechet space $$ {{\mathcal {M}}}(ces (p+)) = \lambda ^\infty (A) , 1 \le p < \infty , $$ for a suitable matrix A.