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

Abstract

The Fréchet sequence spaces ces(p+) are very different to the Fréchet sequence spaces p+ , 1 ≤ p < ∞, that generate them, [3]. 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 Cesàro 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 M(ces(p+)) and its subspace M c (ces(p+)) consisting of the compact multiplier operators are independent of p. Moreover, M c (ces(p+)) can be topologized so that it is the strong dual of the Fréchet-Schwartz space ces(1+) and (M c (ces(p+)) β ces(1+) is a proper subspace of the Köthe echelon Fréchet space M(ces(p+)) = λ ∞ (A), 1 ≤ p < ∞, for a suitable matrix A.