The continuity of superposition operators on some sequence spaces defined by moduli

Kolk, Enno ; Raidjõe, Annemai

Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 777-792.

Résumé

Let $\lambda $ and $\mu $ be solid sequence spaces. For a sequence of modulus functions $\Phi =(\varphi _{k})$ let $ \lambda (\Phi )= \lbrace x=(x_{k}) \: (\varphi _{k}(|x_{k}|))\in \lambda \rbrace $. Given another sequence of modulus functions $\Psi =(\psi _{k})$, we characterize the continuity of the superposition operators ${P_{f}}$ from $\lambda (\Phi )$ into $\mu (\Psi )$ for some Banach sequence spaces $\lambda $ and $\mu $ under the assumptions that the moduli $\varphi _{k}$ $(k \in \mathbb{N})$ are unbounded and the topologies on the sequence spaces $\lambda (\Phi )$ and $\mu (\Psi )$ are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type.

Détail

Classification : 46A45, 47H30
Keywords: sequence space; superposition operator; modulus function; continuity