Geodesic
The continuity of superposition operators on some sequence spaces defined by moduli
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 777-792 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 46A45, 47H30
Keywords: sequence space; superposition operator; modulus function; continuity 
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     author = {Kolk, Enno and Raidj\~oe, Annemai},
     title = {The continuity of superposition operators on some sequence spaces defined by moduli},
     journal = {Czechoslovak Mathematical Journal},
     pages = {777--792},
     year = {2007},
     volume = {57},
     number = {3},
     mrnumber = {2356280},
     zbl = {1174.47048},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a0/}
}
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Kolk, Enno; Raidjõe, Annemai. The continuity of superposition operators on some sequence spaces defined by moduli. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 3, pp. 777-792. http://geodesic.mathdoc.fr/item/CMJ_2007_57_3_a0/