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

Enno Kolk; Annemai Raidjõe

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 3, page 777-792
  • ISSN: 0011-4642

Abstract

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Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = ( ϕ k ) let λ ( Φ ) = { x = ( x k ) ( ϕ k ( | x k | ) ) ∈ λ } . Given another sequence of modulus functions Ψ = ( ψ k ) , we characterize the continuity of the superposition operators P f from λ ( Φ ) into μ ( Ψ ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕ k ( k ∈ ℕ ) are unbounded and the topologies on the sequence spaces λ ( Φ ) and μ ( Ψ ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type.

How to cite

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Kolk, Enno, and Raidjõe, Annemai. "The continuity of superposition operators on some sequence spaces defined by moduli." Czechoslovak Mathematical Journal 57.3 (2007): 777-792. <http://eudml.org/doc/31162>.

@article{Kolk2007,
abstract = {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.},
author = {Kolk, Enno, Raidjõe, Annemai},
journal = {Czechoslovak Mathematical Journal},
keywords = {sequence space; superposition operator; modulus function; continuity; sequence space; superposition operator; modulus function; continuity},
language = {eng},
number = {3},
pages = {777-792},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The continuity of superposition operators on some sequence spaces defined by moduli},
url = {http://eudml.org/doc/31162},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Kolk, Enno
AU - Raidjõe, Annemai
TI - The continuity of superposition operators on some sequence spaces defined by moduli
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 3
SP - 777
EP - 792
AB - 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.
LA - eng
KW - sequence space; superposition operator; modulus function; continuity; sequence space; superposition operator; modulus function; continuity
UR - http://eudml.org/doc/31162
ER -

References

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