Copies of the sequence space ω in F-lattices with applications to Musielak−Orlicz spaces

• Autorzy: Wójtowicz M. , Wiśniewska H.

Identyfikatory

DOI

10.14708/cm.v56i1.1135

• Języki publikacji: EN

Abstrakty

EN

Let E be a fixed real function F-space, i.e., E is an order ideal in L₀(S,Σ,μ) endowed with a monotone F-norm ∥∥ under which E is topologically complete. We prove that E contains an isomorphic (topological) copy of ω, the space of all sequences, if and only if E contains a lattice-topological copy W of ω. If E is additionally discrete, we obtain a much stronger result: W can be a projection band; in particular, E contains a~complemented copy of ω. This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of ω by a Musielak−Orlicz space is characterized as follows. (1) A sequence space ℓ_Φ, where Φ=(φn), contains a copy of ω iff inf_n∈Nφn(∞)=0, where φ_n(∞)=lim_t→∞φn(t). (2) If the measure μ is atomless, then ω embeds isomorphically into L_M(μ) iff the function M_∞ is positive and bounded on some set A∈Σ of positive and finite measure, where M_∞ (s)=lim_n→∞M(n,s), s∈S. In particular, (1)' ℓ_φ does not contain any copy of ω, and (2)' L_φ(μ), with μ atomless, contains a~copy W of ω iff φ is bounded, and every such copy W is uncomplemented in L_φ(μ).

Słowa kluczowe

EN

F-space; F-lattice; Musielak-Orlicz space; sequence space ω

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2016
• Tom: Vol 56, No. 1
• Strony: 103--117
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Wójtowicz M.

• Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, 85-072 Bydgoszcz, Poland

autor

Wiśniewska H.

• Instytut Matematyki, Uniwersytet Kazimierza Wielkiego, 85-072 Bydgoszcz, Poland

Bibliografia

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Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.