Isometric Copies of $$\varvec{l^\infty }$$ l ∞ in Cesàro–Orlicz Function Spaces

Abstract

We characterize Cesàro-Orlicz function spaces Cesϕ containing order isomorphically isometric copy of l ∞ under some mild assumption imposed on the Orlicz function ϕ. We discuss also some useful applicable sufficient conditions for the existence of such a copy.

Rotundity and monotonicity properties of selected Cesàro function spaces

“…In both cases we conclude that Ces ϕ contains an order isomorphically isometric copy of l ∞ (see Corollary 13 from [27] and the proof of Theorems 3 and 4 from [27]). The conclusion follows because l ∞ / ∈ (SM).…”

“…Suppose ϕ / ∈ 2 . Then Ces ϕ contains an order isomorphically isometric copy of l ∞ by Corollary 13 from [27]. Of course, l ∞ / ∈ (R) and consequently Ces ϕ / ∈ (R).…”

“…Proof The implication (i) ⇒ (ii) has been proved in [27,Proposition 8]. Recall only that C : L ϕ → L ϕ if and only if α ϕ > 1 (see [26] for further references).…”

“…Denote by α ϕ the lower Matuszewska-Orlicz index (see [26,27,37]). Recall that α ϕ > 1 iff ϕ * ∈ 2 , where ϕ * is the function complementary to ϕ in the sense of Young (see [37]).…”

“…The Cesàro-Orlicz function spaces Ces ϕ have been studied in [26,27] where the authors, among others, prove the criteria for the existence of order isomorphic (order isomorphically isometric) copy of l ∞ in these spaces. We prove a characterization of strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and rotundity in the spaces Ces ϕ .…”