Non-squareness properties of Orlicz–Lorentz sequence spaces

Abstract

In this paper criteria for non-squareness properties (non-squareness, local uniform non-squareness and uniform non-squareness) of Orlicz-Lorentz sequence spaces λ ϕ,ω and of their n-dimensional subspaces λ n ϕ,ω (n 2) as well as of the subspaces (λ ϕ,ω ) a of all order continuous elements in λ ϕ,ω are given. Since degenerate Orlicz functions ϕ and degenerate weight sequences ω are also admitted, these investigations concern the most possible wide class of Orlicz-Lorentz sequence spaces. Finally, as immediate consequences, criteria for all non-squareness properties of Orlicz sequence spaces, which complete the results of Sundaresan (1966) [53], Hudzik (1985) [23], Hudzik (1985) [24], are deduced. It is worth recalling that uniform non-squareness is an important property, because it implies super-reflexivity as well as the fixed point property (see [31], James (1972) [33] and García-Falset et al. (2006) [19]).

The locally uniformly non-square points of Orlicz-Bochner sequence spaces

“…Definition () X is called a locally uniformly non‐square space if for any , there exists such that …”

“…In 2013, Foralewski et al. presented criteria for uniform non‐squareness and locally uniform non‐squareness of Orlicz–Lorentz sequence spaces. Among a great number of papers concerning this topic, please see , , , , , .…”

“…Therefore, it was natural and interesting to look for criteria of non-squareness properties in various well-known classes of Banach spaces. In 2013, Foralewski et al [2] presented criteria for uniform non-squareness and locally uniform non-squareness of Orlicz-Lorentz sequence spaces. Among a great number of papers concerning this topic, please see [1], [4], [6], [8], [9], [19].…”

Musielak–Orlicz–Bochner function spaces which are uniformly noncreasy

“…As a corollary, the criteria for uniform noncreasiness of Musielak-Orlicz function spaces are given. The topic of this paper is related to the topic of [2,[4][5][6]9,10,13] and [7,8,14].…”

“…on . Pick ⊂ ⧵ such that0 < ∫ ( , ( )) ≤ 1Moreover, since ∈ Δ, there exist disjoint measurable subsets 3 , 4 , 5 ⊂ with 3 > 0,4 > 0 and ≥ 0 satisfying1 = , ) .Pick 0 ∈ ( ). Then, by the Hahn-Banach theorem, there exists * 0 ∈ ( * ) such that * 0…”