Orlicz-Lorentz Sequence Spaces Equipped with the Orlicz Norm

In this article, we consider Orlicz-Lorentz sequence spaces equipped with the Orlicz norm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\lambda _{\varphi ,\omega ,}}\left\| {\, \cdot \,} \right\|_{\varphi ,\omega }^O)$$\end{document} generated by any Orlicz function and any non-increasing weight sequence. As far as we know, research on such a general case is conducted for the first time. After showing that the Orlicz norm is equal to the Amemiya norm in general and giving some important properties of this norm, we study the problem of existence of order isomorphically isometric copies of l∞ in the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\lambda _{\varphi ,\omega ,}}\left\| {\, \cdot \,} \right\|_{\varphi ,\omega }^O)$$\end{document} and we find criteria for order continuity and monotonicity properties of this space. We also find criteria for monotonicity properties of n-dimensional subspaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{\varphi ,\omega }^n\,(n \ge 2)$$\end{document} and the subspace (λφ,ω)a of order continuous elements of λφ,ω. Finally, as an immediate consequence of the criteria considered in this article, the properties of Orlicz sequence spaces equipped with the Orlicz norm are deduced.

(under the convention inf ∅ = ∞). We say that two sequences x, y ∈ l 0 are equimeasurable if µ x (λ) = µ y (λ) for all λ ≥ 0. It is obvious that equimeasurability of x and y gives x * = y * .
Given any Orlicz function ϕ, we define its complementary function in the sense of Young by the formula for all u ∈ R. It is easy to show that ψ is also an Orlicz function. The Orlicz function ϕ and its complementary function ψ satisfy the Young inequality uv ≤ ϕ (u) + ψ (v) for any u, v ≥ 0.
(1. 1) In some proofs in this paper, the case when inequality (1.1) becomes an equality will be important. Let us define a subdifferential ∂ϕ (u) of ϕ at u ≥ 0 as follows: Let ω : N → R be a nonnegative, nontrivial and non-increasing sequence, called a weight sequence.
For any Orlicz function ϕ and any weight sequence ω, the convex semimodular I ϕ,ω : l 0 → R e + = [0, ∞] is defined as follows: ϕ (x (π (i))) ω (i) , where π denotes a permutation of the set N and the supremum is extended over all permutation of N. Note that the second equality in the above formula follows from [11,Theorem 2.2] (see also Remark 1.1). In fact, for the sequence of Orlicz functions (φ i ) ∞ i=1 , where φ i (u) = ϕ(u)ω(i) for any i ∈ N, we have p i (u) = p(u)ω(i) for the same i, where p i (i ∈ N) and p denote the right derivatives of φ i (i ∈ N) and ϕ, respectively. In consequence, the sequence of Orlicz 625 functions (φ i ) ∞ i=1 satisfies condition (L1) from [11] if and only if a ϕ = b ϕ or the sequence ω is non-increasing.
It is well known that the space λ ϕ,ω equipped with the Luxemburg norm · ϕ,ω , defined by is a Banach symmetric space (for basic properties of symmetric spaces, we refer to [2,24,26]).
By analogy to Orlicz spaces, we can define two additional norms (see Remark 1.1), namely the Orlicz norm x * (i)y * (i)ω(i) : I ψ,ω (y) ≤ 1 (1.6) and the Amemiya norm x A ϕ,ω = inf k>0 1 k (1 + I ϕ,ω (kx)) . (1.7) In the past research mainly Orlicz-Lorentz spaces equipped with the Luxemburg norm were studied (see [1, 4, 9-14, 20, 21, 25, 30, 36-38]). Recently, Orlicz-Lorentz spaces equipped with the Orlicz norm have been considered in some papers (see [15,34] and also [6]). However, in those papers only some special cases of Orlicz functions and weight sequences have been investigated. In our paper we admit any Orlicz function and any non-increasing weight sequence. Remark 1.1 Let ϕ be any Orlicz function and ω any nonnegative and nontrivial weight sequence. If a ϕ = b ϕ , then λ ϕ,ω = l ∞ and x ϕ,ω = x A ϕ,ω = x l ∞ /b ϕ for any x ∈ λ ϕ,ω . Let now a ϕ < b ϕ . Then the following conditions are equivalent: the sequence ω is non-increasing, the second equality in formula (1.3) holds and the semimodular I ϕ,ω is convex (see [11,Theorem 2.2]). The convexity of the semimodular I ϕ,ω is in turn essential to prove that the functionals defined by (1.5) and (1.7) are norms. Also, the functional defined by (1.6) is a norm if the sequence ω is non-increasing.

Some Basic Results
We start with the following Lemma 2.1 For any x ∈ λ ϕ,ω , we have Proof We recall the proof of this lemma for completeness. It is obvious that the above formula holds for x = 0. Assume now that x = 0 and take an arbitrary k > 0. If I ϕ,ω (kx) ≤ 1, then x ϕ,ω ≤ 1 k . On the other hand, if I ϕ,ω (kx) > 1, then . Hence, we obtain By the arbitrariness of k > 0, we get It only remains to prove the reversed inequality. If 0 < x ϕ,ω < 1 k0 , then I ϕ,ω (k 0 x) ≤ 1, which means that Since 1 k0 may be arbitrarily close to x ϕ,ω , we conclude that For any x ∈ λ ϕ,ω , we define 3) The function f is continuous on the interval (0, λ ∞ ) and left continuous at λ ∞ if λ ∞ < ∞. Let E ⊂ l 0 be a normed Riesz space, that is, a partially ordered normed vector space over the real numbers, such that (i) x ≤ y implies x + z ≤ y + z for any x, y, z ∈ E; (ii) αx ≥ 0 for any x ≥ 0 in E and any non-negative real α; (iii) for all x, y ∈ E, there exist a least upper bound x ∨ y and a greatest lower bound x ∧ y; (iv) for any x, y ∈ E, we have x ≤ y whenever |x| ≤ |y| (see [35]). We say that E has the Fatou property if for any x ∈ l 0 and (x n ) ∞ n=1 in E + (the positive cone of E) such that x n ր |x| coordinatewise and sup n x n < ∞, we have x ∈ E and x = lim n→∞ x n (see [26]).

Theorem 2.3
The Orlicz and Amemiya norms are equal; that is, for any x ∈ λ ϕ,ω , we have Proof Let us take any x ∈ λ ϕ,ω . In virtue of the Young inequality (1.1), we get, for any k > 0 and any y ∈ λ ψ,ω with I ψ,ω (y) ≤ 1, that By the arbitrariness of k > 0, we obtain x O ϕ,ω ≤ x A ϕ,ω . Now, we will prove the opposite inequality. First, we will prove it for any nonnegative x ∈ λ ϕ,ω such that m (suppx) = n < ∞. We will consider three cases.
Let (x n ) be a sequence in λ ϕ,ω . The following assertions are true: Proof We will give a short proof of the above lemma for the sake of completeness.
ω (i) = ∞ and the function ϕ satisfies the condition ∆ 2 (0). By [31, Theorem 1.6] and Lemma 2.5(i), we only need to show that lim n→∞ I ϕ,ω (2x n ) = 0 whenever lim n→∞ I ϕ,ω (x n ) = 0. Take any fixed ε > 0. Without loss of generality, we may assume that for the same n. By the arbitrariness of ε > 0, we conclude that lim n→∞ I ϕ,ω (2x n ) = 0. Now we shall show the necessity of the condition ∆ 2 (0). Suppose first that a ϕ > 0 and define x n = (a ϕ , 0, 0, . . .) for any n ∈ N. We have that I ϕ,ω (x n ) = 0 and x n O ϕ,ω := a > 0 for all n ∈ N. Assume now that a ϕ = 0 and ϕ does not satisfy the condition ∆ 2 (0). Then, we find a sequence (u n ) ∞ n=1 decreasing to zero and such that u n e i for any n ∈ N. We We may show the necessity of the condition a ϕ = 0 analogously as in (ii). Now we will prove the sufficiency of this condition. In a fashion similar to (ii), we only need to show that lim n→∞ I ϕ,ω (2x n ) = 0 whenever lim n→∞ I ϕ,ω (x n ) = 0. Take any fixed ε > 0.
Since lim n→∞ x * n (1) = 0, we will find k ∈ N such that for any n ≥ k. Therefore, for the same n we obtain By the arbitrariness of ε > 0, it follows that lim n→∞ I ϕ,ω (2x n ) = 0.
For any x ∈ λ ϕ,ω \ {0}, we ask the question if the infimum in formula (1.7) is attained, that means if there exists k > 0 such that In order to answer this question, we define constants as following: First, we shall prove two lemmas.
In the final part of the proof, we will assume that k * < k * * . By the definition of the constants k * and k * * , we have I ψ,ω (p (kx * )) = 1 for any k ∈ (k * , k * * ), and thus we find that for the same k. By the continuity (left continuity at If k * < ∞, then by Lemma 2.10 the function f attains its infimum for any k ∈ K (x). In the second case, if k * = ∞, the function f is decreasing on the interval (0, ∞), whence we get (2.13).
In the next part of the proof, the symbols k * and k * * denote k * (χ A ) and k * * (χ A ), respectively. We will consider three cases.

Case 3 Finally, suppose that lim
On the other hand, if ψ (B) and ψ (u) = ∞ for u > b ψ and the function ψ does not assume the value 1/ Let E ⊂ l 0 be a H Köthe sequence space. An element x ∈ E is said to be order continuous if for any sequence (x n ) in E + (the positive cone of E) with x n ≤ |x| and x n → 0 coordinatewise, there holds x n E → 0. The subspace E a of all order continuous elements in E is an order ideal in E. The space E is called order continuous if E a = E (see [26]).
Since both the Luxemburg and the Orlicz norms are equivalent (see Lemma 2.5(i)), the subspace of order continuous elements for both of these norms is the same (as a subset of the elements of the space λ ϕ,ω ). We will denote it as (λ ϕ,ω ) a . By Theorem 4.2 in [11], we get ω (i) = ∞ and a ϕ = 0, then (λ ϕ,ω ) a = λ θ ϕ,ω (see formula (1.4)). We get also the following

Monotonicity Properties and Copies of l ∞
In this section we study the monotonicity properties of the Orlicz-Lorentz spaces equipped with the Orlicz norm.
This problem was first studied by Gong and Zhang in [15]. However, in that paper, it has been assumed that the Orlicz function ϕ is an N -function (that means that a ϕ = 0, lim ω (i) = ∞. In this paper, we will present respective criteria for Orlicz-Lorentz spaces generated by any Orlicz function and any non-increasing weight sequence. Moreover, we will also study some subspaces of λ ϕ,ω , namely n-dimensional Orlicz-Lorentz spaces λ n ϕ,ω for n ≥ 2 and the subspace (λ ϕ,ω ) a of order continuous elements in λ ϕ,ω . The proof methods, which we apply, are different from most of the methods used in [15]. We use, among other things, some relationships between monotonicity properties and other geometric and topological properties.
Necessity. By Theorem 3.1 the space λ ϕ,ω , · O ϕ,ω is strictly monotone and thus it does not contain an order linearly isometric copy of l ∞ whenever ∞ i=1 ω (i) = ∞ and a ϕ = 0.
ϕ,ω is upper locally uniformly monotone if and only if the function ϕ satisfies the condition ∆ 2 (0).
The necessity of the condition ∆ 2 (0) can be shown in the same way as in the proof of the implication (iii)⇒(i) in Theorem 3.5. Notice that the elements x, y n and z n , n ∈ N defined in that proof belong to (λ ϕ,ω ) a .
(ii) By Theorem 3.7 and [7, Theorem 3], we get that the space (λ ϕ,ω ) a , · O ϕ,ω is strictly monotone and it has the Kadec-Klee property with respect to the coordinatewise convergence, whence by [9,Theorem 4.1] we obtain that it is upper locally uniforlmy monotone.

Remark 3.11 (i) Let us note that if
ω (i) = ∞, a ϕ = 0 and ϕ / ∈ ∆ 2 (0), then the space (λ ϕ,ω ) a , · O ϕ,ω is lower locally uniformly monotone but it is not upper locally uniformly monotone (see Theorems 3.7 and 3.9 and Remark 3.8). A similar problem was considered by Hudzik and Kurc in [18] for the Musielak-Orlicz space (and thereby also for the Orlicz space). It is still unknown if upper local uniform monotonicity implies lower local uniform monotonicity.
The sequence ω is called regular if there exists η ∈ (0, 1] such that for any n ∈ N, the equality The converse implication is not true. We can namely take a sequence ω, where ω (i) = 1 i for i ∈ N, which is not regular although Theorem 3.12 The following conditions are equivalent: (i) the Orlicz function ϕ satisfies the condition ∆ 2 (0) and the weight sequence is regular; (ii) the space λ ϕ,ω , · O ϕ,ω is uniformly monotone; (iii) the space (λ ϕ,ω ) a , · O ϕ,ω is uniformly monotone. Proof (i)⇒(ii). In this part of the proof, we use the idea presented by Hudzik and Kamińska in [16] which was later modified in papers [9] and [15]. We will show the whole proof for clarity and completeness.

Remark 3.13
It is worth noticing that the necessary and sufficient conditions for respective monotonicity properties of the Orlicz-Lorentz space (or its subspaces) equipped with the Orlicz norm are weaker than the analogous conditions for the Orlicz-Lorentz space (or its subspaces) equipped with the Luxemburg norm (see [4,Lemma 1], [9, and [12,Corollary 4.5]). In the case of the Luxemburg norm, we always need to assume that ϕ (b ϕ ) ω (1) ≥ 1. Moreover, the necessary condition for strict monotonicity is the condition ∆ 2 (0) for the space (λ ϕ,ω , · ϕ,ω ) and the condition a ϕ = 0 for the space λ n ϕ,ω , · ϕ,ω (see Example 3.10).