Generalized Orlicz-Lorentz sequence spaces λϕ generated by Musielak-Orlicz functions ϕ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ λ 2 for ϕ is defined in such a way that it guarantees many positive topological and geometric properties of λϕ. The problems of the Fatou property, the order continuity and the Kadec-Klee property with respect to the uniform convergence of the space λϕ are considered. Moreover, some embeddings between λϕ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λϕ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. PreliminariesThroughout this paper R, R + and N denote the sets of reals, nonnegative reals and natural numbers, respectively. The triple (N, 2 N , m) stands for the counting measure space, while l 0 = l 0 (m) denotes the space of all sequences x : N → (−∞, ∞). For every x = (x n ) ∞ n=1 ∈ l 0 (if it is more convenient the notation x(n) is used in place of x n ) we define supp x = {n ∈ N : x n = 0} and |x|(n) = |x(n)| for all n ∈ N , that is, |x| denotes the absolute value of x.The letter e stands for a Banach sequence lattice over the measure space (N, 2 N , m), that is, e is a nontrivial Banach subspace of l 0 (that is e = {0}) satisfying the following conditions (see [23] and [27]):(i) if x ∈ e, y ∈ l 0 and |y| ≤ |x| (i.e., |y n | ≤ |x n | for all n ∈ N ), then y ∈ e and y ≤ x .(ii) there exists a sequence x in e that is strictly positive on the whole N . The positive cone in e is denoted by e + . A Banach lattice e is said to have the Fatou property (e ∈ (FP) for short) if for any x ∈ l 0 and (x m ) ∞ m=1 in e + such that x m x coordinatewise and sup m x m < ∞, we have x ∈ e and x = lim m x m (see [23] and [27]).An element x ∈ e is said to be order continuous if for any sequence (x m ) in e + such that x m ≤ |x| and x m → 0 coordinatewise there holds x m → 0. The subspace e a of all order continuous elements in e is an order ideal of e. A Banach sequence lattice e is said to be order continuous (e ∈ (OC) for short) if e a = e (see [23] and [27]). Notice that e a satisfies condition (ii) from the definition of a Banach sequence lattice, that is, supp e a = N .We say that e has the coordinatewise Kadec-Klee property (e ∈ (H c ) for short) if for any x ∈ e and any sequence (x m ) in e the conditions x m → x and x m (n) → x(n) for all n ∈ N , yield x − x m → 0. We *

Generalized Orlicz-Lorentz function spaces Λϕ generated by Musielak-Orlicz functions ϕ satisfying some growth and regularity conditions (cf. [34] and [38]) are investigated. A regularity condition Δ Λ 2 for ϕ is defined in such a way that it guarantees many positive topological and geometric properties of Λϕ . The problems of the Fatou property, order continuity (separability) and the Kadec-Klee property with respect to the local convergence in measure of Λϕ are considered. Moreover, some embeddings between Λϕ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of Λϕ are presented. This paper generalizes the results from [20]. Analogous results in the sequence case were presented in [10] and [11], but the techniques in the function case are different. Investigations of structural topological and geometric properties of the generalized Orlicz-Lorentz function spaces Λ ϕ generated by special Musielak-Orlicz functions, not necessarily being weighted Orlicz functions, are initiated in this paper. Examples 2.6-2.9 on pages 1012-1014 and Examples 6.2-6.3 on page 1019 show that the class of generalized Orlicz-Lorentz spaces is much more wide than the class of Orlicz-Lorentz spaces. One of the main problems in the investigations of generalized Orlicz-Lorentz spaces Λ ϕ is to find a regularity condition of "Δ 2 -type" for the generating Musielak-Orlicz function ϕ which guarantees "good properties" of Λ ϕ . In this paper we introduce the condition Δ Λ 2 , essentialy weaker than the condition Δ 2 used in the theory of MusielakOrlicz spaces (see [7]). Unfortunately, it is not judged if this condition is the simplest possible among these ones that can guarantee the desired "good properties". The essential difficulty here is the fact that the generalized Orlicz-Lorentz spaces, as opposed to the Orlicz-Lorentz spaces, are not Calderón-Lozanovskiǐ spaces (see [16] and [36]). In consequence, some new techniques were developed in this paper.Recall that investigations of generalized Orlicz-Lorentz sequence spaces were initiated in [10] and [11]. This paper is organized as follows. In Section 1 condition (L1), which is necessary and sufficient for convexity of the functional ϕ x = γ 0 ϕ(t, x * (t)) dt is defined. Thanks to this property the generalized Orlicz-Lorentz function space Λ ϕ is a symmetric Banach space. It is also shown that if a Musielak-Orlicz function ϕ does not satisfy condition (L2), then the space Λ ϕ contains an order linearly isometric copy of l

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]).

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In this paper, criteria for non-squareness and uniform non-squareness of Orlicz-Lorentz function spaces ϕ,ω are given. Since degenerated Orlicz functions ϕ and degenerated weight functions ω are also admitted, this investigation concerns the most possible wide class of Orlicz-Lorentz function spaces.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 James in Ann. Math. 80:542-550, 1964; Pacific J. Math. 41:409-419, 1972 and García-Falset et al. in J. Funct. Anal. 233:494-514, 2006). MSC: 46B20; 46B42; 46A80; 46E30Keywords: uniform non-squareness; non-squareness; Orlicz-Lorentz space; Lorentz space; Orlicz function; Luxemburg norm; strict monotonicity; uniform monotonicity; reflexivity; super-reflexivity; fixed point property Uniform non-squareness of Banach spaces has been defined by James as the geometric property which implies super-reflexivity (see [, ]). So, after proving this property for a Banach space, we know, without any characterization of the dual space, that it is superreflexive, so reflexive as well. Recently, García-Falset, Llorens-Fuster and Mazcuñan-Navarro have shown that uniformly non-square Banach spaces have the fixed point property (see []).Therefore, it was natural and interesting to look for criteria of non-squareness properties in various well-known classes of Banach spaces. Among a great number of papers concerning this topic, we list here [-].The problem of uniform non-squareness of Calderón-Lozanovskiȋ spaces was initiated by Cerdà, Hudzik and Mastyło in []. Since the class of Orlicz-Lorentz spaces is a subclass of Calderón-Lozanovskiȋ spaces, we can say that also the problem of uniform nonsquareness of Orlicz-Lorentz spaces was initiated in []. However, the results of our paper show that those results were only some sufficient conditions for uniform non-squareness which were very far from being necessary and sufficient. Analogous results for OrliczLorentz sequence spaces were presented in [], but the techniques of the proofs in the function case are different (in some parts completely different) than in the sequence case.