On some geometric properties of generalized Orlicz–Lorentz sequence spaces

• Published in: Indagationes mathematicae Vol. 24; no. 2; pp. 346 - 372
• Main Author: Foralewski, Paweł
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2013
• Subjects: Generalized Orlicz–Lorentz space; Kadec–Klee properties; Lower local uniform monotonicity; Luxemburg norm; Non-squareness; Orlicz function; Orlicz–Lorentz space; Strict monotonicity; Uniform monotonicity; Uniform non-squareness; Upper local uniform monotonicity; Strict monotonicity; Lower local uniform monotonicity; Upper local uniform monotonicity; Luxemburg norm; Kadec–Klee properties; Uniform monotonicity; Orlicz function; Generalized Orlicz–Lorentz space; Uniform non-squareness; Non-squareness; Orlicz–Lorentz space
• ISSN: 0019-3577; 1872-6100
• DOI: 10.1016/j.indag.2012.11.007

In this paper, we continue investigations concerning generalized Orlicz–Lorentz sequence spaces λφ initiated in the papers of Foralewski et al. (2008) [16,17] (cf. also Foralewski (2011) [11,12]). As we will show in Examples 1.1–1.3 the class of generalized Orlicz–Lorentz sequence spaces is much more wider than the class of classical Orlicz–Lorentz sequence spaces. Moreover, it is shown that if a Musielak–Orlicz function φ satisfies condition δ2λ, then λφ has the coordinatewise Kadec–Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λφ, a strong condition of δ2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λφ, of their subspaces of order continuous elements (λφ)a as well as of finite dimensional subspaces λφn of λφ are presented.