Henryk Hudzik

Some remarks ons-convex functions

SummaryTwo kinds ofs-convexity (0 <s ≤ 1) are discussed. It is proved among others thats-convexity in the second sense is essentially stronger than thes-convexity in the first, original, sense whenever 0 <s < 1. Some properties ofs-convex functions in both senses are considered and various examples and counterexamples are given.

Approximative Compactness and Full Rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces

We prove that approximative compactness of a Banach space X is equivalent to the conjunction of reflexivity and the Kadec-Klee property of X. This means that approximative compactness coincides with the drop property defined by Rolewicz in Studia Math. 85 (1987), 25 – 35. Using this general result we find criteria for approximative compactness in the class of Musielak–Orlicz function and sequence spaces for both (the Luxemburg norm and the Amemiya norm) as well as critria for this property in the class of Lorentz–Orlicz spaces. Criteria for full rotundity of Musielak–Orlicz spaces are also presented in the case of the Luxemburg norm. An example of a reflexive strictly convex Köthe function space which is not approximatively compact and some remark concerning the compact faces property for Musielak–Orlicz spaces are given.

Geometric Properties of Symmetric Spaces with Applications to Orlicz–Lorentz Spaces

We deal with the basic convexity properties –rotundity, and uniform, local uniform and full rotundity –- for symmetric spaces. A characterization of Orlicz–Lorentz spaces with the Kadec–Klee property for pointwise convergence is given. These results are applied to obtain criteria of convexity properties for Orlicz–Lorentz sequence spaces, and some new proofs of the sufficiency part of criteria for rotundity and uniform rotundity for Orlicz–Lorentz function spaces.

Monotonicity properties of Lorentz spaces

Criteria for uniform monotonicity, local uniform monotonicity and strict monotonicity of Lorentz spaces are given. Let LO denote the space of all (equivalence classes of ) Lebesgue measurable real-valued functions defined on the interval [0, y), 7 t}, and we denote by f* the decreasing rearrangement of If l, that is, f*(t) = inf{s> 0: df(s) Let (E, 11 IIE) be a Banach function lattice over the measure space (T, X, ,u) [2,13,14]. E is said to be strictly monotone if for every x, y E E+ (the positive part of E), x > y, IIX IE = 1 and y :

Banach lattices with order isometric copies of l

0 imply that IIX YIIE e imply IIX YIIE E. Obviously uniform monotonicity implies local uniform monotonicity and local uniform monotonicity implies strict monotonicity. Received by the editors January 14, 1994. 1991 Mathematics Subject Classification. Primary 46E30. This research was done while the first named author was visiting Memphis State University. ( 1995 American Mathematical Society

ON GEOMETRIC PROPERTIES OF ORLICZ-LORENTZ SPACES

Abstract A criterion in order that a Banach lattice contains an order isometric copy of l∞ is given. The criterion is strongly connected with the theory of M-ideals.

On the geometry of some Calderón-Lozanovskiǐ interpolation spaces

Criteria for local uniform rotundity and midpoint local uniform rotun- dity in Orlicz-Lorentz spaces with the Luxemburg norm are given. Strict K-monotonicity and Kadec-Klee property are also discussed. Introduction. A function : R+ R+ is said to be an Orlicz function if is convex, (0) = 0, and (u) 0f or all u 0. Let (Ω Σ ) denote a complete -finite measure space and let L 0 = L 0 (Ω Σ ) denote the space of all (equivalence classes of) -measurable real-valued functions, equipped with the topology of convergence in measure on -finite sets. For any f L 0 the nonincreasing rearrangementof f is the function f defined by

Basic topological and geometric properties of Cesàro-Orlicz spaces

Abstract Geometric properties of Calderon-Lozanovskiǐ spaces between Banach function lattices and L∝ are considered. It is shown that under some general conditions these spaces posses certain monotonicity, rotundity and uniform nonsquareness properties. Some applications to renorming of Lorentz-Orlicz and Orlicz spaces are given.

Geometric properties of Kothe-Bochner spaces

Necessary and sufficient conditions under which the Cesàro-Orlicz sequence spacecesϕ is nontrivial are presented. It is proved that for the Luxemburg norm, Cesàro-Orlicz spacescesϕ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements incesϕ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spacescesϕ are given.

Matrix multiplication operators on Banach function spaces

Convexity, monotonicity and smoothness properties of Kothe spaces of vectorvalued functions are described.