Mieczysław Mastyło

On the dual of Orlicz-Lorentz space

A description of the Kothe dual of the Orlicz-Lorentz space A φ,ω generated by an Orlicz function φ and a regular weight function ω is presented. It is also shown that in the case of separable Orlicz-Lorentz spaces the regularity condition on w is necessary and sufficient for the coincidence of the Banach dual space with the described Kothe dual space.

ON GEOMETRIC PROPERTIES OF ORLICZ-LORENTZ 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

Summing inclusion maps between symmetric sequence spaces

In 1973/74 Bennett and (independently) Carl proved that for 1 < u < 2 the identity map id: l u → l 2 is absolutely (u, 1)-summing, i.e., for every unconditionally summable sequence (x n ) in l u the scalar sequence (∥x n ∥l 2 ) is contained in l u , which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a 2-concave symmetric Banach sequence space E the identity map id: E → l 2 is absolutely (E,1)-summing, i.e., for every unconditionally summable sequence (x n ) in E the scalar sequence (∥x n ∥ l2 ) is contained in E. Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator T on l 2 with values in a 2-concave symmetric Banach sequence space E is a multiplier from l 2 into E. Furthermore, we prove an asymptotic formula for the k-th approximation number of the identity map id: l n 2 → E n , where E n denotes the linear span of the first n standard unit vectors in E, and apply it to Lorentz and Orlicz sequence spaces.

THE SCHUR AND (WEAK) DUNFORD-PETTIS PROPERTIES IN BANACH LATTICES

We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that £i, c0 and loo are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class A. We also present examples of weighted Orlicz spaces with the Schur property which are not S£\ -spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.

Summing inclusion maps between symmetric sequence spaces, a survey

Abstract For 1 ≤ p ≤ 2 let E be a p-concave symmetric Banach sequence space, so in particular contained in l p . It is proved in [14] and [15] that for each weakly 2 -summable sequence ( x n ) in E the sequence (║ x n ║ p ) of norms in l p is a multiplier from l p into E. This result is a proper improvement of well-known analogues in l p -spaces due to Littlewood, Orlicz, Bennett and Carl, which had important impact on various parts of analysis. We survey on a series of recent articles around this cycle of ideas, and prove new results on approximation numbers and strictly singular operators in sequence spaces. We also give applications to the theories of eigenvalue distribution and interpolation of operators.

The Dunford-Pettis Property for Symmetric Spaces

A complete description of symmetric spaces on a separable measure space with the Dunford-Pettis property is given. It is shown that l1, c0 and l ∞ are the only symmetric sequence spaces with the DunfordPettis property, and that in the class of symmetric spaces on (0, α), 0 < α ≤ ∞, the only spaces with the Dunford-Pettis property are L1, L, L1 ∩ L, L1 + L, (L) and (L1 + L), where X denotes the norm closure of L1 ∩ L in X. It is also proved that all Banach dual spaces of L1 ∩ L and L1 + L have the Dunford-Pettis property. New examples of Banach spaces showing that the Dunford-Pettis property is not a three-space property are also presented. As applications we obtain that the spaces (L1 + L) and (L) have a unique symmetric structure, and we get a characterization of the Dunford-Pettis property of some Kothe-Bochner spaces. Received by the editors November 12, 1998; revised January 25, 2000. Research supported by KBN Grant 2 P03A 05009. AMS subject classification: 46E30, 46B42. c ©Canadian Mathematical Society 2000. 789

Orlicz norm estimates for eigenvalues of matrices

AbstractLet φ be a supermultiplicative Orlicz function such that the function

Eigenvalue estimates for operators on symmetric Banach sequence spaces

t \mapsto \varphi \left( {\sqrt t } \right)