Relationships between K-monotonicity and rotundity properties with application

Abstract

In this paper we investigate a relationship between fully k-rotundity properties, uniform K-monotonicity properties, reflexivity and K-order continuity in a symmetric spaces E. We also answer a crucial question whether fully k-rotundity properties might be restricted in definition to E d the positive cone of all nonnegative and decreasing elements of E. We present a complete characterization of decreasing uniform K-monotonicity and K-order continuity in E. It is worth mentioning that we also establish several auxiliary results describing reflexivity in Lorentz spaces Γp,w and K-order continuity in Orlicz spaces L ψ . Finally, we show an application of discussed geometric properties to the approximation theory.2000 Mathematics Subjects Classification: 46E30, 46B20, 46B28.

Sequence Lorentz Spaces and Their Geometric Structure

“…where φ E is the fundamental sequence of a symmetric space E on N. Next, in view of Remark 3.2 in [4] and assuming that E has the Fatou property, we may show that φ E (∞) = ∞ if and only if x * (∞) = 0 for any x ∈ E. Proof. Let (x m ) ⊂ γ + p,w , x ∈ ℓ 0 and x m ↑ x pointwise and sup m∈N x m γp,w < ∞.…”

On approximation properties of $l_{1}$ -type spaces

“…It is necessary to recall some investigations devoted to relationships between substochastic operators and semigroups, among other used to explore properties of Markov processes or Markov chains, which have a substantial history dating back also to the 1960s (see [14,15,16]). The next motivation for this paper we find in [4,7,8,11], where authors presented equivalent conditions for K-monotonicity properties with applications in symmetric spaces. In the spirit of the previous investigations we give a complete answer for the essential problem about a relation between uniform K-monotonicity properties.…”