A criterion of weak compactness for operators on subspaces of Orlicz spaces

Abstract

We give a criterion of weak compactness for the operators on the Morse-Transue spaceMΨ, the subspace of the Orlicz spaceLΨgenerated byL∞.

Some new properties of composition operators associated with lens maps

“…By [8], Lemma 11, that implies that the sequence (f n ) has a subsequence equivalent to the canonical basis of c 0 and hence B ψ is not reflexive.…”

Reflexive subspaces of some Orlicz spaces

“…We shall say (see [6] and [7]) that ψ satisfies the ∆ 0 condition (and write ψ ∈ ∆ 0 ) if there exists a constant β > 1 such that…”

“…We are going to give a new proof of this result, using a criterion of weak compactness proved by P. Lefèvre, D. Li, H. Queffélec and L. Rodríguez-Piazza (see [6,Theorem 4]). …”

“…In this paper, we show that for such Orlicz functions φ, the reflexive subspaces of L φ are actually closed in the L 1 -norm (and so the L φ -topology is the same as the L 1 -topology). In order to prove this, we shall use a result of J. Alexopoulos (Theorem 1), saying that, when φ * ∈ ∆ 0 , the weakly compact subsets of L φ have equi-absolutely continuous norm, and we shall begin by giving a new proof of this result, using a recent characterization, due to P. Lefèvre, D. Li, H. Queffélec and L. Rodríguez-Piazza (see [6, Theorem 4]), of the weakly compact operators defined on a subspace of the Morse-Transue space M ψ , when ψ ∈ ∆ 0 . …”

Weak compactness of operators acting on o–O type spaces

“…More recently, characterizations in the same spirit have been given for operators acting on H ∞ ( [9]) and certain subspaces of Orlicz spaces [10]. In [14] it was proven that M * * 0 ≃ M in a canonical way.…”