The near Radon-Nikodym property in Lebesgue-Bochner function spaces

Abstract

Let X be a Banach space and (Ω, Σ, λ) be a finite measure space, 1 ≤ p < ∞. It is shown that L p (λ, X) has the Near Radon-Nikodym property if and only if X has it. Similarly if E is a Köthe function space that does not contain a copy of c 0 , then E(X) has the Near Radon-Nikodym property if and only if X does. 1991 Mathematics Subject Classification. 46E40, 46G10; Secondary 28B05, 28B20. Key words and phrases. Lebesgue-Bochner spaces, Representable operators. DEFINITIONS AND PRELIMINARY RESULTSThroughout this note, I n,k = [ k−1 2 n , k 2 n ) will be the sequence of dyadic intervals in [0, 1] and Σ n is the σ-algebra generated by the finite sequence (I n,k ) k=1,2 n . The word operator will always mean linear bounded operator and L(E, F ) will stand for the space of all operators from E into F . For any given Banach space E, its closed unit ball will be denoted by E 1 . Definition 1. Let X be a Banach space. An operator T : L 1 [0, 1] → X is said to be representable if there is a Bochner integrable function g ∈ L ∞ ([0, 1], X) such that T(f )= f g dµ for all f in L 1 [0, 1]. Definition 2. An operator D : L 1 [0, 1] → X is called a Dunford-Pettis operator if D sends weakly compact sets into norm compact sets.It is well known ([7] Example 5,III.2.11) that all representable operators from L 1 [0, 1] are Dunford-Pettis; but the converse is not true in general.Definition 3. An operator T : L 1 [0, 1] → X is said to be nearly representable if for each Dunford-Pettis operator D :The notion of nearly representable operators was introduced by Kaufman, Petrakis, Riddle and Uhl in [17]. It should be noted that since the class of Dunford-Pettis operators from L 1 [0, 1] into L 1 [0, 1] is a Banach lattice ([3]), if an operator T ∈ L(L 1 [0, 1], X) fails to be nearly representable then one can find a positive Dunford-Pettis operator D ∈ L(L 1 [0, 1], L 1 [0, 1]) such that T • D is not representable.[26] P. Wojtaszczyk. Banach spaces for analysts . Cambridge University Press, first edition, (1991).

Some properties of the projective tensor product UX derived from those of U and X

“…When A is a Riesz set, then U{X) has II-A-complete continuity property whenever X has II-A-complete continuity property [24]. (3) U(X) has the near Radon-Nikodym property whenever X has the near Radon-Nikodym property [27]. An application of Theorem 2 now yields COROLLARY 6 .…”