Positivity (2021) 25:801–818 https://doi.org/10.1007/s11117-020-00787-1 Positivity Nuclear operators on Banach function spaces Marian Nowak1 Received: 13 March 2020 / Accepted: 11 September 2020 / Published online: 6 October 2020 © The Author(s) 2020 Abstract Let X be a Banach space and E be a perfect Banach function space over a finite measure space (,,λ)such that L∞ ⊂ E ⊂ L1.LetE denote the Köthe dual of E and τ(E, E) stand for the natural Mackey topology on E. It is shown that every nuclear operator T : E → X between the locally convex space (E,τ(E, E)) and a Banach space X is Bochner representable. In particular, we obtain that a linear operator T : L∞ → X between the locally convex space (L∞,τ(L∞, L1)) and a Banach space X is nuclear if and only if its representing measure mT : → X has the Radon-Nikodym property and |mT |() =T nuc (= the nuclear norm of T ). As an application, it is shown that some natural kernel operators on L∞ are nuclear. Moreover, it is shown that every nuclear operator T : L∞ → X admits a factorization ϕ ϕ through some Orlicz space L , that is, T = S ◦ i∞, where S : L → X is a Bochner ∞ ϕ representable and compact operator and i∞ : L → L is the inclusion map. Keywords Banach function spaces · Mackey topologies · Mixed topologies · Vector measures · Nuclear operators · Bochner representable operators · Kernel operators · Radon–Nikodym property · Orlicz spaces · Orlicz-Bochner spaces Mathematics Subject Classification 47B38 · 47B10 · 46E30 1 Introduction and preliminaries We assume that (X, ·X ) is a real Banach space. For terminology concerning Riesz spaces and function spaces, we refer the reader to [9,13,27]. We assume that (,,λ)is a finite measure space. Let L0 denote the correspond- ing space of λ-equivalence classes of all -measurable real functions on . Then L0 is a super Dedekind complete Riesz space, equipped with the topology To of con- B Marian Nowak [email protected] 1 Institute of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65–516 Zielona Góra, Poland 123 802 M. Nowak vergence in measure. By S() we denote the space of all real -simple functions = n 1 ∈ s i=1 ci Ai , where the sets Ai are pairwise disjoint. 0 Let (E, ·E ) be a Banach function space, where E is an order ideal of L such ∞ 1 that L ⊂ E ⊂ L , and ·E is a Riesz norm on E.ByTE we denote the ·E -norm topology on E.ByE we denote the Köthe dual of E, that is, E := v ∈ L0 : |u(ω) v(ω)| dλ<∞ for all u ∈ E . The associated norm ·E on E is defined for v ∈ E by vE = sup |u(ω) v(ω)| dλ : u ∈ E, uE ≤ 1 . We will assume that E is perfect, that is, E = E and uE =uE . The order ∼ ∼ continuous dual En of E separates the points of E and En can be identified with E v → ∈ ∼ through the Riesz isomorphism E Fv En , where Fv(u) = u(ω)v(ω) dλ for u ∈ E and Fv=vE (see [13, Theorem 6.1.1]). The Mackey topology τ(E, E) is a locally convex- solid Hausdorff topology with the Lebesgue property (see [9, Corollary 82H]). Then τ(E, E ) ⊂ TE and τ(E, E ) = TE if the norm ·E is order continuous. The most important classes of Banach function spaces are Lebesgue spaces L p (1 ≤ p ≤∞) and Orlicz spaces Lϕ (see [19]). Now we present a characterization of (τ(E, E ), ·X )-continuous linear operators T : E → X (see [17, Proposition 2.2]). Proposition 1.1 For a bounded linear operator T : E → X the following statements are equivalent: (i) Tis(τ(E, E ), ·X )-continuous. (ii) T (un)X → 0 if un(ω) → 0 λ-a.e. and |un(ω)|≤|u(ω)| λ-a.e. for some u ∈ E and all n ∈ N. ∈ ( 1 ) → λ( ) → (iii) For each u E, T u An X 0 whenever An 0. For terminology and basic facts concerning vector measure, we refer the reader to [4,6,7,22]. For a finitely additive measure m : → X,by|m|(A) we denote the variation of m on A ∈ . A measure m : → X is said to be λ-continuous if m(An)X → 0 whenever λ(An) → 0. Let L1(X) denote the Banach space of λ-equivalence classes of all X-valued Bochner integrable functions g defined on , equipped with the norm g1 := g(ω)X dλ. Recall that a λ-continuous measure m : → X of finite variation is said to have the Radon-Nikodym property with respect to λ if there exists a function g ∈ L1(X) 123 Nuclear operators on Banach function spaces 803 ( ) = (ω) λ ∈ = λ such that m A A g d for all A . Then we write m g and a function g is called the density of m with respect to λ. Assume that m : → X is a λ-continuous measure. Following [6, §13] for A ∈ , we put n |m|E (A) := sup |ci |m(Ai )X , i=1 = n 1 ∈ S() where the supremum is taken for all functions s i=1 ci Ai such that Ai ⊂ A for 1 ≤ i ≤ n and sE ≤ 1. The set function |m|E will be called a E-variation of the measure m. = ∞ | | ( ) =| |( ) ∈ If, in particular, E L , then m L1 A m A for A . Let L0(X) stand for the linear space of λ-equivalence classes of all strongly - measurable functions g : → X.Let 0 E(X) = g ∈ L (X) :g(·)X ∈ E . Then E(X) equipped with the norm gE(X) := · g(·)X E is a Banach space, called a Köthe-Bochner space (see [14]). Definition 1.1 A bounded linear operator T : E → X is said to be Bochner repre- sentable, if there exists g ∈ E(X) such that T (u) = u(ω)g(ω) dλ for u ∈ E. The concept of nuclear operators between Banach spaces in due to Ruston [21]. Grothendieck carried over the concept of nuclear operators to locally convex spaces [10,11](seealso[26, p. 289], [18], [23, Chap. 3, §7], [4, Chap. 6], [5,22]). Following [23, Chap. 3, §7] (see also [2, Chap. 4], [12, 17.3, p. 379]), we have Definition 1.2 A linear operator T : E → X is said to be τ(E, E)-nuclear if (v ) { : ∈ N} τ( , ) there exist a sequence n in E such that the family Fvn n is E E - 1 equicontinuous, a bounded sequence (xn) in X and a sequence (αn) ∈ such that ∞ T (u) = αn u vn dλ xn for u ∈ E. (1.1) n=1 Let ∞ T nuc := inf |αn|vnE xnX , n=1 1 where the infimum is taken over all sequences (vn) in E , (xn) in X and (αn) ∈ such that T admits a representation (1.1). 123 804 M. Nowak It is known that a τ(E, E )-nuclear operator T : E → X is (τ(E, E ), ·X )- continuous and τ(E, E)-compact, that is, T (V ) is relatively norm compact in X for some τ(E, E)-neighborhood V of 0 in E (see [23, Chap. 3, §7, Corollary 1], [12, Corollary 4, p. 379]). In this paper we study τ(E, E)-nuclear operators T : E → X. In Section 2 it is shown that every τ(E, E)-nuclear operator T : E → X is Bochner representable (see Theorem 2.3 below). In particular, we obtain that a linear operator T : L∞ → X ∞ 1 is τ(L , L )-nuclear if and only if its representing measure mT : → X has the Radon-Nikodym property and |mT |() =T nuc (see Theorem 2.5 below). As an application, we obtain that some natural kernel operators on L∞ are τ(L∞, L1)- nuclear (see Proposition 2.9 below). In Section 3 it is shown that every τ(L∞, L1)- nuclear operator T : L∞ → X admits a factorization through some Orlicz space Lϕ, ϕ that is, T = S ◦ i∞, where S : L → X is a Bochner representable, compact operator ∞ ϕ and i∞ : L → L denotes the inclusion map (see Corollary 3.5). 2 Nuclear operators on Banach function spaces Assume that T : E → X is a linear operator. Then the measure mT : → X defined by mT (A) := T (1A) for A ∈ is called a representing measure of T . If, in particular, T is (τ(E, E ), ·X )-continuous, then using Proposition 1.1 we obtain that mT is countably additive. Since mT (A) = 0ifλ(A) = 0, by the Pettis theorem mT is λ-continuous, that is, mT λ. The following lemma will be useful. Lemma 2.1 Let T : E → Xbea(τ(E, E ), ·X )-continuous linear operator. If |mT |E () < ∞ and mT has the Radon-Nikodym property with respect to λ with a density g ∈ L1(X), then g ∈ E(X) and 1AgE(X) =|mT |E (A) for all A ∈ , and T (u) = u(ω) g(ω) dλ for all u ∈ E. Proof First we shall show that for A ∈ , |mT |E (A) = sup |s(ω)|g(ω)X dλ : s ∈ S(), sE ≤ 1 . (2.1) A 123 Nuclear operators on Banach function spaces 805 | |( ) = (ω) λ = k 1 ∈ S() Note that mT A A g X d .Fors i=1 ci Ai ,wehave k k ci mT (A ∩ Ai ) = ci g(ω) dλ = s(ω) g(ω) dλ. ∩ i=1 i=1 A Ai A = k 1 ∈ S() ≤ We now show that for s i=1 ci Ai and s E 1, we have k k |ci | g(ω)X dλ = |ci ||mT |(A ∩ Ai ) ≤|mT |E (A).