Nuclear operators on Banach function spaces
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Positivity
Nuclear operators on Banach function spaces Marian Nowak1 Received: 13 March 2020 / Accepted: 11 September 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 ⊂ L 1 . Let E 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 ∞ , L 1 )) and a Banach space X is nuclear if and only if its representing measure m T : → X has the Radon-Nikodym property and |m T |() = 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 through some Orlicz space L ϕ , that is, T = S ◦ i ∞ , where S : L ϕ → X is a Bochner representable and compact operator and i ∞ : L ∞ → L ϕ 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 L 0 denote the corresponding space of λ-equivalence classes of all -measurable real functions on . Then L 0 is a super Dedekind complete Riesz space, equipped with the topology To of con-
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Marian Nowak [email protected] Institute of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65–516 Zielona Góra, Poland
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M. Nowak
vergence n in measure. By S() we denote the space of all real -simple functions ci 1 Ai , where the sets Ai ∈ are pairwise disjoint. s = i=1 Let (E, · E ) be a Banach function space, where E is an order ideal of L 0 such that L ∞ ⊂ E ⊂ L 1 , and · E is a Riesz norm on E. By T E we denote the · E -norm topology on E. By E we denote the Köthe dual of E, that is, 0 |u(ω) v(ω)| dλ < ∞ for all u ∈ E . E := v ∈ L :
The associated norm · E on E is defined for v ∈ E by v E = sup
|u(ω) v(ω)| dλ : u ∈ E, u E ≤ 1 .
We will assume that E is perfect, that is, E = E and u E = u E . The order continuous dual E n∼ of E separates the points of E and E n∼ can be identified with E through the Riesz isomorphism E v → Fv ∈ E n∼ , where Fv (u) =
u(ω)v(ω) dλ for u ∈ E and Fv = v E
(see [13, Theorem 6.1.1]). The Mackey topology τ (E, E ) is a locally convexsolid Hausdorff topology with the Lebesgue property (see [9, Corollary 82H]). Then τ (E, E ) ⊂ T E and τ (E, E ) = T
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