Nuclear Operators on Banach Function Spaces

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 corresponding 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 representable, 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 ∈ ,

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). ∩ i=1 A Ai i=1

Indeed, let ε>0 be given. Then for each 1 ≤ i ≤ k, there exists a -partition ( ) ji ∩ Ai, j j=1 of A Ai such that

ji ε | |( ∩ ) ≤ ( ) + . mT A Ai mT Aij X k|ci | j=1

Hence ⎛ ⎞ k k ji ⎝ ⎠ |ci ||mT |(A ∩ Ai ) ≤ ci mT (Aij)X + ε ≤|mT |E (A) + ε, i=1 i=1 j=1 k ji 1 = k 1 . because i=1 j=1 ci Aij i=1 ci Ai Then we have

k k |ci |mT (A ∩ Ai )X ≤ |ci ||mT |(A ∩ Ai ) i=1 i=1 k k = | | (ω) λ = | | 1 (ω) (ω) λ ci g X d ci Ai g X d ∩ i=1 A Ai A i=1

= |s(ω)|g(ω)X dλ ≤|mT |E (A). A

Taking supremum on the left side, we get

|mT |E (A) = sup |s(ω)|g(ω)X dλ : s ∈ S(), sE ≤ 1 . A

We shall now show that g ∈ E (X), that is, g(·)X ∈ E . Indeed, let u ∈ E. Then there exists a sequence (sn) in S() such that 0 ≤ sn(ω) ↑ |u(ω)| λ-a.e. (see [13, Corollary I.6]). Choose c > 0 such that cuE ≤ 1. Then by the Fatou lemma, 123 806 M. Nowak

c |u(ω)|g(ω)X dλ ≤ sup csn(ω)g(ω)X dλ ≤|mT |E (), n

and this means that g(·)X ∈ E , that is, g ∈ E (X). Moreover, for A ∈ and u ∈ E with uE ≤ 1, using (2.1) we get

|u(ω)|1A(ω) g(ω)X dλ ≤ sup sn(ω)1A(ω) g(ω)X dλ ≤|mT |E (A) n and it follows that 1AgE(X) ≤|mT |E (A).Inviewof(2.1) we get |mT |E (A) ≤ 1 | | ( ) =1 Ag E (X). Hence mT E A Ag E (X). = k 1 ∈ S() Note that for s i=1 ci Ai ,wehave

k T (s) = ci mT (Ai ) = s(ω) g(ω) dλ. i=1

T (sn) − T (u)X =T (sn − u)X → 0. Hence T (u) = u(ω)g(ω) dλ. As a consequence of Lemma 2.1,wehave Proposition 2.2 Assume that T : E → X is a Bochner representable operator, that is, there exists g ∈ E(X) such that T (u) = u(ω) g(ω) dλ forallu∈ E.

Then the following statements hold: (i) T is (τ(E, E ), ·X )-continuous. (ii) For every A ∈ , |mT |E (A) =1AgE(X).

Proof (i) Assume that un(ω) → 0 λ-a.e. and |un(ω)|≤|u(ω)| λ-a.e. for some u ∈ E and all n ∈ N. Since for n ∈ N,

T (un)X ≤ |un(ω)|g(ω)X dλ, 123 Nuclear operators on Banach function spaces 807

1 where u g(·)X ∈ L , by the Lebesgue dominated convergence theorem, we get T (un)X → 0. Hence in view of Proposition 1.1 T is (τ(E, E ), ·X )- continuous. = k 1 ∈ S() ≤ (ii) Assume that s i=1 ci Ai and s E 1. Then

k k |ci |mT (Ai )X = |ci | g(ω)X dλ = |s(ω)|g(ω)X dλ i=1 i=1 Ai ≤sE gE(X) ≤gE(X).

Hence |mT |E () ≤gE(X). Using (i) and Lemma 2.1, we get |mT |e (A) = 1AgE(X) for all A ∈ . The following result shows a relationship between τ(E, E)-nuclear operators and Bochner representable operators T : E → X. Theorem 2.3 Assume that T : E → Xisaτ(E, E)-nuclear operator. Then T is Bochner representable and |mT |E () ≤T nuc. Proof Let ε>0 be given. Then there exists a bounded sequence (vn) in E , a bounded 1 sequence (xn) in X and a sequence (αn) ∈ such that ∞ T (u) = αn u vn dλ xn for u ∈ E n=1 and ∞ |αn|vnE xnX ≤T nuc + ε. (2.2) n=1

Hence we have ∞ mT (A) = αn vn dλ xn. (2.3) n=1 A | | () ≤ = k 1 ∈ S() We shall now show that mT E T nuc. Indeed, let s i=1 ci Ai with sE ≤ 1. Then using (2.2) we get

k k ∞ |ci |mT (Ai )X = ci αn vn(ω) dλ xnX = = = Ai i 1 i 1 n1 ∞ k ≤ |αn| |ci | |vn(ω)| dλ xnX n=1 i=1 Ai ∞ = |αn|xnX |s(ω)||vn(ω)| dλ n=1 123 808 M. Nowak

∞ ≤ |αn|xnX vnE ≤T nuc + ε. n=1

Hence we get

|mT |E () ≤T nuc. (2.4) ∈ N := n α v ⊗ , ∈ N > For n ,letgn i=1 i i xi . Then for n k with n k,wehave n gn(ω)− gk (ω)X dλ = αi vi (ω)xi dλ i=k+1 X n n ≤ |αi ||vi (ω)|xi X dλ = |αi |xi X |vi | dλ i=k+1 i=k+1 n n ≤ |αi |xi X 1E vi E ≤ sup x j X sup v j E 1E |αi |. ∈N ∈N i=k+1 j j i=k+1

1 This follows that (gn) is a Cauchy sequence in the Banach space L (X), so there 1 exists g ∈ L (X) such that gn − g1 → 0. One can easily show that v dλ x = v(ω) xdλ for v ∈ E , x ∈ X, A ∈ . A A

Hence for A ∈ ,wehave n n αi vi dλ xi − gdλ = αi vi (ω) xi dλ − g(ω) dλ A A A A i=1 X i=1 X n ≤ αi vi (ω) xi − g(ω) dλ = gn(ω) − g(ω)X dλ → 0. A i=1 X A

Then in view of (2.3)forA ∈ , we get

mT (A) = g(ω) dλ. A

Using Lemma 2.1and (2.4) we see that g ∈ E (X) with gE(X) =|mT |E () ≤ T nuc and T (u) = u(ω) g(ω) dλ for all u ∈ E.

Now we shall study τ(L∞, L1)-nuclear operators T : L∞ → X. Making use of the Dunford–Pettis theorem (see [3, Theorem, p. 93]) we have

Theorem 2.4 For a subset H of L1 the following statements are equivalent: (i) H is relatively weakly compact. v < ∞ (ii) supv∈H 1 and H is uniformly integrable. ∞ (iii) {Fv : v ∈ H} is τ(L , L1)-equicontinuous. 123 Nuclear operators on Banach function spaces 809

Note that in view of Theorem 2.4 and Deﬁnition 1.2, a linear operator T : L∞ → X ∞ 1 is τ(L , L )-nuclear if there exist a bounded uniformly integrable sequence (vn) in 1 1 L , a bounded sequence (xn) in X and a sequence (αn) ∈ such that ∞ ∞ T (u) = αn u vn dλ xn for all u ∈ L (2.5) n=1 and then ∞ T nuc = inf |αn|vn1 xnX , n=1

1 1 where the inﬁmum is taken over all sequences (vn) in L , (xn) in X and (αn) ∈ such that T admits a representation (2.5). Now we can state our main result. Theorem 2.5 For a linear operator T : L∞ → X the following statements are equivalent:

(i) mT has the Radon-Nikodym property with respect to λ. (ii) T is Bochner representable. (iii) Tisaτ(L∞, L1)-nuclear operator.

In this case |mT |() =T nuc. Proof (i)⇒(ii) Assume that (i) holds with the density g ∈ L1(X). Note that for s ∈ S(),wehave

sdmT = s(ω) g(ω) dλ.

∞ Let u ∈ L . Choose a sequence (sn) in S() such that u − sn∞ → 0. Then we have

T (u) = udmT = lim sn dmT = lim sn(ω) g(ω) dλ = u(ω) g(ω) dλ.

(ii)⇒(i) This is obvious. (ii)⇒(iii) Assume that (ii) holds, that is, there exists g ∈ L1(X) such that ∞ T (u) = u(ω) g(ω) dλ for all u ∈ L .

Let L1⊗ˆ X denote the projective tensor product of L1 and X, equipped with the norm π deﬁned for w ∈ L1⊗ˆ X,by ∞ π(w) := inf |αn|vn1 xnX , n=1 123 810 M. Nowak

(v ) 1 ( ) v = where the inﬁmum is taken over all sequences n in L , xn in X with lim n 1 = (α ) ∈ 1 w = ∞ α v ⊗ 0 lim xn X and n such that n=1 n n xn (see [22, Proposition 2.8, pp. 21–22]). It is known that L1⊗ˆ X is isometrically isomorphic to L1(X) through the isometry J, where

J(v ⊗ x) := v(·) x for v ∈ L1, x ∈ X

(see [4, Example 10, p. 228], [22, Example 2.19, p. 29], [5, Theorem 1.1.10, p. 14]). Let 1 ε>0 be given. Then there exist sequence (vn) in L and (xn) in X with limn vn1 = 1 0 = lim xnX and (αn) ∈ such that

∞ −1 1 ˆ J (g) = αn vn ⊗ xn in L ⊗ X,π n=1 and ∞ −1 |αn|vn1 xnX ≤ π(J (g)) + ε =g1 + ε. (2.6) n=1

Hence ∞ ∞ 1 g = J αnvn ⊗ xn = αn vn(·) xn in L (X), ·1 . n=1 n=1

Note that n u(ω) g(ω) dλ − αi u(ω) vi (ω) dλ xi i=1 X n ≤ |u(ω)| g(ω) − αi vi (ω) xi dλ i=1 X n n ≤u∞ g(ω) − αi vi (ω) xi dλ =u∞ · g − αi vi ⊗ xi i=1 X i=1 1

Hence ∞ ∞ T (u) = αn u vn dλ xn for u ∈ L . n=1

Since lim vn1 = 0, the set {vn : n ∈ N} is uniformly integrable and it follows that T is τ(L∞, L1)-nuclear. In view of (2.6) we get

T nuc ≤g1 =|mT |(). (2.7) 123 Nuclear operators on Banach function spaces 811

(iii)⇒(ii) Assume that (iii) holds. Then by Theorem 2.3 T is Bochner representable and

|mT |() =g1 ≤T nuc. (2.8)

Thus (i)⇔(ii)⇔(iii) hold and using (2.7) and (2.8), we get |mT |() =T nuc.

Let baλ() denote the Banach space of all bounded ﬁnitely additive real measures μ on such that μ(A) = 0ifλ(A) = 0, equipped with norm μ:=|μ|(). ∞ ∗ ∞ The Banach dual (L ) of L can be identiﬁed with baλ() through the integration ∞ ∗ ∞ mapping baλ() μ → Fμ ∈ (L ) , where Fμ(u) = udμ for all u ∈ L and |μ|() =Fμ. Recall (see [26, p. 279]) that a linear operator T : L∞ → X is said to be nuclear if there exist a bounded sequence (μn) in baλ(), a bounded sequence (xn) in X and 1 a sequence (αn) ∈ such that

∞ ∞ T (u) = αn udμn xn for all u ∈ L . n=1

∞ ∞ 1 It is well known that a bounded linear operator T : L → X is (τ(L , L ), ·X )- continuous if and only if its representing measure mT : → X is λ-continuous (see [17, Proposition 3.1]). Hence due to Swartz (see [24, Theorem 1 and Theorem 3]) we have

∞ ∞ 1 Theorem 2.6 Assume that T : L → Xisa(τ(L , L ), ·X )-continuous linear operator. Then the following statements are equivalent: (i) T is nuclear. (ii) mT has the Radon-Nikodym property with respect to λ.

Combining Theorem 2.5 and Theorem 2.6, we get

Corollary 2.7 For a linear operator T : L∞ → X the following statements are equivalent: (i) Tisτ(L∞, L1)-nuclear. ∞ 1 (ii) Tis(τ(L , L ), ·X )-continuous and nuclear.

As an application of Theorem 2.5 we show that some natural kernel operators on L∞ are τ(L∞, L1)-nuclear. Assume that K1 and K2 are compact Hausdorff spaces and k(·, ·) ∈ C(K2 × K1). Let Bo be the σ -algebra of Borel sets in K1 and λ : Bo →[0, ∞) be a countably additive measure. We will need the following lemma.

∞ Lemma 2.8 For every u ∈ L (λ), the mapping u : K1 s → u(s) k(·, s) ∈ C(K2) is continuous. 123 812 M. Nowak

Proof Let s0 ∈ K1 and ε>0 be given. Then for every t ∈ K2 there exist a neighborhood Vt of t and a neighborhood Wt of s0 such that ε | k(z, s) − k(t, s0)|≤ for all z ∈ Vt , s ∈ Wt . u∞ ,..., ∈ = n := Hence there exist t1 tn K2 such that K2 i=1 Vti . Let us put W n ∈ ≤ ≤ ∈ ∈ i=1 Wti .Fort K2, choose i0 with 1 i0 n such that t Vti . Then for s W, | ( , ) − ( , |≤ ε 0 we have k t s k t s0 u∞ . Hence

u(s) − u(s0)∞ = sup |k(t, s) − k(t, s0)|u∞ ≤ ε. t∈K2

This means that u is continuous.

Note that the Banach space C(K1, C(K2)) can be embedded in the Banach space 1 L (C(K2)) such that with each function from C(K1, C(K2)) is associated its λ- 1 equivalence class in L (C(K2)). ∞ In view of Lemma 2.8 we can deﬁne a kernel operator Tk : L (λ) → C(K2) by ∞ Tk(u) := u(s) k(·, s) dλ(s) for all u ∈ L (λ). K1

∗ For t ∈ K2,letδt (w) := w(t) for all w ∈ C(K2). Then δt ∈ C(K2) and according to the Hille’s theorem (see [DU, Theorem 6, p. 47]), we have ∞ Tk(u)(t) = δt (Tk(u)) = u(s) k(t, s) dλ(s) for all u ∈ L (λ), t ∈ K2. K1

Then ( ) = (1 ) = (·, ) λ( ) ∈ B , mTk A Tk A k s d s for all A o A

1 where the mapping K1 s → k(·, s) ∈ C(K2) belongs to L (C(K2)). Hence for A ∈ Bo, | |( ) = (·, ) λ( ) = | ( , )| λ( ). mTk A k s ∞ d s sup k t s d s A A t∈K2

Hence as a consequence of Theorem 2.5 we have

∞ ∞ 1 Proposition 2.9 The kernel operator Tk : L (λ) → C(K2) is τ(L (λ), L (λ))- nuclear and

Tknuc = sup |k(t, s)| dλ(s). K1 t∈K2 123 Nuclear operators on Banach function spaces 813

3 Application of the theory of Orlicz spaces to vector measures

First we recall terminology and basic facts concerning Orlicz spaces and Orlicz- Bochner spaces (see [19] for more details). By a Young function we mean here a convex continuous mapping ϕ :[0, ∞) →[0, ∞) that vanishes only at 0 and ϕ(t)/t → 0ast → 0 and ϕ(t)/t →∞as t →∞.Byϕ∗ we denote the complementary function of ϕ in the sense of Young, that is, ϕ∗(t) = sup{ts − ϕ(s) : s ≥ 0} for t ≥ 0. Note that ϕ∗∗ = ϕ. The corresponding Orlicz space Lϕ is an ideal of L0 deﬁned by ϕ L := u ∈ L0 : ϕ(α|u (ω)| ) dλ<∞ for some α>0 , and equipped with the topology Tϕ, deﬁned by two equivalent norms: uϕ := inf α>0 : ϕ(|u(ω)| /α)dλ ≤ 1 , 0 ϕ∗ ∗ uϕ := sup |u(ω) v(ω)| dλ : v ∈ L , ϕ (|v(ω)|) dλ ≤ 1 ,

called the Luxemburg norm and the Orlicz norm. Then we have: uϕ ≤ 1 if and only if ϕ(|u(ω)|) dλ ≤ 1, unϕ → 0 if and only if ϕ(α|un(ω)|) dλ → 0 for every α>0, ϕ Eϕ := u ∈ L : ϕ(α | u(ω)|) dλ<∞ for all α>0 = ∈ ϕ :1 → λ( ) → , u L An u ϕ 0if An 0

∗ (Lϕ) = Lϕ . The Orlicz-Bochner space Lϕ(X) is deﬁned by ϕ 0 L (X) := g ∈ L (X) : ϕ(α g(ω)X ) dλ<∞ for some α>0 0 ϕ = g ∈ L (X) :g(·)X ∈ L .

For g ∈ Lϕ(X),let

0 0 gϕ := g(·)X ϕ and gϕ := g(·)X ϕ.

Then

ϕ 0 ϕ E (X) := g ∈ L (X) :g(·)X ∈ E .

Assume that ϕ is Young function. If m : → X is a λ-continuous measure, we write |m|ϕ∗ instead of |m|(Lϕ ) . 123 814 M. Nowak

Lemma 3.1 Assume that m : → Xisaλ-continuous measure and ϕ is Young function. Then for A ∈ , we have

−1 −1 1 |m|(A) ≤ ϕ |m|ϕ∗ (A). λ(A)

Proof ∈ ε> ( )n Let A and 0 be given. Then there is a -partition Ai i=1 of A such that

n − 1 1 |m|(A) ≤ m(A ) + ϕ−1 ε. i X λ(A) i=1 −1 1 = ϕ−1 1 It is known that A ϕ λ(A) (see [RR, Chap. 3.4, Corollary 7, p. 79]). Hence n ϕ−1 1 1 = ϕ−1 1 1 = Ai A 1 λ(A) ϕ λ(A) ϕ i=1 and

n 1 1 ϕ−1 |m|(A) ≤ ϕ−1 m(A ) + ε. λ(A) λ(A) i X i=1 − ϕ 1 1 | |( ) ≤| | ∗ ( ) It follows that λ(A) m A m ϕ A ,so

−1 −1 1 |m|(A) ≤ ϕ |m|ϕ∗ (A). λ(A) ϕ γ [Tϕ, T ] γϕ By 0 Lϕ (in brief, ) we denote the natural mixed topology on L (in the sense of Wiweger), that is, γϕ is the ﬁnest linear topology that agrees with T0 ϕ on Tϕ-bounded sets in L (see [1,25] for more details). Then γϕ is a locally convex- solid Hausdorff topology and γϕ and Tϕ have the same bounded sets. This means that ϕ ϕ (L ,γϕ) is a generalized DF-space (see [20]) and it follows that (L ,γϕ) is quasi- normable (see [20, p. 422]). Hence as a consequence of the Grothendieck’s classical result (see [20, p. 429]), we have

Proposition 3.2 For a linear operator S : Lϕ → X the following statements are equivalent:

(i) S is (γϕ, ·X )-continuous and compact. ϕ (ii) S is γϕ-compact, that is, there exists a γϕ-neighborhood V of 0 in L such that T (V ) is relatively norm compact in X. 123 Nuclear operators on Banach function spaces 815

We say that a Young function ϕ increases essentially more rapidly than another ψ (in symbols, ψ ϕ) if for arbitrary c > 0, ψ(ct)/ϕ(t) → 0ast → 0 and t →∞. Note that Lϕ ⊂ Eψ if ψ ϕ. The following result will be useful (see [16, Theorem 2.1]).

Theorem 3.3 Let ϕ be a Young function. Then γϕ { · ψ : ψ ϕ} (i) is generated by the family of norms Lϕ . ϕ ∗ ϕ∗ ϕ (ii) (L ,γϕ) ={Fv : v ∈ E }, where Fv(u) = u(ω) v(ω) dλ for all u ∈ L . According to [15, Corollary 1.6] we have the following identity: ϕ L1 = E , where ϕ runs over the family of all Young functions. (3.1) ϕ

Now we can state the main result in this section.

Theorem 3.4 Assume that a measure m : → X has the Radon-Nikodym property ∗ with the density g ∈ L1(X). Then there exists a Young function ϕ such that g ∈ Eϕ (X) and the following statements hold: ϕ (i) The operator Sg : L → X deﬁned by ϕ Sg(u) := u(ω) g(ω) dλ for all u ∈ L

is (γϕ, ·X )-continuous and compact. (ii) Sg is γϕ-compact, that is, there exists a Young function ψ with ψ ϕ such that ϕ { u(ω)g(ω) dλ : u ∈ L , uψ ≤ 1} is a relatively norm compact subset of X. 0 (iii) |m|ϕ∗ () < ∞ and |m|ϕ∗ (A) =1Agϕ∗ for every A ∈ . (iv) |m|ϕ∗ is λ-continuous, that is, |m|ϕ∗ (An) → 0 if λ(An) → 0. Proof In view of (3.1) there exists a Young function ψ such that g ∈ Eψ (X), that is, ∗ g(·) ∈ Eψ . Hence |u|g(·) ∈ L1 for every u ∈ Lψ . Let us put ϕ = ψ∗. Then X X∗ ∗ ∗ ϕ∗ = ψ∗∗ = ψ and (Lϕ) = Lϕ = Lψ and Lψ = Lϕ, and so g ∈ Eϕ (X). Then for u ∈ Lϕ,wehave ( ) ≤ | (ω)| (ω) λ = (| |), Sg u X u g X d Fg(·)X u

ϕ∗ ϕ ϕ∗ where g(·)X ∈ E . The inequality shows that Sg is (|σ |(L , E ), ·X )- ∗ continuous, where |σ| denotes the absolute weak topology. Since (|σ |(Lϕ, Eϕ ) is the ∗ coarsest locally convex-solid topology on Lϕ with dual Eϕ , and by Theorem 3.3(ii) γϕ is such a topology, it follows that Sg is (γϕ, ·X )-continuous. To show that Sg is compact, choose a sequence ( fn) of X-valued simple functions on such that fn(ω) − g(ω)X → 0 and fn(ω)X ≤g(ω)X λ-a.e. and for all ∗ n ∈ N (see [7, Theorem 6, p. 4]). Hence for every α>0, we have that ϕ (α fn(ω) − ∗ ∗ g(ω)X ) → 0 λ-a.e. and ϕ (α fn(ω)−g(ω)X ) ≤ ϕ (2αg(ω)X )λ-a.e. and for all 123 816 M. Nowak ∈ N ϕ∗( α (ω) ) λ<∞ n , where 2 g X d . By the Lebesgue dominated convergence ∗ 0 theorem, ϕ (α fn(ω) − g(ω)X ) dλ → 0 and this means that fn − gϕ∗ → 0. ϕ For each n ∈ N,letSn : L → X be a linear operator deﬁned by ϕ Sn(u) := u(ω) fn(ω) dλ for all u ∈ L .

Note that the range of each Sn is contained in the span of ﬁnite set of values of fn. ϕ Therefore Sn is compact and by the Hölder’s inequality for each u ∈ L ,wehave (Sn − Sg)(u)X = u(ω)( fn(ω) − g(ω)) dλ X 0 ≤ |u(ω)|fn(ω) − g(ω)X dλ ≤uϕ fn − gϕ∗ .

It follows that Sn − Sg→0, so Sg is a compact operator. (ii) In view of (i) and Proposition 3.2 Sg is γϕ-compact. Using Theorem 3.3 we obtain that (ii) holds. = k 1 ∈ S() ≤ (iii) Let s i=1 ci Ai and s ϕ 1. Then

k k 0 |ci |m(Ai )X = |ci | g(ω)X dλ = |s(ω)|g(ω)X dλ ≤gϕ∗ i=1 i=1 Ai

| | ∗ () ≤ 0 = and hence m ϕ g ϕ∗ . Note that m mSg . Hence using Lemma 2.1 we have 0 that |mϕ∗ |(A) =1Agϕ∗ for A ∈ . ∗ (iv) This follows from (iii) because g ∈ Eϕ (X).

∞ ϕ ∞ Let i∞ : L → L denotes the inclusion map. Note that i∞ is (σ(L , L1), ϕ ϕ∗ ∞ ϕ ϕ∗ σ(L , L ))-continuous, and it follows that i∞ is (τ(L , L1), τ(L , L ))-continuous ϕ ϕ∗ ∞ (see [8, Theorem 8.6.1]). Since γϕ ⊂ τ(L , L ), we obtain that i∞ is (τ(L , L1), γϕ)- continuous. As a consequence of Theorem 3.4 and Theorem 2.5, we can show that every τ(L∞, L1)-nuclear operator T : L∞ → X admits a factorization through some Orlicz space Lϕ.

Corollary 3.5 Assume that T : L∞ → Xisaτ(L∞, L1)-nuclear operator. Then there exists a Young function ϕ such that T = S ◦ i∞, where ϕ (i) S : L → X is a Bochner representable and γϕ-compact linear operator. (ii) |mS|ϕ∗ (An) → 0 if λ(An) → 0.

123 Nuclear operators on Banach function spaces 817

Proof (i) In view of Theorem 2.5 the representing measure mT : → X has the 1 Radon-Nikodym property with respect to λ, that is, mT = gλ, where g ∈ L (X). Hence according to Theorem 3.4 there exists a Young function ϕ such that g ∈ ∗ Eϕ (X) and an operator S : Lϕ → X deﬁned by

ϕ S(u) := u(ω) g(ω) dλ for all u ∈ L ,

∞ is γϕ-compact. Note that for u ∈ L , T (u) = udmT = u(ω)g(ω) dλ = S(u). ( ) = (ω) λ ∈ (ii) Since mS A A g d for all A , using Theorem 3.4 we have that |mS|ϕ∗ (An) → 0ifλ(An) → 0.

Acknowledgements The author wishes to express his thanks to the referee for remarks and suggestions leading to an improvement of the paper.

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