COMMENTATIONES MATHEMATICAE Vol. 48, No. 2 (2008), 113-119

Marian Nowak

Bochner representable operators on K¨othe-Bochner spaces

Abstract. Let E be a Banach function space and X be a real . We study Bochner representable operators from a K¨othe-Bochner space E(X) to a Banach space Y . We consider the problem of compactness and weak compactness of Bochner representable operators from E(X) (provided with the natural mixed topology) to Y . 2000 Mathematics Subject Classification: 47B38, 47B05, 46E40, 46A70. Key words and phrases: K¨othe-Bochner spaces, Generalized DF-spaces, Bochner rep- resentable operators, Weakly compact operators, Compact operators, Mixed topolo- gies, Mackey topologies.

1. Introduction and preliminaries. The representation of linear operators on function spaces (in particular, Lebesgue spaces and Orlicz spaces) has been the object of much study (see [Ph], [DP], [G1], [G2], [DS], [D], [U1], [U2], [A1], [A2], [Na1], [Na2]). In particular, K. Andrews [A1, Theorems 2 and 5], [A2, Theorem 3] and V.G. Navodnov [Na2, Corollary] obtained a Dunford-Pettis-Phillips type theorem for compact operators (resp. weakly compact operators) from the space L1(X) of Bochner integrable functions to a Banach space Y. Moreover, V.G. Navodnov ([Na2], [Na1]) has considered Bochner representable operators from a K¨othe-Bochner space E(X) to a Banach space Y . In Section 2 we study the problem of compactness and weak compactness of Bochner representable operators from a K¨othe-Bochner space E(X) (provided with the natural mixed topology γE(X) ) to a Banach space Y . The space (E(X), γE(X)) is a generalized DF-space, so we can apply the Grothendieck’s DF techniques (see [G1], [G2], [Ru]). Throughout the paper (X, ) and (Y, ) are real Banach spaces k · kX k · kY with the Banach duals X∗ and Y ∗. Let BX and BY denote the closed unit balls in X and Y . Let (X,Y ) stand for the space of all bounded linear operators L 114 Bochner representable operators on K¨othe-Bochner spaces

from X to Y provided with the uniform convergence X Y . The strong operator topology (briefly SOT) is the topology on (X,Y ) definedk · k → by the family of seminorms p : x X , where p (U) := U(x)L for U (X,Y ) (see [DS, { x ∈ } x k kY ∈ L p. 475–477]). Let N stand for the set of natural numbers. Let (Ω, Σ, µ) be a complete finite space. Assume that (E, ) k · kE is a Banach function space and let E0 stand for the K¨othedual of E. Then the associated norm on E0 can be defined for v E0 by k · kE0 ∈

v E0 = sup u(ω) v(ω) dµ : u E, u E 1 . k k Ω ∈ k k ≤  Z 

A Banach function space (E, E) is said to be perfect, if E = E00 and u = u for u E. It is wellk · known k that (E, ) is perfect if and only k kE00 k kE ∈ k · kE if E satisfies both the σ-Fatou property and the σ-Levy property (see [KA, Theoremk · k 6.1.7]). 1 From now we will assume that L∞ E L , where the inclusion maps ⊂ ⊂ are continuous. Moreover, we will assume that E0 on E0 is order continuous. 1 1 k · k Since (L )0 = L∞, the space L is excluded. By L0(X) we denote the set of µ-equivalence classes of all strongly Σ- measurable functions f :Ω X. Let (X) stand for the topology on L0(X) → T0 of the F -norm L0(X) that generates convergence in measure on sets of finite measure. k · k For f L0(X) let f(ω) = f(ω) for ω Ω. Then the space ∈ k kX ∈ E(eX) = f L0(X): f E ∈ ∈  equipped with the norm f E(X) := f E ise a Banach space and is usually called a K¨othe-Bochner spacek(seek [CM], [L]k k for more details). We will denote by E(X) the topology of the norm E(X). Recalle that the algebraic tensor product ET X is the subspace of E(X)k·k spanned by the functions of the form u x, (u ⊗ x)(ω) = u(ω)x, where u E, x X and ω Ω. For r > 0 denote ⊗ ⊗ ∈ ∈ ∈ B (r) = f E(X): f r . E(X) ∈ k kE(X) ≤  Let τ be a linear topology on E(X). A linear operator T : E(X) Y → is said to be (τ, Y )-compact (resp. (τ, Y )-weakly compact) if there exists a neighbourhood Uk ·of k 0 for τ such that Tk( ·U k) is relatively norm compact (resp. relatively weakly compact) in Y . By Bd(E(X), τ) we will denote the collection of all τ-bounded subsets of E(X).

2. Bochner representable operators on K¨othe-Bochner spaces. We start by recalling terminology concerning Bochner representable operators T : E(X) Y (see [A ], [A ], [Na ], [Na ] for more details). → 1 2 1 2 A function :Ω (X,Y ) is said to be SOT-measurable if for every x X the functionK :Ω→ Lω (ω)(x) Y is strongly Σ-measurable. We ∈ Kx 3 7→ K ∈ M. Nowak 115

say that two SOT-measurable functions 1 and 2 are SOT-equivalent (briefly, ) if (ω)(x) = (ω)(x) for allK x XKand µ-almost all ω Ω (see K1 ≈ K2 K1 K2 ∈ ∈ [Na1], [Na2]).

Recall that a bounded linear operator T : E(X) Y is said to be Bochner representable if there exists a SOT-measurable function→ :Ω (X,Y ) (called the representing kernel for T ) such that for each f E(KX) the→ function L f, : Ω ω f(ω), (ω) Y is Bochner integrable and∈ h Ki 3 7→ h K i ∈ T (f) = f(ω), (ω) dµ for all f E(X). h K i ∈ ZΩ The following theorem will be of importance (see [Na1, Theorem 1]).

Theorem 2.1 For a SOT-measurable function :Ω (X,Y ) the following statements are eqiuvalent: K → L (i) is the representing kernel for a Bochner representable operator T : E(X) YK. → (ii) There exists a SOT-measurable function :Ω (X,Y ) such that K0 → L K0 ≈ K and the µ-equivalence class of the function 0( ) X Y belongs to E0. kK · k →

Recall that the mixed topology γ[ E(X), 0(X) E(X)] (briefly γE(X)) on E(X) is the finest Hausdorff locally convexT topologyT on E(X) which agrees with 0(X) on E(X)-bounded subsets of E(X) (see [C, Chap. III], [N2], [F, 3] for moreT details).k · k Then § (X) γ (X). T0 E(X) ⊂ E(X) ⊂ TE Since BE(X)(1) is closed in (E(X), 0(X) E(X)) (see [KA, Lemma 4.3.4]), by [W, Theorem 2.4.1] we get T (1) Bd (E(X), γ ) = Bd (E(X), ). E(X) k · kE(X)

This means that (E(X), γE(X)) is a generalized DF-space (see [Ru]). Hence using the Grothendieck classical results (see [Ru, p. 429], [G1, Corollary 1 of The- orem 11], [G2, Chap. IV, 4.3, Corollary 1 of Theorem 2]) we get:

Theorem 2.2 Let T : E(X) Y be a (γ , )-continuous linear operator → E(X) k · kY which transforms γE(X)-bounded sets into relatively norm compact (resp. relatively weakly compact) sets in Y . Then T is (γE(X), Y )-compact (resp. ((γE(X), )-weakly compact). k · k k · kY

A linear operator T : E(X) Y is called (γ, )-linear if T (f ) 0 → k · kY k n kY → whenever f 0 0 and sup f < (see [W]). It is known that a k nkL (X) → n k nkE(X) ∞ linear operator T : E(X) Y is (γ, Y )-linear if and only if T is (γE(X), Y )- continuous (see [W, Theoem→ 2.6.1(iii)]).k·k k·k 116 Bochner representable operators on K¨othe-Bochner spaces

Proposition 2.3 Assume that (E, ) is a perfect Banach function space such k · kE that the associated norm E0 on E0 is order continuous. Then every Bochner representable operator T :kE ·( kX) Y is (γ , )-continuous. → E(X) k · kY Proof Assume that T : E(X) Y is a Bochner representable operator. Then, by Theorem 2.1, there exists a SOT-measurable→ function :Ω (X,Y ) such K0 → L that the µ-equivalence class v0 = [ 0( ) X Y ] belongs to E0 and k K · k → T (f) = f(ω), (ω) dµ for all f E(X). h K0 i ∈ ZΩ Hence for f E(X) we get ∈

T (f) Y = f(ω), 0(ω) dµ k k Ωh K i Y Z

f(ω), (ω) dµ ≤ k h K0 ikY ZΩ

f(ω) X 0(ω) X Y dµ . ≤ k k · k K k → ZΩ Putting ϕ (u) = u(ω) v (ω) dµ for u E, v0 0 ∈ ZΩ we see that ϕv0 is a γ-linear functional on E (see [N1, Theorem 3.1]). It follows that T is a (γ, )-linear operator, i.e., T is (γ , )-continuous. k · kY E(X) k · kY Before stating our main result we require a preliminary definition (see [A1, p. 258]). A SOT-measurable function :Ω (X,Y ) is said to have its essential range in the uniformly norm compact operatorsK →(resp. L uniformly weakly compact op- erators) if there exists a relatively norm compact (resp. relatively weakly compact) set C in Y such that x, (ω) C for µ-almost all ω Ω and all x B . h K i ∈ ∈ ∈ X First we recall the well known Dunford-Pettis-Phillips type theorem for compact operators (resp. weakly compact operators) from the space L1(X) to a Banach space Y (see [Na2, Corollary]).

Theorem 2.4 For a bounded linear operator T : L1(X) Y the following state- ments are equivalent: →

(i) T is ( 1 , )-compact (resp. ( 1 , )-weakly compact ). k · kL (X) k · kY k · kL (X) k · kY (ii) T is Bochner representable operator with the representing kernel having its essential range in the uniformly norm compact operators (resp. uniformlyK weakly compact operators).

Now we are in position to extend and strengthen the implication (ii)= (i) of Theorem 2.4 for the case of linear operators from a K¨othe-Bochner space E⇒(X) to a Banach space Y . M. Nowak 117

Theorem 2.5 Assume that (E, ) is a perfect Banach function space such k · kE that the associated norm E0 on E0 is order continuous. Let T : E(X) Y be a Bochner representablek · k operator with the representing kernel having its→ essential range in the uniformly norm compact operators (resp. uniformlyK weakly compact operators). Then T is (γE(X), Y )-compact (resp. (γE(X), Y )-weakly compact ). k·k k·k

Proof In view of Proposition 2.3 T is (γE(X), Y )-continuous. Since Bd (E(X), γ ) = Bd (E(X), ) (see (1)), byk · Theorem k 2.2 it is enough E(X) k · kE(X) to show that for every r > 0, the set T (BE(X)(r)) is relatively norm compact (resp. relatively weakly compact) in Y . Indeed, let r > 0. Since the inclusion map 1 (E, E) , (L , L1 ) is supposed to be continuous, there exists r0 > 0 such k · k → k · k 1 that BE(X)(r0) BL1(X)( 2 ). Moreover,⊂ by our assumption there exists a relatively norm compact (resp. relatively weakly compact) set C in Y such that x, (ω) C for µ-almost all ω Ω and all x B . Now we shall show thath TK(f) i ∈conv C for all f ∈ ∈ X ∈ ∈ BE(X)(r0), where conv C stands for the norm closed (=weakly closed) convex hull of C in Y . Indeed, let f BE(X)(r0). Then by [DS, p. 117] there exists a sequence (s ) of X-valued Σ-simple∈ functions such that s (ω) 2 f(ω) µ-a.e. on n k n kX ≤ k kX Ω and sn f in µ-measure. Then sn f L0(X) 0 and supn sn E(X) 2 f →2 r . Hence T (s ) T (f)k − k0, because−→ T is (γ, k )-linear.k ≤ k kE(X) ≤ 0 k n − kY −→ k · kY kn 1 Since sn = i=1 An,i xn,i and sn BE(X)(2 r0) for n N, we get kn ⊗ ∈ ∈ s B 1 (1), i.e., x µ(A ) 1 for n . Hence, using [DU, n L (X) P i=1 n,i X n,i N Corollary∈ 2.2.8, p. 48] we getk k ≤ ∈ P

kn T (s ) = T (1 x ) n An,i ⊗ n,i i=1 X kn = x , (ω) dµ h n,i K i i=1 ZAn,i Xk n x = x n,i , (ω) dµ k n,ikX x K i=1 An,i n,i X X Z Dk k E kn x µ(A ) conv C conv C. ∈ k n,ikX n,i ⊂ i=1  X 

Hence T (f) conv C, that is, T (BE(X)(r0)) is relatively norm compact (resp. relatively weakly∈ compact) in Y , because conv C is norm compact in Y by Mazur’s theorem (resp. conv C is weakly compact in Y by Krein-Smulian’sˇ theorem). Since T (B (r)) r conv C , we obtain that T (B (r)) is relatively norm compact E(X) r0 E(X) (resp. relatively⊂ weakly compact) in Y , as desired.

Now we will consider Bochner representable operators on the space L∞(X). By a Young function we mean here a continuous convex mapping Φ : [0, ) [0, ) that vanishes only at 0 and Φ(t)/t 0 as t 0 and Φ(t)/t as t∞ →. ∞ → → → ∞ → ∞ 118 Bochner representable operators on K¨othe-Bochner spaces

The Orlicz-Bochner space LΦ(X) := f L0(X): Φ(λ f(ω) )dµ < { ∈ Ω k kX ∞ for some λ > 0 is a Banach space under the norm f LΦ(X) := inf λ > 0 : Φ( f(ω) /λ)}dµ 1 (see [RR] for more details). Rk k { Ω k kX ≤ } R We will need the following characterization of the mixed topology γL∞(X) on L∞(X) (see [N2, Theorem 4.5]).

Theorem 2.6 Then mixed topology γL∞(X) on L∞(X) is generated by the family of norms Φ , where Φ runs over the family of all Young functions. k · kL (X)

As a consequence of Theorems 2.5 and 2.6 we get:

Corollary 2.7 Let T : L∞(X) Y be a Bochner representable operator and assume that the representing kernel→ for T has its range in the uniformly norm compact operators (resp. uniformly weaklyK compact operators). Then there exists a Young function Φ such that the set

f(ω), (ω) dµ : f L∞(X), f Φ 1 h K i ∈ k kL (X) ≤  ZΩ  is relatively norm compact (resp. relatively weakly compact ) in Y .

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Marian Nowak Faculty of Mathematics, Computer Science and Econometrics ul. Szafrana 4A, 65–516 Zielona Gora,´ Poland E-mail: [email protected]

(Received: 5.02.2008)