On interpolation of Asplund operators

Math. Z. 250, 267–277 (2005)

Mathematische Zeitschrift

DOI: 10.1007/s00209-004-0748-7

On interpolation of Asplund operators Fernando Cobos1 , Luz M. Fern´andez-Cabrera2 , Antonio Manzano3 , Ant´on Mart´ınez4 1 2

3 4

Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad Complutense de Madrid, 28040 Madrid, Spain (e-mail: [email protected]) Secci´on Departamental de Matem´atica Aplicada, Escuela de Estad´ıstica, Universidad Complutense de Madrid, 28040 Madrid, Spain (e-mail: luz [email protected]) Departamento de Matem´aticas y Computaci´on, Escuela Polit´ecnica Superior, Universidad de Burgos, 09001 Burgos, Spain (e-mail: [email protected]) Departamento de Matem´atica Aplicada, E.T.S. Ingenieros Industriales, Universidad de Vigo, 36200 Vigo, Spain (e-mail: [email protected])

Received: 20 February 2003 / Published online: 7 January 2005 – © Springer-Verlag 2005

Abstract. We study the interpolation properties of Asplund operators by the complex method, as well as by general J - and K-methods. Mathematics Subject Classification (2000): 46B70, 47B10 1. Introduction Let A, B be Banach spaces. A bounded linear operator T ∈ L(A, B) is said to be a Radon-Nikod´ym operator if for any probability measure µ, T maps each µ-continuous A-valued measure of finite variation into a µ-differentiable B-valued measure (see [22] and [11]). An operator T ∈ L(A, B) is called an Asplund operator if T ∗ is a Radon-Nikod´ym operator. Asplund operators have been studied widely. See, for example, the papers by Edgar [12], Stegall [24] and the book by Diestel, Jarchow and Tonge [10]; see also the book by Pietsch [22] and the paper by Heinrich [15] where they are referred to as decomposing operators. Asplund operators A form a closed injective and surjective operator ideal in the sense of Pietsch [22]. A Banach space A is said to be an Asplund space if its identity operator IA belongs to A. This class of Banach spaces has attracted considerable attention in recent years. It originated in the work of Asplund [1] where these spaces are called strong differentiability space. In fact, a Banach space A is Asplund if every continuous convex function on A is Fr´echet differentiable at each point of a dense Gδ subset of A.Any reflexive space isAsplund. Further, c0 and some other non-reflexive spaces are Asplund, but 1 and ∞ are not Asplund. We refer to the books by Giles

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[14], Bourgin [3] and Fabian [13] for further details and relevant references. Let us only point out here that every closed subspace of an Asplund space is also Asplund, and that every Asplund operator factors through an Asplund space. This last result is due to Heinrich [15], Cor. 2.4, and independently to Reˇınov [23] and Stegall [24], Thm. 1.14. The proofs of Heinrich and Stegall use certain interpolation techniques based on ideas developed by Davis, Figiel, Johnson and Pelczy´nski in their famous paper on the factorization property of weakly compact operators [9]. More precisely, focus on Heinrich’s paper, first he showed that if the embedding i : A0 ∩ A1 −→ A0 + A1 is an Asplund operator, then the real interpolation space (A0 , A1 )θ,q is Asplund for 0 < θ < 1, 1 < q < ∞, and from this result he derived easily the factorization property of Asplund operators. In this paper we continue the research on interpolation properties of Asplund spaces and Asplund operators. In Section 2 we prove that the complex interpolation space (A0 , A1 )[θ] is Asplund when any of A0 or A1 has this property. However, similarly to the case of weak compactness, it is not true in general that if i : A0 ∩ A1 −→ A0 + A1 is Asplund then (A0 , A1 )[θ] is Asplund. Afterwards, in Section 3, we investigate the behaviour of Asplund operators under general J - and K-methods. Many authors have worked on these interpolation methods, which are extensions of the real method (see, for example, Peetre [20], Brudnyˇı and Krugljak [4], Cwikel and Peetre [8] and Nilsson [19]). These methods arise when one wants to describe all interpolation spaces with respect to the couple (L1 , L∞ ). The usual weighted Lq norm of the real method is not enough and one should work with general lattice norms. Here we prove that if the lattice  that defines the methods is an Asplund space and T : A0 ∩ A1 −→ B0 + B1 is an Asplund operator, then the interpolated operators by the K- and the J -method are Asplund operators as well. We end the paper by returning to the complex method to show that if T : A0 −→ B0 is Asplund and T : A1 −→ B1 is bounded, then T : (A0 , A1 )[θ] −→ (B0 , B1 )[θ] is Asplund. 2. Complex interpolation ¯ be the space of all functions f Let A¯ = (A0 , A1 ) be a Banach couple and let F(A) from the closed strip S = {z ∈ C : 0 ≤ Re z ≤ 1} into A0 + A1 such that (a) f is bounded and continuous on S and analytic on the interior of S, and (b) the functions t −→ f (j + it) (j = 0, 1) are continuous from R into Aj and tend to zero as |t| → ∞.
 We put f F = maxj =0,1 supt∈R f (j + it)Aj . The complex interpolation space (A0 , A1 )[θ] (where 0 < θ < 1) consists of all a ∈ A0 + A1
such that ¯ a = f (θ) for some f ∈ F(A). The norm of (A0 , A1 )[θ] is a[θ] = inf f F :  f (θ) = a , f ∈ F(A0 , A1 ) . Full details on complex interpolation can be found in [2] and [25]. We only recall that A0 ∩ A1 is dense in (A0 , A1 )[θ] , and that, if A0 ∩ A1 is dense in A0 and in A1 , then the dual space (A0 , A1 )∗[θ] of (A0 , A1 )[θ] coincides with the upper complex space (A∗0 , A∗1 )[θ] .

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269

In order to determine whether (A0 , A1 )[θ] is Asplund, we shall need the following characterization of Asplund operators. Given any Banach space A, we denote by UA the closed unit ball of A. If D ⊆ A is a bounded set, let µD denote the seminorm on A∗ given by µD (f ) = sup {|f (x)| : x ∈ D} ,

f ∈ A∗ .

Theorem 2.1 (see [3], Thm. 5.2.11). Let T ∈ L(A, B) be a bounded linear operator from the Banach space A into the Banach space B. Then T is Asplund if and only if the seminormed space (B ∗ , µT (D) ) is separable whenever D ⊆ UA is countable. Now we are ready to describe the behaviour of Asplund spaces under complex interpolation. Theorem 2.2. Let A¯ = (A0 , A1 ) be a Banach couple and let 0 < θ < 1. Then the complex interpolation space (A0 , A1 )[θ] is Asplund whenever A0 or A1 is Asplund. Proof. Let A◦j be the closure of A0 ∩ A1 in Aj . Since (A0 , A1 )[θ] = (A◦0 , A◦1 )[θ] and any closed subspace of an Asplund space is Asplund, without loss of generality we may assume that A0 ∩ A1 is dense in A0 and in A1 . Hence (A0 , A1 )∗[θ] = (A∗0 , A∗1 )[θ] = (A∗0 , A∗1 )[θ] . The last equality holds because A∗j has the Radon-Nikod´ym property for j = 0 or j = 1 (see [21]). In particular, we get that (A0 + A1 )∗ = A∗0 ∩ A∗1

is dense in

(A0 , A1 )∗[θ] .

(2.1)

On the other hand, in the following diagram (A0 , A1 )[θ]

A0  Tw TTT * A1

A0 + A 1 '  jjjj4

any of these two embeddings is Asplund. Whence, by [15], Prop. 1.7, we obtain that the embedding i : (A0 , A1 )[θ] −→ A0 + A1

is Asplund.

(2.2)

Consequently, given any D ⊆ U(A0 ,A1 )[θ ] with D countable, it follows from (2.2)   that  (A0 + A1 )∗ , µD is separable. Combining this fact with (2.1), we conclude that (A0 , A1 )∗[θ ] , µD is separable. Using Theorem 2.1, this implies that (A0 , A1 )[θ] is Asplund. 

Remark 2.1. If we only assume that the embedding i : A0 ∩ A1 −→ A0 + A1 is Asplund, then it is not true in general that (A0 , A1 )[θ] is Asplund. Indeed, Mastylo has constructed in [18], page 161, a couple of Lorentz spaces (ϕ , ψ ) such that the embedding i : ϕ ∩ ψ −→ ϕ + ψ is weakly compact and so it is Asplund, but (ϕ , ψ )[θ ] contains a subspace isomorphic to 1 and therefore (ϕ , ψ )[θ] is not an Asplund space. We will come back to Theorem 2.2 at the end of the next section and we will ¯ B¯ and any operator T with show that it can be extended to general couples A, T : A0 −→ B0 Asplund and T : A1 −→ B1 bounded.

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3. Real interpolation In this section we will show that the behaviour under interpolation of Asplund operators improves when one works with the extensions of the real method. By a Z-lattice  we mean a Banach space of real valued sequences with Z as index set, such that it contains all sequences with only finitely many non-zero coordinates, and moreover  satisfies that whenever |ξm | ≤ |µm | for each m ∈ Z and {µm } ∈ , then {ξm } ∈  and {ξm } ≤ {µm } . A Z-lattice  is said to be regular if for any {τn }n∈N ⊆  with τn ↓ 0 it follows that τn  −→ 0 . The associated space  of  is formed by all sequences {ηm } for which  ∞   {ηm } = sup |ηm ξm | : {ξm } ≤ 1 < ∞. m=−∞

The space  is also a Z-lattice. The Z-lattice  is said to be K-non-trivial if {min(1, 2m )} ∈ . It is called J -non-trivial if  ∞   −m sup min(1, 2 )|ξm | : ξ  ≤ 1 < ∞. m=−∞

Let A¯ = (A0 , A1 ) be a Banach couple and let  be a K-non-trivial Z-lattice. The K-space A¯ ;K = (A0 , A1 );K consists of all a ∈ A0 + A1 such that {K(2m , a)} ∈ . Here K(2m , a) = K(2m , a; A0 , A1 ) = inf{a0 A0 + 2m a1 A1 : a = a0 + a1 , aj ∈ Aj }. The norm in (A0 , A1 );K is given by aA¯ ;K = {K(2m , a)} . If  is J -non-trivial, the J -space A¯ ;J = (A0 , A1 );J consists of all sums a = ∞  um (convergence in A0 + A1 ) where {um } ⊆ A0 ∩ A1 and {J (2m , um )} ∈ . m=−∞

Here J (2m , um ) = J (2m , um ; A0 , A1 ) = max{um A0 , 2m um A1 }. We put

 aA¯ ;J = inf {J (2m , um )} : a =

∞ 

 um .

m=−∞

Spaces A¯ ;K and A¯ ;J are Banach spaces. ¯ B) ¯ to mean Let B¯ = (B0 , B1 ) be another Banach couple. We write T ∈ L(A, that T is a linear operator from A0 + A1 into B0 + B1 whose restriction to each Aj defines a bounded operator from Aj into Bj (j = 0, 1). We put T A, ¯ B¯ =  max T A0 ,B0 , T A1 ,B1 .

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271

¯ B), ¯ then it is clear that the restriction of T to A¯ ;K defines a If T ∈ L(A, bounded operator T : A¯ ;K −→ B¯ ;K with T A¯ ;K ,B¯ ;K ≤ T A, ¯ B¯ . A similar estimate holds for the J -method. In other words, K- and J -method are exact interpolation methods. If  is K- and J -non-trivial Z-lattice, then A¯ ;K → A¯ ;J . But it is not true in general that A¯ ;K coincides with A¯ ;J . We refer to [19] and [4] for full details on these interpolation methods. For  = q (2−θm ), the space q with the weight {2−θm }, K- and J -methods coincide with the real method (A0 , A1 )θ,q = (A0 , A1 )q (2−θ m );K = (A0 , A1 )q (2−θ m );J

(see [2] and [25]).

Here 0 < θ < 1, 1 ≤ q ≤ ∞. In a more general way, if f is a function parameter and  = q (1/f (2m )) then we recover the real method with a function parameter (A0 , A1 )f,q = (A0 , A1 )q (1/f (2m ));K = (A0 , A1 )q (1/f (2m ));J

(see [20] and [16]).

For t > 0, let tR be R with the norm λt R = t|λ|. The characteristic function ϕK of the K-method is defined by (R, (1/t)R);K = (1/ϕK (t))R

(see [16]).

The characteristic function ϕJ of the J -method is defined analogously. It turns out that ϕK (t) = {min(1, 2m /t)}−1  and  ∞   m ϕJ (t) = sup min(1, t/2 ) |ξm | : {ξm } ≤ 1 , t > 0, m=−∞

(see [6], Lemma 2.4). Characteristic functions are quasiconcave. Later on we will need to know the behaviour of these functions at 0 and ∞. We write ϕ ∈ P0 if min{1, 1/t}ϕ(t) −→ 0

as

t →0

or

as

t → ∞.

We put ϕ ∗ (t) = 1/ϕ(1/t) . We also recall that the functions (see [5]) ψA¯ ρA¯

;J

(t) = sup{K(t, a) : aA¯ ;K = 1},

;K

(t) = inf{J (t, a) : a ∈ A0 ∩ A1 , aA¯ ;J = 1},

are related with the characteristic functions by the inequalities ψA¯

;K

ρ ∗¯

A;J

(t) ≤ ϕK (t) for all

t > 0,

(3.1)

(t) ≤ ϕJ∗ (t) for all

t > 0,

(3.2)

(see [6], Lemma 2.1). Given any sequence of Banach spaces {Em }, we put
  (Em ) = {xm } : xm ∈ Em and {xm }(Em ) =  xm Em  < ∞ . We denote by Qk : (Em ) −→ Ek the projection Qk {xm } = xk , and by Pr : r x} where δ r is the Kronecker delta. Er −→ (Em ) the embedding Pr x = {δm m In the following results we use again the characterization of Asplund operators given in Theorem 2.1.

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Theorem 3.1. Let  be a K-non-trivial Asplund Z-lattice with ϕK ∈ P0 , let A¯ = ¯ B). ¯ Then T : (A0 , A1 ), B¯ = (B0 , B1 ) be Banach couples and let T ∈ L(A, ¯ ¯ A;K −→ B;K is Asplund if and only if T : A0 ∩ A1 −→ B0 + B1 is Asplund. Proof. Factorization A0 ∩ A1



/ A¯ ;K

 / B¯ ;K 

T

/ B0 + B 1

yields that T : A0 ∩A1 −→ B0 +B1 is Asplund if T : A¯ ;K −→ B¯ ;K is Asplund. Conversely, assume that T : A0 ∩ A1 −→ B0 + B1 is Asplund. Since ϕK ∈ P0 it follows from (3.1) that lim ψA¯

t→0

;K

(t) = 0 = lim (1/t)ψA¯ t→∞

;K

(t).

Then, by [7], Thm. 3.3, we get that T : A¯ ;K −→ B0 + B1 is Asplund. Let Fm be the space B0 + B1 normed by K(2m , ·), m ∈ Z, and let T : A¯ ;K −→ (Fm ) be the operator defined by T a = {· · · , T a, T a, T a, · · · }. Since any two norms of the family {K(2m , ·)} are equivalent on B0 + B1 , and since Qm T = T , we have that Qm T : A¯ ;K −→ Fm is Asplund for each m ∈ Z. We claim that T : A¯ ;K −→ (Fm )

is Asplund.

(3.3)

Indeed, let D ⊆ UA¯ ;K be a countable set. For each m ∈ Z, write Wm = Qm T (D). We know that (Fm∗ , µWm ) is separable. Let m be a countable set that is dense in (Fm∗ , µWm ). To establish (3.3) it suffices to show that the countable set  N   = Pk gk : gk ∈ k , N ∈ N 

k=−N

)∗ , µ



. Since  is Asplund,  does not contain a copy either is dense in (Fm T (D) of ∞ or 1 . Hence, by [26], Thms. 117.3 and 117.2, the spaces  and  are regular. By regularity of  and [18], Prop. 3.1, we get that the dual space (Fm )∗ of (Fm ) is isometrically isomorphic to  (Fm∗ ). Take any h ∈ (Fm )∗ =  (Fm∗ ) and put hm = Qm h ∈ Fm∗ . Usingthat  is regular, given any ε > 0, we can find    N ∈ N such that h − Pk hk  ∗ ≤ ε/2T A, ¯ B¯ . Choose gk ∈ k such that |k|≤N

 (Fm )

µWk (hk − gk ) ≤ ε/(4N + 2). Then



   µT (D) h − Pk gk ≤ µT (D) h − Pk h k + µT (D) (Pk (hk − gk )) |k|≤N

     − P h ≤ T A, h ¯ B¯ k k |k|≤N

|k|≤N  (Fm∗ )

+



|k|≤N

|k|≤N

µWk (hk − gk ) ≤ ε.

This establishes (3.3). The operator T can be factorized as T = j T , where j : B¯ ;K −→ (Fm ) is the isometric embedding defined by j b = {· · · , b, b, b, · · · }. Using that Asplund operators forms an injective operator ideal, it follows from (3.3) that T : A¯ ;K −→ B¯ ;K is Asplund and completes the proof. 

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273

The corresponding result for J -spaces reads: Theorem 3.2. Let  be a J -non-trivial Asplund Z-lattice with ϕJ∗ ∈ P0 , let A¯ = ¯ B). ¯ Then T : (A0 , A1 ), B¯ = (B0 , B1 ) be Banach couples and let T ∈ L(A, A¯ ;J −→ B¯ ;J is Asplund if and only if T : A0 ∩ A1 −→ B0 + B1 is Asplund. Proof. Clearly, if T : A¯ ;J −→ B¯ ;J is Asplund then T : A0 ∩ A1 −→ B0 + B1 is Asplund because A0 ∩ A1 → A¯ ;J and B¯ ;J → B0 + B1 . Conversely, assume that T : A0 ∩ A1 −→ B0 + B1 is Asplund. Since ϕJ∗ ∈ P0 , (3.2) implies that ρ ∗¯ ∈ P0 . Using [7], Thm. 3.1, we obtain that T : A0 ∩ A1 −→ A;J

B¯ ;J is Asplund. Moreover, since  is an Asplund space, we know by [26], Thms. 117.3 and 117.2, that  and  are regular. Let Gm be the space A0 ∩ A1 endowed with the norm J (2m , ·), m ∈ Z, and let T : (Gm ) −→ B¯ ;J be the operator given ∞     by T {um } = T um . For each m ∈ Z, the operator TPm = T : Gm −→ m=−∞

B¯ ;J is Asplund because any two norms of the family {J (2m , ·)} are equivalent on A0 ∩ A1 . We claim that T : (Gm ) −→ B¯ ;J

is Asplund.

(3.4)

Indeed, take any D ⊆ U(Gm ) countable.  For each m ∈ Z, put Dm = Qm (D) ⊆ ∗ ), µ UGm . We have that T∗ (B¯ ;J Pm (Dm ) is separable, so there is a countable set m that is dense. Put     = Pk Qk gk : gk ∈ k , N ∈ N .   |k|≤N

We set  ⊆  (G∗m ) = (Gm )∗ is dense in  ∗ are∗going to show that the countable ∗ ¯ ¯  T (B;J ), µD . Take any f ∈ B;J and any ε > 0. Using the regularity of  we can find N ∈ N such that       Pk Qk f T ∗ ≤ ε/2. f T −  (Gm )

|k|≤N

Now, for each |k| ≤ N , choose gk ∈ k such that µPk Qk (D) (f T−gk ) ≤ ε/(4N +2). Then we obtain

 µD f T − P k Q k gk |k|≤N

     Pk Qk f T ≤ f T − |k|≤N

≤ ε/2 +



|k|≤N

This proves (3.4).

 (G∗m )

+

 |k|≤N

µPk Qk (D) (f T − gk ) ≤ ε.

µD (Pk Qk (f T − gk ))

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The operator T is the composition of T with the metric surjection π :(Gm ) −→ ∞  A¯ ;J defined by π {um } = um . Consequently, using surjectivity of Asplund m=−∞

operators, we derive that T : A¯ ;J −→ B¯ ;J is Asplund.



Corollary 3.1. Let  be an Asplund Z-lattice and let A¯ = (A0 , A1 ) be a Banach couple with the embedding i : A0 ∩ A1 −→ A0 + A1 being an Asplund operator. (i) If  is K-non-trivial with ϕK ∈ P0 , then the space A¯ ;K is Asplund. (ii) If  is J -non-trivial with ϕJ∗ ∈ P0 , then the space A¯ ;J is Asplund. If one of the spaces in the couple is Asplund, then we can weaken the assumptions on ϕK and ϕJ . Indeed, repeating the proofs of Theorems 3.1 and 3.2 but using now [5], Thms. 3.1 and 3.2 instead of [7], Thms. 3.3 and 3.1, we obtain: Corollary 3.2. Let  be an Asplund Z-lattice and let A¯ = (A0 , A1 ) be a Banach couple. Assume that A0 is Asplund. (i) If  is K-non-trivial and lim ϕK (t)/t = 0, then the space A¯ ;K is Asplund. t→∞ (ii) If  is J -non-trivial and lim t/ϕJ (t) = 0, then the space A¯ ;J is Asplund. t→0

The behaviour at 0 and ∞ of the functions ϕK , ϕJ can be controlled by the norms of shift operators on . For k ∈ Z, the shift operator τk is defined by τk {ξm }m∈Z = {ξm+k }m∈Z . It turns out (see [6], Lemma 2.5) that if 2−n τn , −→ 0

and τ−n , −→ 0

as

n → ∞,

then ϕK ∈ P0 and ϕJ∗ ∈ P0 . In particular, these conditions are satisfied for  = q (2−θm ) or  = q (1/f (2m )), f being a function parameter. Writing down Corollary 3.1 for  = q (2−θm ) with 0 < θ < 1 and 1 < q < ∞, we recover a result of Heinrich [15], Cor. 2.5/(ii): Corollary 3.3. Let 0 < θ < 1, 1 < q < ∞ and let A¯ = (A0 , A1 ) be a Banach couple. Then the following are equivalent: (a) (A0 , A1 )θ,q is Asplund. (b) The embedding i : A0 ∩ A1 −→ A0 + A1 is an Asplund operator. As we have seen in Remark 2.1, a similar result to Corollary 3.3 does not hold for the complex method. The next result refers to the limit cases q = 1 and q = ∞. Proposition 3.1. Let 0 < θ < 1 and q = 1 or q = ∞. Let A¯ = (A0 , A1 ) be a Banach couple. Then the following are equivalent: (a) (A0 , A1 )θ,q is Asplund. (b) The embedding i : A0 ∩ A1 −→ A0 + A1 is an Asplund operator and its range is closed.

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Proof. If (A0 , A1 )θ,q is Asplund then it does not contain a subspace isomorphic to q (because q = 1 or q = ∞) and so, using [17], Thm. 1, we have that A0 ∩ A1 is closed in A0 +A1 . In other words, i : A0 ∩A1 −→ A0 +A1 has closed range. Moreover, i factorizes through the identity of (A0 , A1 )θ,q , therefore i : A0 ∩ A1 −→ A0 + A1 is Asplund. This shows that (a) implies (b). The converse implication follows from [6], Prop. 4.7. 

We end the paper by returning to the complex method to establish Theorem 2.2 in its general form. Corollary 3.4. Let 0 < θ < 1, let A¯ = (A0 , A1 ), B¯ = (B0 , B1 ) be Banach couples ¯ B). ¯ If T : A0 −→ B0 is Asplund, then T : (A0 , A1 )[θ] −→ and let T ∈ L(A, (B0 , B1 )[θ ] is also Asplund. Proof. We can factorize T : A0 −→ B0 as T

A0

- B 0 6 IB0

 ? A0 /Ker(T )

j0

- B 0

where (x) = [x] is the quotient mapping and j0 [x] = T x. Put E = A0 /Ker(T ). The map j0 : E −→ B0 is a continuous embedding, and so (E, B0 ) is a Banach couple. Since the ideal of Asplund operators is surjective, it follows from the fact that T : A0 −→ B0 is Asplund that j0 : E −→ B0 is Asplund. Hence, by Corollary 3.3, we get that W = (E, B0 )1/2,2 is anAsplund space. The operator T : A0 −→ B0 admits the factorization T

A0

- B . 0

@



@ T@

@ R @

I W

Consequently, the interpolated operator by the complex method can be factorized as

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T

(A0 , A1 )[θ]

- (B , B ) 0 1 [θ]

@



@ T@

@ R @

I (W, B1 )[θ]

and Theorem 2.2 implies that T : (A0 , A1 )[θ] −→ (B0 , B1 )[θ] is Asplund.



Combining Corollary 3.4 with the reiteration theorem for the complex method, we obtain: Corollary 3.5. Let A¯ = (A0 , A1 ), B¯ = (B0 , B1 ) be Banach couples and let T ∈ ¯ B). ¯ If there exits 0 < θ0 < 1 such that T : (A0 , A1 )[θ0 ] −→ (B0 , B1 )[θ0 ] is L(A, Asplund, then T : (A0 , A1 )[θ] −→ (B0 , B1 )[θ] is Asplund for all 0 < θ < 1. Interpolation properties of other operator ideals have been investigated by the authors in [6]. There, we have studied the case of weakly compact operators, Rosenthal operators and Banach-Saks operators. Acknowledgements. Authors have been supported in part by Ministerio de Ciencia y Tecnolog´ıa (BFM2001-1424).

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