REGULAR MULTILINEAR OPERATORS ON C(K) SPACES

FERNANDO BOMBAL AND IGNACIO VILLANUEVA

Abstract. The purpose of this paper is to characterize the class of regular
continuous multilinear operators on a product of C(K) spaces, with values in
an arbitrary Banach space. This class has been considered recently by several
authors (see, f.i., [3], [8], [9]) in connections with problems of factorization
of polynomials and holomorphic mappings. We also obtain several charac-
terizations of the compact dispersed spaces K in terms of polynomials and
multilinear forms defined on C(K).

1991 Mathematics Subject Classification: 46E15, 46B25.

1. Introduction and Notations

Let K be a compact Hausdorff space. C(K) will be the space of scalar
valued continuous functions on K, Σ will denote the σ-algebra of the Borel sets of
K and B(Σ) will stand for the space of Σ-measurable functions on K which are the
uniform limit of elements of Σ-simple functions.

As it is well known, the Riesz representation theorem gives a representation of
the operators on C(K) as integrals with respect to Radon measures, and this has
been very fruitfully used in the study of the properties of C(K) spaces. In a series
of papers (see specially [6], [7]), Dobrakov developed a theory of polymeasures ,
functions defined on a product of σ-algebras which are separately measures, that
can be used to obtain a Riesz-type representation theorem for multilinear operators
defined on a product of C(K) spaces.

Before going any further, we shall clear out our notation: If X is a Banach space,
X∗ will denote its topological dual and BX its closed unit ball. Lk(E1 . . . , Ek;Y )
will be the Banach space of all the continuous k-linear mappings from E1×· · ·×Ek

into Y , and P(kX; Y ) the space of continuous k-homogeneous polynomials from
X to Y , i.e., the class of mappings P : X → Y of the form P (x) = T (x, . . . , x),
for some T ∈ Lk(X, . . . X;Y ). When Y = K, we will omit it. We shall use the
convention [i]. . . to mean that the i-th coordinate is not involved.

We shall denote the semivariation of a measure µ by ‖µ‖ and also the semivari-
ation of a polymeasure γ by ‖γ‖ (for the general theory of polymeasures see [6], or
[14]). It seems convenient to recall here that a polymeasure is called regular if it
is separately regular and it is called countably additive if it is separately countably
additive. We will denote the set of the bounded semivariation polymeasures defined
in Σ1 × · · · × Σk with values in X as bpm(Σ1, . . . , Σk;X). rcapm(Σ1, . . . , Σk; X)
stands for the subset of the regular countably additive polymeasures and bsv-
ω∗ − rcapm(Σ1, . . . , Σk;X∗) for the subset of bpm(Σ1, . . . , Σk;X∗) composed of
those polymeasures that verify that for each x ∈ X, x ◦ γ ∈ rcapm(Σ1, . . . , Σk;K).

Both authors were partially supported by DGICYT grant PB97-0240.

1



2 FERNANDO BOMBAL AND IGNACIO VILLANUEVA

As customary we will call rca(Σ; X) the set of regular countably additive measures
from Σ into X.

With these notations at hand we can state for further references the following
theorem from [4], which extends and completes previous results of Pelczynski ([11])
and Dobrakov ([7]):

Theorem 1.1. ([4]) Let K1, . . . ,Kk be compact Hausdorff spaces, let X be a
Banach space and let T ∈ Lk(C(K1), . . . , C(Kk); X). Then there is a unique
T ∈ Lk(B(Σ1), . . . , B(Σk), X∗∗) which extends T and is ω∗−ω∗ separately continu-
ous (the ω∗- topology that we consider in B(Σi) is the one induced by the ω∗-topology
of C(Ki)∗∗ ). Besides, we have

1. ‖T‖ = ‖T‖.
2. For every (g1,

[i]. . ., gk) ∈ B(Σ1)× [i]. . . ×B(Σk) there is a unique X∗∗-valued
bounded ω∗-Radon measure γ

g1,[i]...,gk
on Ki (i.e., a X∗∗-valued finitely additive

bounded vector measure on the Borel subsets of Ki, such that for every x∗ ∈ X∗,
x∗ ◦ γ

g1,[i]...,gk
is a Radon measure on Kk), verifying

∫
gidγ

g1,[i]...,gk
= T (g1, . . . , gi−1, gi, gi+1, . . . , gk), ∀gi ∈ B(Σi).

3. T is and ω∗ − ω∗ sequentially continuous (i.e., if (gn
i )n∈N ⊂ B(Σi), for i =

1, . . . k, and gn
i

ω∗→ gi, then limn→∞ T (gn
1 , . . . , gn

k ) = T (g1, . . . , gk) in the σ(X∗∗, X∗)
topology.

Besides, if we define γ : B(Σ1)× · · · ×B(Σk) 7→ X∗∗ as

γ(A1, . . . , Ak) := T (χA1 , . . . χAk
),

then γ is a polymeasure of bounded semivariation that verifies
(a) ‖T‖ = ‖γ‖.
(b) T (f1, . . . , fk) =

∫
(f1, . . . , fk)dγ (fi ∈ C(Ki))

(c) For every x∗ ∈ X∗, x∗◦γ is a regular (scalar) polymeasure and the map x∗ 7→
x∗◦γ is continuous for corresponding weak-∗ topologies in X∗ and (C(K1)⊗̂ · · · ⊗̂C(Kk))∗.

Conversely, if γ : Σ1 × · · · ×Σk 7→ X∗∗ is a polymeasure which verifies (c), then
it has finite semivariation and formula (b) defines a k-linear continuous operator
from C(K1)× · · · × C(Kk) into X for which (a) holds.

Therefore the correspondence T ↔ γ is an isometric isomorphism between Lk(C(K1), . . . , C(Kk); X)
and the polymeasures in bsv-ω∗- rcapm(Σ1×· · ·×Σk; X∗∗) that verify condition c).

Our aim now is to exploit both representation theories, measures and polymea-
sures, to study the multilinear operators on C(K) spaces. In this paper we present
some results in this direction.

2. The Main Results

The following definition can be found in [6] or in [14]

Definition 2.1. A polymeasure γ : Σ1× · · ·×Σk 7→ X is said to be uniform in the
ıth variable if it is countably additive and the measures{

γ(A1, . . . , Ai−1, ·, Ai+1, . . . Ak) ∈ ca(Σi; X) : (A1,
[i]. . ., Ak) ∈ Σ1×

[i]· · · ×Σk

}

are uniformly countably additive.
A polymeasure is said to be uniform if it is uniform in every variable.



REGULAR MULTILINEAR OPERATORS ON C(K) SPACES 3

It is easy to check that given a natural number r, 1 < r < k and r indexes
1 ≤ j(1) < j(2) < . . . < j(r) ≤ k, and given fixed hj(p) ∈ B(Σj(p)), p = 1 . . . r, we
can construct the multilinear operator

Thj(1),...hj(r) :
∏

1≤q≤k
q 6∈(j(1)...j(r))

C(Kq) 7→ X

defined as Thj(1),...hj(r)(hq(1), . . . , hq(k−r)) := T (h1, . . . , hk) whose associated poly-
measure we will call γhj(1),...hj(r) .

Given a bounded polymeasure γ : Σ1 × · · · ×Σk 7→ X and a fixed number i, 1 ≤
i ≤ k, we can construct in a natural way the measure φi : Σi 7→ bpm(Σ1,

[i]· · ·, Σk; X)
defined as φi(Ai) := γAi

. The fact that φi is bounded, indeed ‖φi‖ = ‖γ‖, and the
following lemma are easy to check.

Lemma 2.2. With the above notation, a countably additive polymeasure γ is uni-
form in the ıth variable if and only if φi is countably additive. The same is true if
in this statement “countably additive” is replaced by “regular”.

Let E1, . . . , Ek, X be Banach spaces. Each T ∈ Lk(E1, . . . , Ek;X) generates in
a natural way k linear operators

Ti : Ei 7→ Lk−1(E1,
[i]. . ., Ek;X), i = 1, . . . , k

defined as Ti(xi)(x1,
[i]. . ., xk) := T (x1, . . . , xk) for each xj ∈ Ej , j = 1, . . . , k.

We will state now a definition:

Definition 2.3. A k-linear mapping T ∈ Lk(E1, . . . , Ek;X) is said to be regular
if every mapping Ti above defined is weakly compact.

When X is the scalar field, the above definition was given in [3]. In general,
given an operator ideal U , we can define the U-regular k-linear mappings as those
such that the corresponding Ti belong to U for every 1 ≤ i ≤ k. When U is the
ideal of compact operators, such mappings have been considered in [8], and for a
general closed injective operator ideal U in [9]. In every case a non-linear version
of the factorization theorem of Davies, Figiel, Johnson and Pelczynsky (see [5,
pgs. 250, 259]) through operators in U is obtained for such multilinear mappings.
These results are then applied to get some factorization theorems for holomorphic
mappings.

We are ready now to prove the following characterization of the uniform poly-
measures.

Theorem 2.4. Let K1, . . . ,Kk be compact Hausdorff spaces, let X be a Banach
space and let T ∈ Lk(C(K1), . . . , C(Kk); X). Let γ : Σ1 × · · · × Σk → X∗∗ be the
polymeasure associated to it according to theorem 1.1. Then γ is uniform if and
only if T is regular. Besides, in that case the measures φi defined before lemma 2.2
are the measures canonically associated to the operators Ti.

Proof. Let us first assume that γ is uniform (in particular this means that γ is
regular countably additive and therefore X-valued, see [7]). According to lemma
2.2 this means that for each i = 1, . . . , k, φi ∈ rca(Σi; rcapm(Σ1,

[i]. . . Σk; X)).
Since rcapm(Σ1,

[i]. . . Σk;X) ⊂ Lk−1(C(K1), [i]. . ., C(Kk); X) (cfr. theorem 1.1) we



4 FERNANDO BOMBAL AND IGNACIO VILLANUEVA

get that φi ∈ rca(Σi;Lk−1(C(K1), [i]. . ., C(Kk); X)). Then we can consider the op-
erator Hφi

∈ L(C(Ki);Lk−1(C(K1), [i]. . ., C(Kk); X)) associated to φi by the Riesz
representation theorem (vector valued case; see, f.i. [5, Theorem VI.2.1]). Since φi

is countably additive we know that Hφi is weakly compact ([5, Theorem VI.2.5]).
We consider now H∗∗

φi
, the bitranspose of Hφi

. Since Hφi
is weakly compact we

get that H∗∗
φi

is Lk−1(C(K1), [i]. . ., C(Kk); X)-valued. It is easy to see that for every

Ai ∈ Σi, and for every (f1,
[i]. . ., fk) ∈ C(K1)× [i]. . . ×C(Kk),

H∗∗
φi

(Ai)(f1,
[i]. . ., fk) = < φi(χAi

), (f1,
[i]. . ., fk) >=

∫
(f1,

[i]. . ., fk)dγAi

= T (f1, . . . fi−1, χAi
, fi+1, . . . fk).

Therefore,

H∗∗
φi

(gi)(f1,
[i]. . ., fk) = T (f1, . . . fi−1, gi, fi+1, . . . fk),

for every Σi-simple function gi and for every (f1,
[i]. . ., fk) ∈ C(K1)× [i]. . . ×C(Kk).

From continuity, we get the same relation for every gi ∈ B(Σi). In particular, when
we choose fi ∈ C(Ki) we get

H∗∗
φi

(fi)(f1,
[i]. . ., fk) = T (f1, . . . fi−1, fi, fi+1, . . . fk)

= T (f1, . . . fi−1, fi, fi+1, . . . fk) = Ti(fi)(f1,
[i]. . ., fk).

Obviously this means that Ti = Hφi and, therefore, that Ti is weakly compact.
Let us now assume that T is regular. Then, for every i = 1 . . . k, Ti ∈ L(C(Ki);Lk−1(C(K1), [i]. . .

, C(Kk); X) is weakly compact and so the measure µi associated to it by the Riesz
representation theorem is countably additive and Lk−1(C(K1), [i]. . ., C(Kk); X)-valued
([5, Theorem VI.2.5]). We will check now that for every i = 1 . . . k, µi = φi. Then,
the proof will be finished just by looking at lemma 2.2.

Let T ∗∗i be the bitranspose of Ti. For each Ai ∈ Σi let (fα
i )α∈I be a net in

C(Ki) such that fα
i

ω∗→ χAi . T ∗∗i is known to be ω∗-ω∗ continuous; being Ti

weakly compact we get that T ∗∗i is Lk−1(C(K1), [i]. . ., C(Kk); X)-valued. Both of
these facts together imply that (T ∗∗i (fα

i ))α∈I converges weakly to T ∗∗i (χAi). For
fixed (f1,

[i]. . ., fk) ∈ C(K1)× [i]. . . ×C(Kk) and x∗ ∈ X∗, the linear form

θ : Lk−1(C(K1), [i]. . ., C(Kk); X) 7→ K

defined as θ(S) := 〈S(f1,
[i]. . ., fk), x∗〉 is clearly continuous and therefore

θ(T ∗∗i (fα
i )) → θ(T ∗∗i (χAi)) = 〈T ∗∗i (χAi)(f1,

[i]. . ., fk), x∗〉.
Besides,

θ(T ∗∗i (fα
i )) = 〈T ∗∗i (fα

i )(f1,
[i]. . ., fk), x∗〉 = 〈T (f1, . . . fi−1, f

α
i , fi+1, . . . fk), x∗〉.

Since T is separately ω∗-ω∗ continuous we get that this last expression converges
to 〈T (f1, . . . fi−1, χAi , fi+1, . . . , fk), x∗〉. So we have obtained that for every x∗ ∈
X∗, 〈T (f1, . . . fi−1, χAi , fi+1, . . . , fk), x∗〉 = 〈T ∗∗i (χAi)(f1,

[i]. . ., fk), x∗〉. Therefore

for every Ai ∈ Σi and for every (f1,
[i]. . ., fk) ∈ C(K1)×

[i]· · · ×C(Kk),

T (f1, . . . fi−1, χAi , fi+1, . . . , fk) = T ∗∗i (χAi)(f1,
[i]. . ., fk) = µi(Ai)(f1,

[i]. . ., fk).



REGULAR MULTILINEAR OPERATORS ON C(K) SPACES 5

But clearly

T (f1, . . . fi−1, χAi
, fi+1, . . . , fk) =

∫
(f1,

[i]. . ., fk)dγAi
= φi(Ai)(f1,

[i]. . ., fk).

From here it follows that µi = φi and the proof is over. ¤

Since every operator from C(K1) to C(K2)∗ is weakly compact (cfr. [5, Theorem
VI-2-15], f.i.), we get immediately the following result (see [6]):

Corollary 2.5. Every regular countably additive scalar bimeasure γ : Σ1×Σ2 → K
is uniform.

From the above theorem we can derive the following propositions, useful to decide
whether a polymeasure is or is not uniform. Previously we will need a lemma.

Lemma 2.6. Let T : C(K1) × · · · × C(Kk) 7→ X be a regular k- linear operator.
Let (fn

i )n∈N ⊂ C(Ki) be a weakly null sequence and let ((gn
1 )n∈N, [i]. . ., (gn

k )n∈N) ⊂
B(Σ1)× [i]. . . ×B(Σk) be bounded sequences. Then, with the notation of theorem 1.1,
T (gn

1 , . . . gn
i−1, f

n
i , gn

i+1, . . . , g
n
k ) converges in norm to zero.

Proof. If T is regular, then the above defined operator Ti is weakly compact and
therefore completely continuous, by the Dunford-Pettis property of C(Ki). This
means that ‖Ti(fn

i )‖ → 0. We observe now that, due to the uniqueness of the
extension (1.1), for every (g1,

[i]. . ., gk) ⊂ B(Σ1)× [i]. . . ×B(Σk) and for every fi ∈
C(Ki), we have Ti(fi)(g1,

[i]. . ., gk) = T (g1, . . . gi−1, fi, gi+1, . . . , gk). By the equality
of the norms of the operator and its extension, we can write ‖Ti(fn

i )‖ → 0. This
can also be written as

sup
gj∈BB(Σj)

‖Ti(fn
i )(g1,

[i]. . ., gk)‖ → 0,

which means that

sup
gj∈BB(Σj)

‖T (g1, . . . gi−1, f
n
i , gi+1, . . . , gk)‖ → 0

and finishes the proof. ¤

Proposition 2.7. A regular countably additive polymeasure γ : Σ1×· · ·×Σk 7→ X
is uniform in the ıth variable if and only if the measures

{γ
g1,[i]...,gk

: (g1,
[i]. . ., gk) ∈ B(Σ1)×

[i]· · · ×B(Σk), ‖gj‖ ≤ 1}
are uniformly countably additive.

Proof. One of the implications is clear. For the other, let us suppose that γ is

uniform in the ıth variable. Were the measures {γ
g1,[i]...,gk

; (g1,
[i]. . ., gk) ∈ B(Σ1)×

[i]· · ·
×B(Σk)} not uniformly countably additive, then there would exist ε > 0, a se-
quence (An

i )n∈N ⊂ Σi of disjoint open sets and sequences ((gn
1 )n∈N, [i]. . ., (gn

k )n∈N) ⊂
B(Σ1)× [i]. . . ×B(Σk) with ‖gn

j ‖ ≤ 1 for each n ∈ N and for each j = 1 [i]. . . k, such
that ‖γ

g1,[i]...,gk
(An

i )‖ > ε. Then for each n ∈ N there would exist fn
i ∈ C(Ki) with

suppfn
i ⊂ An

i and ‖fn
i ‖ ≤ 1 such that ‖ ∫

fn
i dγ

g1,[i]...,gk
‖ > ε, and this in contradic-

tion with lemma 2.6, since the sequence fn
i converges weakly to 0. ¤



6 FERNANDO BOMBAL AND IGNACIO VILLANUEVA

Proposition 2.8. A regular countably additive polymeasure γ : Σ1×· · ·×Σk 7→ X
is uniform in the ıth variable if and only if the measures

{γ
f1,[i]...,fk

; (f1,
[i]. . ., fk) ∈ C(K1)×

[i]· · · ×C(Kk), ‖fj‖ ≤ 1}
are uniformly countably additive.

Proof. In one direction the result follows from the previous proposition. For the
other, we will suppose without loss of generality that i = k. Let us suppose that
the measures {γf1,...,fk−1 ; (f1, . . . , fk−1) ∈ C(K1) × · · · × C(Kk−1), ‖fj‖ ≤ 1} are
uniformly countably additive. If γ is not uniform in the kth variable then there
exist a sequence An

k ⊂ Σk of disjoint open sets and sequences (An
j )n∈N ⊂ Σj for

j = 1 . . . k−1 such that ‖γ(An
1 , . . . , An

k , )‖ > ε. Since γ is regular, γ(·, An
2 , . . . An

k ) is
regular for each n ∈ N and therefore there exists a function fn

1 ∈ C(K1) with
‖fn

1 ‖ ≤ 1 such that ‖ ∫
fn
1 dγAn

2 ,...,An
k
‖ > ε. Now γfn

1 ,·,χAn
3

,...,χAn
k

is also regu-
lar and therefore there exists a function fn

2 ∈ C(K2) with ‖fn
2 ‖ ≤ 1 such that

‖ ∫
fn
2 dγfn

1 ,χAn
3

,...,χAn
k
‖ > ε. Continuing in the same way we obtain k− 1 sequences

of norm one functions fn
j ⊂ C(Kj), j = 1 . . . k − 1 such that ‖γfn

1 ,...,fn
k−1

(An
k )‖ > ε

which contradicts the hypothesis. ¤

3. Polymeasures on compact dispersed spaces

Recall that a compact Hausdorff space is said to be dispersed if it does not
contain any non empty perfect set. In [12] a deep insight is given into the structure
of dispersed spaces, proving among other results that K is dispersed if and only if
C(K) contains no copy of `1, if and only if C(K)∗ contains no copy of L1. Also, in
this case C(K)∗ can be identified with `1(Γ) for some Γ.

Some (if not all) of the following results are probably known, but we have not
been able to find an explicit reference.

Theorem 3.1. For a compact Hausdorff space K, the following statements are
equivalent:

a) K is dispersed.
bo) For every k ≥ 1, the space Lk(C(K)) is Schur.
b1) For some k ≥ 2, the space Lk(C(K)) is Schur.
b2) For some k ≥ 2, the space P(kC(K)) is Schur.
b3) For every k ≥ 2, the space P(kC(K)) is Schur.
co) For every k ≥ 1, the space Lk(C(K)) is weakly sequentially complete.
c1), c2), c3): Same statements as b1), b2), b3), replacing Schur by weakly se-

quentially complete.
do) For every k ≥ 1, Lk(C(K)) contains no copy of `∞.
d1), d2), d3): Same statements as b1), b2), b3), replacing Schur by the non

containment of `∞.
e) For every k ≥ 1, Lk(C(K)) contains no copy of co.
e1), e2), e3): Same statements as d1), d2), d3), replacing `∞ by co.

Proof. Since Lk(C(K)) is a dual space for every k ≥ 1, every (d) statement is
equivalent to the corresponding (e) statement. Also clearly bi) ⇒ ci) ⇒ di), for
every i, bo) ⇒ b1) ⇒ b2) and bo) ⇒ b3) ⇒ b2). Therefore, it rests to prove a) ⇒
bo) and e2) ⇒ a).



REGULAR MULTILINEAR OPERATORS ON C(K) SPACES 7

a) ⇒ bo): We shall prove it by induction on k. For k = 1, it is clear since
C(K)∗ ≈ `1(Γ). Suppose now that

Lk(C(K)) =

(⊗̂k

π
C(K)

)∗

:= X∗

(cfr. [5, Corollary VIII.2.2]) is Schur. Then

Lk+1(C(K)) = L(C(K); X∗) =
(
C(K)⊗̂πX

)∗
.

Since C(K) contains no copy of `1 and has the Dunford-Pettis property, by the
induction hypothesis it follows that all members of the last space are compact
operators. Hence, since C(K)∗ has the approximation property,

Lk+1(C(K)) = C(K)∗⊗̂εX
∗

([5, Theorem VIII.3.6]), which is a Schur space, since this property is stable by
taking injective tensor products (cfr. f.i. [13]).

e2) ⇒ a): If K is not dispersed, C(K)∗ ⊃ L1 ⊃ `2. Consequently

`2⊗̂ε`2 ⊂ C(K)∗⊗̂εC(K)∗ ⊂ (
C(K)⊗̂πC(K)

)∗

(topological inclusions), and it is well known that if (en) is the canonical basis of
`2, then (en ⊗ en) is equivalent to the canonical basis of co (cfr., f.i., [10]). This
means that P(2C(K)) contains a copy of co. Since P(2C(K)) is a (complemented)
subspace of P(kC(K)), for every k ≥ 2, it follows that the latter space contains a
copy of co, too. ¤

As we mention in corollary 2.5, every scalar regular bimeasure on a compact
Hausdorff space is always uniform. This is not true for arbitrary polymeasures, as
the following example from [2] shows: The 3-linear map T : C([0, 1]) × C([0, 1]) ×
C([0, 1]) → C defined by

T (f, g, h) :=
∞∑

i=1

f(
1
2i

)
∫ 1

0

gri dx

∫ 1

0

hri dx,

where ri is the standard ith Rademacher function, is not regular. See [2] for details.
In the next theorem we show that the uniformity of all the k-polymeasures for

some (every) k ≥ 3, characterizes the compact dispersed spaces. We shall denote
by K(X; Y ) and W(X;Y ) the compact and weakly compact operators between X
and Y , respectively.

Theorem 3.2. For a compact Hausdorff space K the following statements are
equivalent

a) K is dispersed.
f) For every (some) k ≥ 2, L (

C(K);Lk(C(K))
)

= K (
C(K);Lk(C(K))

)
.

g) For every (some) k ≥ 2, L (
C(K);Lk(C(K))

)
= W (

C(K);Lk(C(K))
)
.

h) For every (some) k ≥ 3, any scalar regular k-polymeasure on the product of
the Borel σ-algebra of K, is uniform.

Proof. a) ⇒ f) was included in the proof of a) ⇒ bo) in theorem 3.1, and clearly
f) ⇒ g). The equivalence of (g) and (h) follows from theorem 2.4. Finally, let
us prove that (g) implies (a): Let k ≥ 3. If K is not dispersed, C(K) is infinite
dimensional and thus contains a copy of co ([5, Corollary VI.2.16]). On the other
hand, by theorem 3.1, Lk−1(C(K)) contains a copy of `∞. By the injectivity of this



8 FERNANDO BOMBAL AND IGNACIO VILLANUEVA

space, the inclusion map from co into `∞ can be extended to the whole space C(K),
providing in this way a non weakly compact operator in L (

C(K);Lk−1(C(K))
)
. ¤

The equivalence of (a), (f) and (g) has been also obtained in [1], although with
a different and, in our opinion, more involved proof.

References

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[3] Aron, R. and Galindo, P., Weakly compact multilinear mappings, Proc. of the Edinburgh
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[4] Bombal, F. and Villanueva, I. , Multilinear operators in spaces of continuous functions, to
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