Integral Operators on the Product of C(K) Spaces

Abstract

We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the representing polymeasure of the operator. Some applications are given. In particular, we characterize the Borel polymeasures that can be extended to a measure in the product sigma -algebra, generalizing previous results for bimeasures. We also give necessary conditions for the weak compactness of the extension of an integral multilinear operator on a product of C(K) spaces. (C) 2001 Elsevier Science.

Full text

Journal of Math. Anal. and Appl., 264, No. 1 (2001), 107-121.
M.R. 2003e:47106
INTEGRAL OPERATORS ON THE PRODUCT OF C(K)SPACES
FERNANDO BOMBAL AND IGNACIO VILLANUEVA
Abstract. We study and characterize the integral multilinear operators on
a product of C(K) spaces in terms of the representing polymeasure of the
operator. Some applications are given. In particular, we characterize the
Borel polymeasures that can be extended to a measure in the product σ-
algebra, generalizing previous results for bimeasures. We also give necessary
conditions for the weak compactness of the extension of an integral multilinear
operator on a product of C(K) spaces.
1. Introduction
The modern theory of Banach spaces is greatly indebted to the work of A.
Grothendieck. In his papers [11] and [12] he introduced the most important classes
of operator ideals, whose study and characterization in different concrete classes
of Banach spaces has been a permanent subject of interest since then. One of the
classes defined in [11] and now intensively studied, is the class of integral operators
(see below), whose definition establishes a first connection between the linear and
the multilinear (bilinear, in fact) theory.
Grothendieck himself started the study of several classes of operators on C(K)
spaces in [12]. As a consequence of the Riesz representation Theorem, every con-
tinuous linear map Tfrom C(K) into another Banach space Xhas a representing
measure, i.e., a finitely additive measure mof bounded semi variation defined on
the Borel σ-field of K, with values in X∗∗ (the bidual of X), in such a way that
T(f):=Zf dm, for each f∈C(K).
(see, e.g. [5] or [6]). The study of the relationships between Tand its representing
measure plays a central role in this research.
When Tis a continuous k-linear map from a product C(K1)× · · · × C(Kk)
(where Kiare compact Hausdorff spaces) into a Banach space X, there exists also
an integral representation theorem with respect to the representing polymeasure
of T(see below for the definitions). If k= 1, the integral operators (G-integral
in our notation; see Definition 2.3 below) are precisely those whose representing
measure has bounded variation (see e.g. [16, p. 477] and [5, Th. VI.3.3]). The
aim of this paper is to study and characterize the multilinear vector valued integral
Both authors were partially supported by DGICYT grant PB97-0240.
1

2 FERNANDO BOMBAL AND IGNACIO VILLANUEVA
operators on a product of C(K) spaces in terms of the corresponding representing
polymeasure. As an application we obtain an intrinsic characterization of the Borel
polymeasures than can be extended to measures in the product Borel σ-algebra,
extending some previous results for the case of bimeasures. We also study the
relationship between the weak compactness of an integral multilinear map on a
product C(K1)× · · · × C(Kk) and that of its linear extension to C(K1× · · · × Kk).
Some other applications are given.
2. Definitions and Preliminaries
The notation and terminology used throughout the paper will be the standard in
Banach space theory, as for instance in [5]. However, before going any further, we
shall establish some terminology: Lk(E1. . . , Ek;X) will be the Banach space of all
the continuous k-linear mappings from E1×· · · ×Ekinto Xand Lk
wc(E1. . . , Ek;X)
will be the closed subspace of it formed by the weakly compact multilinear opera-
tors. When X=Kor k= 1, we will omit them. If T∈ Lk(E1. . . , Ek;X) we shall
denote by ˆ
T:E1⊗ · · · ⊗ Ek→Xits linearization. As usual, E1ˆ
⊗· · · ˆ
⊗Ekwill
stand for the (complete) injective tensor product of the Banach spaces E1, . . . , Ek.
We shall use the convention [i]
. . . to mean that the i-th coordinate is not involved.
If T∈ Lk(E1, . . . , Ek;X) we denote by Ti(1 ≤i≤k) the operator Ti∈
L(Ei;Lk−1(E1,[i]
. . ., Ek;X) defined by
Ti(xi)(x1,[i]
. . ., xk) := T(x1, . . . , xk),
Let now Σi(1 ≤i≤k) be σ-algebras (or simply algebras) of subsets on some non
void sets Ωi. A function γ: Σ1× · ·· × Σk−→ Xor γ: Σ1× · ·· × Σk−→ [0,+∞]
is a (countably additive) k-polymeasure if it is separately (countably) additive.
([8, Definition 1]) . A countably additive polymeasure γis uniform in the ith
variable if the measures {γ(A1, . . . , Ai−1,·, Ai+1, . . . , Ak) : Aj∈Σj(j6=i)}are
uniformly countably additive. As in the case k= 1 we can define the variation of
a polymeasure γ: Σ1× · · · × Σk−→ X, as the set function
v(γ) : Σ1× · ·· × Σk−→ [0,+∞]
given by
v(γ)(A1, . . . , Ak) = sup 


n1
X
j1=1
· · ·
nk
X
jk=1 

γ(Aj1
1, . . . Ajk
k)




where the supremum is taken over all the finite Σi-partitions (Aji
i)ni
ji=1 of Ai(1 ≤
i≤k). We will call bvpm(Σ1, . . . , Σk;X) the Banach space of the polymeasures
with bounded variation defined on Σ1× · · · × Σkwith values in X, endowed with
the variation norm.
We can define also its semivariation
kγk: Σ1× · ·· × Σk−→ [0,+∞]
by
kγk(A1, . . . , Ak) = sup 








n1
X
j1=1
· · ·
nk
X
jk=1
aj1
1. . . ajk
kγ(Aj1
1, . . . , Ajk
k)









INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 3
where the supremun is taken over all the finite Σi-partitions (Aji
i)ni
ji=1 of Ai(1 ≤
i≤k), and all the collections (aji
i)ni
ji=1 contained in the unit ball of the scalar
field. We will call bpm(Σ1, . . . , Σk;X) the Banach space of the polymeasures with
bounded semivariation defined on Σ1× · · · × Σkwith values in X, endowed with
the semivariation norm.
If γhas finite semivariation, an elementary integral R(f1, f2, . . . fk)dγ can be
defined, where fiare bounded, Σi-measurable scalar functions, just taking the
limit of the integrals of k-uples of simple functions (with the obvious definition)
uniformly converging to the fi’s (see [8]).
If K1, . . . , Kkare compact Hausdorff spaces, then every multilinear operator
T∈ Lk(C(K1), . . . , C(Kk); X) has a unique representing polymeasure γ: Bo(K1)×
· · · Bo(Kk)→X∗∗ (where Bo(K) denotes the Borel σ-algebra of K) with finite
semivariation, in such a way that
T(f1, . . . , fk) = Z(f1, . . . , fk)dγ for fi∈C(Ki),
and such that for every x∗∈X∗,x∗◦γ∈rcapm(Bo(K1), . . . , Bo(Kk)), the set
of all regular, countably additive scalar polymeasures on Bo(K1)× · · · × Bo(Kk).
(Cfr. [2]).
Given a polymeasure γwe can consider the set function γmdefined on the semi-
ring of all measurable rectangles A1× · ·· × Ak(Ai∈Σi) by
γm(A1× · ·· × Ak) := γ(A1, . . . , Ak)
It follows, f. i. from [7, Prop. 1.2] that γmis finitely additive and then it can be
uniquely extended to a finitely additive measure on the algebra a(Σ1× · · · × Σk)
generated by the measurable rectangles. In general , this finitely additive measure
cannot be extended to the σ-algebra Σ1⊗ · · · ⊗ Σkgenerated by Σ1× · · · × Σk. But
if there is a countably additive measure µof bounded variation on Σ1⊗ · · · ⊗ Σk
that extends γm, then by standard measure theory (see e.g. [6, Th. I.5.3] and [7])
we have
v(γm)(A1× · · · × Ak) = v(γ)(A1, . . . , Ak) = v(µ)(A1× · · · ×Ak),for Ai∈Σi.(∗)
The next definition extends Grothendieck’s notion of multilinear integral forms
to the multilinear integral operators:
Definition 2.1. A multilinear operator T∈ Lk(E1, . . . , Ek;X)is integral if ˆ
T
( i.e., its linearization) is continuous for the injective ()topology on E1⊗ · · · ⊗
Ek. Its norm (as an element of L(E1ˆ
⊗· · · ˆ
⊗Ek;X)) is the integral norm of T,
kTkint := kˆ
Tk.
Proposition 2.2. T∈ Lk(E1, . . . , Ek;X)is integral if and only if x∗◦Tis integral
for every x∗∈X∗.
Proof. For the non-trivial part, let us consider the map X∗3x∗7→ x∗◦ˆ
T∈
(E1ˆ
⊗· · · ˆ
⊗Ek)∗, well defined by hypothesis. A simple application of the closed
graph theorem proves that this linear map is continuous. Hence,
sup
kx∗k≤1
kx∗◦ˆ
Tk=M < ∞.
But it is easy to see that kTkint := supkuk≤1kˆ
T(u)k=M.
The next definition is well known.

4 FERNANDO BOMBAL AND IGNACIO VILLANUEVA
Definition 2.3. An operator S∈ L(E;X)is G-integral (the “G” comes from
“Grothendieck”) if the associated bilinear form
BS:E×X∗−→ K
(x, y)7→ y(T(x))
is integral. In that case the integral norm of S,kSkint := kBSkint.
Let us recall that a bilinear form T∈ L2(E1, E2) is integral if and only if any of
the two associated linear operators T1∈ L(E1;E∗
2) and T2∈ L(E2;E∗
1) is G-integral
in the above sense (cfr., e.g. [5, Ch- VI]).
Proposition 2.4. Let k≥2,E1, . . . , Ekbe Banach spaces and T∈ Lk(E1, . . . , Ek).
Then Tis integral if and only if there exists i,1≤i≤k, such that
a) For every xi∈Ei,Ti(xi)is integral.
b) The mapping
˜
Ti:Ei→(E1⊗
[i]
· · · ⊗Ek)∗
defined by
˜
Ti(xi) := \
Ti(xi)
is a G-integral operator.
If (a) and (b) are satisfied for some i, then the same happens for any other index
j,1≤j≤k. Moreover, in this case, kTkint =k˜
Tikint.
Proof. If (a) and (b) are satisfied and we put Fi:= E1⊗
[i]
· · · ⊗Ek, the bilinear map
BTi:Ei×Fi→Kis integral and kBTikint =k˜
Tik([5, Corollary VIII.2.12]). From
the associativity, the commutativity of the -tensor product and the definitions, it
follows that Tis integral and the norms are equal.
Conversely, suppose that Tis integral. We shall prove that (a) and (b) hold for
i= 1: From the hypothesis and the associativity of the injective tensor product, it
follows that the bilinear map
BT:E1×(E2ˆ
⊗· · · ˆ
⊗Ek)−→ K
(x, u)7→ ˆ
T(x⊗u)
is integral. By [5, Corollary VIII.2.12], the associated linear operator from E1into
(E2ˆ
⊗· · · ˆ
⊗Ek)∗is G-integral. Clearly, this operator coincides with ˜
T1, and this
proves (a) and (b) for i= 1. 
3. Integral forms on C(K)spaces
Let now K, K1, . . . , Kkbe compact Hausdorff spaces.
Recall that, for every Banach space X,C(K, X), the Banach space of all the
X-valued continuous functions on Kendowed with the sup norm, is canonically
isometric to C(K)ˆ
⊗X([5, Example VIII.1.6]). Moreover, if X=C(S) (Sa
compact Hausdorff space) then C(K, C(S)) is canonically isometric to C(K×S).
Thus, we have the following identifications
C(K1)ˆ
⊗· · · ˆ
⊗C(Kk)≈C(Ki, C(K1)ˆ
⊗
[i]
· · · ˆ
⊗C(Kk)≈C(K1× · ·· × Kk).
Suppose that T∈ Lk(C(K1), . . . , C (Kk)) with representing polymeasure γ. If there
exists a regular measure µon the Borel σ-algebra of K1× · · ·× Kkthat extends γm,

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 5
then, by the Riesz representation theorem, µis the representing measure of some
continuous linear form ˆ
Ton C(K1× · · · × Kk)≈C(K1)ˆ
⊗· · · ˆ
⊗C(Kk) and clearly
ˆ
T(f1⊗ · ·· ⊗ fk) = ZK1×···×Kk
f1· · · fkdµ =(1)
=T(f1, . . . , fk) = ZK1×···×Kk
(f1, . . . , fk)dγ.
Consequently Tis integral. Note also that, as follows from the introduction,
kˆ
Tk=v(µ) = v(γ).
Conversely, if Tis such that its linearization ˆ
Ton C(K1)⊗ · · · ⊗ C(Kk) is
continuous for the -topology (i.e., Tis integral), another application of the Riesz
representation theorem yields a measure µon Bo(K1× · · · × Kk) such that (1)
holds. By the uniqueness of the representation theorem for k-linear maps, we have
µ(A1× · ·· × Ak) = γ(A1, . . . , Ak) (for every Ai∈Σi)
and so µextends γ. Summarizing, we have proved
Proposition 3.1. Let k≥2and T∈ Lk(C(K1), . . . , C (Kk)) with representing
polymeasure γ. Then Tis integral if and only if γcan be extended to a regular
measure µon Bo(K1× · · · × Kk)in such a way that
µ(A1× · ·· × Ak) = γ(A1, . . . , Ak)for every Ai∈Σi,(1 ≤i≤k).
In this case, kˆ
Tk=kTkint =v(γ) = v(µ).
Consequently, (C(K1)ˆ
⊗· · · ˆ
⊗C(Kk))∗can be isometrically identified with a
subspace of the space of all regular polymeasures on Bo(K1)× · · · × Bo(Kk) with
finite variation, endowed with the variation norm.
Now we are going to obtain an intrinsic characterization of the extendible Radon
polymeasures which will allow us to see that the previous isometry is onto.
If Σ1,. . . , Σkare σ-algebras, Xis a Banach space and γ∈bpm(Σ1, . . . , Σk;X),
then we can define a measure
ϕ1: Σ1−→ bpm(Σ2, . . . , Σk;X)
by
ϕ1(A1)(A2, . . . , Ak) = γ(A1, A2, . . . , Ak).
It is known that kϕ1k=kγk(see [3]). Related to this we have the following lemma,
whose easy proof we include for completeness:
Lemma 3.2. Let Xbe a Banach space, Ω1, . . . , Ωksets and Σ1,..., Σkσ-algebras
defined on them. Let now γ: Σ1× · · · × Σk−→ Xbe a polymeasure. Then
v(γ)<∞if and only if ϕ1takes values in bvpm(Σ2, . . . Σk;X)and v(ϕ1)<∞
when we consider the variation norm in the image space. In that case, v(ϕ1)(A1) =
v(γ)(A1,Ω2, . . . , Ωk)and v(ϕ1(A1))(A2, . . . Ak)≤v(γ)(A1, A2, . . . , Ak). Of course
the role played by the first variable could be played by any of the other variables.
Proof. Let us first assume that v(γ)<∞. In the following we will adopt the
convention that supj2...,jkmeans the supremum over all the finite Σi-partitions
(Aji
i)ni
ji=1 of Ai(2 ≤i≤k). Then, with this notation,
v(ϕ1(A1)) = v(ϕ1(A1))(Ω2, . . . , Ωk) = sup
j2,...,jkX
j2
· · · X
jk
kϕ1(A1)(Ai2
2, . . . , Aik
k)k=

6 FERNANDO BOMBAL AND IGNACIO VILLANUEVA
(2) = sup
j2...,jkX
j2
· · · X
jk
kγ(A1, Aj2
2, . . . , Ajk
k)k ≤ v(γ)(A1,Ω2, . . . , Ωk)≤
≤v(γ)(Ω1,Ω2, . . . , Ωk) = v(γ).
Therefore, ϕ1is bvpm(Σ2, . . . , Σk;X)-valued. Let us now see that it has bounded
variation when we consider the variation norm in the image space:
(3) v(ϕ1) = v(ϕ1)(Ω1) = sup
j1X
j1
kϕ1(Aj1
1)k= sup
j1X
j1
v(ϕ1(Aj1
1)) ≤
≤sup
j1X
j1
v(γ)(Aj1
1,Ω2, . . . , Ωk)≤v(γ)(Ω1,Ω2, . . . , Ωk) = v(γ)<∞.
In the next to last inequality we have used that the variation of a polymeasure is
itself separately countably additive ([8, Theorem 3]).
Conversely, if ϕ1is bvpm(Σ2, . . . , Σk;X)-valued and with bounded variation
when we consider the variation norm in the image space, then
(4) v(γ) = v(γ)(Ω1, . . . , Ωk) = sup
j1,j2,...,jkX
j1X
j2
· · · X
jk
kγ(Aj1
1, Aj2
2, . . . , Ajk
k)k ≤
≤sup
j1X
j1
sup
j2,...,jkX
j2
· · · X
jk
kϕ1(Aj1
1)(Aj2
2, . . . , Ajk
k)k= sup
j1X
j1
kϕ1(Aj1
1)k=
=v(ϕ1)(Ω1) = v(ϕ1)<∞.
Putting together both inequalities we get that
v(γ) = v(ϕ1).
To prove the first of the last two statements of the lemma we replace Ω1by A1
in (3), (4). To prove the last statement we replace (Ω2,. . . ,Ωk) by (A2, . . . , Ak) in
(2). 
Let T∈ Lk(C(K1), . . . , C(Kk)) with representing polymeasure γ. Let us con-
sider T1:C(K1)−→ Lk−1(C(K2), . . . , C (Kk)) and let
ϕ1:Bo(K1)−→ rcapm(Bo(K2), . . . , B o(Kk))
be defined as above.
It is known that ϕ1is countably additive if and only if γis uniform ([3, Lemma
2.2]) and in this case ϕ1is the representing measure of T1([3, Theorem 2.4]). From
the definitions, it is easy to check that every polymeasure with finite variation is
uniform.
Now we can prove the first of our main results.
Theorem 3.3. Let T∈ Lk(C(K1), . . . , C(Kk)) with representing polymeasure γ.
Then the following are equivalent:
a) v(γ)<∞.
b) Tis integral.
c) γcan be extended to a regular measure µon Bo(K1× · · · × Kk).
d) γcan be extended to a countably additive (not necessarily regular) measure
µ2on Bo(K1)⊗ · · · ⊗ Bo(Kk).

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 7
Proof. The equivalence between (b) and (c) is just Proposition 3.1. If (c) holds,
defining µ2:= µ|Bo(K1)⊗···⊗Bo(Kk)proves (d). Since a countably additive scalar
measure has bounded variation, from (∗) in the previous Section, we get that (d)
implies (a). Finally let us prove that (a) implies (b): By Proposition 2.4 we have
to show that
i) \
T1(f1)∈(C(K2)ˆ
⊗· · · ˆ
⊗C(Kk))∗:= F∗
and
ii) ˜
T1:C(K1)−→ F∗is G-integral.
We shall proceed by induction on k. For k= 1 there is nothing to prove.
Let k= 2. In this case we only have to prove (ii). By the discussion following
Lemma 3.2, the representing measure of T1is ϕ1:Bo(K1)−→ C(K2)∗and v(ϕ1) =
v(γ)<∞. Since every dual space is 1-complemented in its bidual, by Corollary
VIII.2.10 and Theorems VI.3.3 and VI.3.12 of [5], T1is integral and
kT1kint =kˆ
Tk=v(ϕ) = v(γ)
by Lemma 3.2.
Let us now suppose the result true for k−1. Let T∈ Lk(C(K1), . . . , C(Kk)).
i) For f1∈C(K1), the representing polymeasure of T1(f1) is γf1, defined by
γf1(A2, . . . , Ak) = Z(f1, χA2, . . . , χAk)dγ,
as can be easily checked. Since |γf1(A2, . . . , Ak)| ≤ kf1k∞v(γ)(K1, A2, . . . , Ak),
it follows that γf1has finite variation and v(γf1)≤v(γ)kf1k∞. Hence, by the
induction hypothesis, T1(f1) is integral.
ii) As before, we have to prove that the representing measure of
˜
T1:C(K1)−→ (C(K2)ˆ
⊗· · · ˆ
⊗C(Kk))∗
has finite variation. The representing measure of ˜
T1is ˜ϕ1, where
˜ϕ1(A1)(f2⊗ · ·· ⊗ fk) = Z(χA1, f2, . . . , fk)dγ
and, clearly, ˜ϕ1is just ϕ1of Lemma 3.2 considering the integral (equivalently
variation) norm in the image space. Therefore, Lemma 3.2 proves that v( ˜ϕ1) =
v(γ)<∞, and so ˜
T1is G-integral. 
Remark 3.4.The equivalence (a)⇔(d) was proved for bimeasures in [14, Corol-
lary 2.9]. The techniques used in that paper, essentially different from ours, do not
seem to extend easily to the case of k-polymeasures when k≥3.
To the best of our knowledge, it was unknown when a polymeasure could be
decomposed as the sum of a positive and a negative polymeasure. It is clear now
that, for the polymeasures in rcapm(Bo(K1), . . . , Bo(Kk)), this happens only in
the most trivial case, that is, when γcan be extended to a measure, and then
decomposed as such.
Corollary 3.5. Given γ∈rcapm(Bo(K1), . . . , Bo(Kk)),γcan be decomposed as
the sum of a positive and a negative polymeasure if and only if v(γ)<∞.
Proof. If v(γ)<∞, then γcan be extended to µas in Theorem 3.3. Let us
now decompose this measure µas the sum of a positive and a negative measure
µ=µp+µn. Clearly now γ=µp+µn, considering µpand µnas polymeasures.

8 FERNANDO BOMBAL AND IGNACIO VILLANUEVA
Conversely, if γ=γp+γn, where γp(resp. γn) is a positive (resp. negative)
polymeasure, then
v(γ) = v(γ)(K1, . . . , Kk)≤γp(K1, . . . , Kk)−γn(K1, . . . , Kk)<∞.

4. Vector-valued integral maps on C(K)spaces
We will use now the results of the preceding section to characterize the vector
valued integral operators. First we will need a new definition: Let Ω1,. . . ,Ωkbe non-
empty sets and Σ1,. . . ,Σkbe σ-algebras defined on them. If γ: Σ1×· · ·× Σk−→ X
is a Banach space valued polymeasure, we can define its quasivariation
kγk+: Σ1× · ·· × Σk−→ [0,+∞]
by
kγk+(A1, . . . , Ak) = sup 








n1
X
j1=1
· · ·
nk
X
jk=1
aj1,...,jkγ(A1,j1, . . . , Ak,jk)








where (Ai,ji)ni
ji=1 is a Σi-partition of Ai(1 ≤i≤k) and |aj1,...,jk| ≤ 1 for all
(j1, . . . , jk).
It is not difficult to see that the quasivariation is separately monotone and sub-
additive and that, for every (A1, . . . , Ak)∈Σ1× ·· · × Σk,
kγk(A1, . . . , Ak)≤ kγk+(A1, . . . , Ak)≤v(γ)(A1, . . . , Ak).
It can also be checked that kγk+= sup{v(x∗◦γ); x∗∈BX∗}.
We can consider the space of polymeasures γ∈bpm(Σ1, . . . , Σk;X) such that
kγk+<∞. Standard calculations show that k·k+is a Banach space norm in this
space. This space has been recently considered in [7], where the authors develop
a theory of integration for these polymeasures. We will prove in this section that,
for the polymeasures representing multilinear operators on C(K1)× · · · × C(Kk),
the ones with finite quasivariation are precisely those which can be extended to
a measure on Bo(K1× · · · × Kk), and thus the previously mentioned integration
theory can be dispensed with.
Theorem 4.1. Let k≥2,T:C(K1)×· · · ×C(Kk)−→ Xbe a multilinear operator
and let γ:Bo(K1)× · ·· × Bo(Kk)−→ X∗∗ be its associated polymeasure. Then
the following are equivalent:
a) kγk+<∞.
b) Tis integral.
c) γcan be extended to a bounded ω∗-regular measure µ:Bo(K1× · · · × Kk)−→
X∗∗ (in such a way that
µ(A1× · ·· × Ak) = γ(A1, . . . , Ak)for every Ai∈Σi,(1 ≤i≤k)).
d) γcan be extended to a bounded ω∗-countably additive (not necessarily regular)
measure µ2:Bo(K1)⊗ · · · ⊗ Bo(Kk)−→ X∗∗.
Proof. Let us first prove that (a) implies (b): If kγk+<∞, then, for every x∗∈X∗,
v(x∗◦γ)<∞. Since x∗◦γis clearly the representing polymeasure of x∗◦T, using
Theorem 3.3, we obtain that x∗◦Tis integral for every x∗∈X∗and now we can
apply Proposition 2.2 to finish the proof.

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 9
Let us now prove that (b) implies (c): Let T∈ Lk(C(K1), . . . , C (Kk); X) be
integral, and call T∈ L(C(K1× · · · × Kk); X) its extension. If
µ:Bo(K1× · · · × Kk)−→ X∗∗
is the representing measure of T, it is clear that µsatisfies (c).
Clearly (c) implies (d). Now, if (d) is true, then kµ2k= supx∗∈BX∗v(x∗◦µ) =
supx∗∈BX∗v(x∗◦γ) = kγk+, and (a) holds. 
Corollary 4.2. Let T:C(K1)× · · · × C(Kk)−→ Xbe a multilinear operator with
representing polymeasure γ. Then the following statements are equivalent:
a) v(γ)<∞.
b) Tis integral and its extension T:C(K1× · ·· × Kk)−→ Xis G-integral.
Proof. If Tis integral and µis the representing measure of its extension Tthen,
as we saw in Section 2, v(γ) = v(µ). Hence, the equivalence between (a) and
(b) follows from kγk+≤v(γ) and the fact (already used) that a linear operator
on a C(K)-space is G-integral if and only if its representing measure has finite
variation. 
The fact that every integral multilinear map T:C(K1)× · · · × C(Kk)−→ X
can be extended to a continuous linear map T:C(K1)ˆ
⊗· · · ˆ
⊗C(Kk)≈C(K1×
· · · × Kk)−→ Xhas some immediate consequences:
Proposition 4.3. Let T:C(K1)× · · · × C(Kk)−→ Xbe an integral multilinear
operator and for each i, 1≤i≤k, let (fn
i)⊂C(Ki)be bounded sequences.
a) If at least one of the sequences (fn
i)is weakly null and (x∗
n)⊂X∗is a weakly
Cauchy sequence, then
lim
n→∞hT(fn
1, . . . , f n
k), x∗
ni= 0 (†)
b) If all the sequences (fn
i)are weakly Cauchy and (x∗
n)is a weakly null sequence
in X∗, then (†)holds.
Proof. From the well known characterization of the weak topology in C(K)-spaces,
it follows that the sequence (fn
1⊗ · · ·⊗ fn
k)⊂C(K1)ˆ
⊗· · · ˆ
⊗C(Kk)≈C(K1× · · ·×
Kk) is, respectively, weakly null (under (a)) or weakly Cauchy (under (b)), and
hT(fn
1), . . . , f n
k), x∗
ni=hfn
1⊗ · ·· ⊗ fn
k, T ∗(x∗
n)i
The result follows from the Dunford-Pettis property of C(K1× ·· · × Kk). 
If k= 2, it can be proved that the sequence (fn
1⊗fn
2) is also weakly null or weakly
Cauchy, respectively, in the projective tensor product C(K1)ˆ
⊗πC(K2) ([4, Lemma
2.1]). Hence, when Xhas the Dunford-Pettis Property the above result is true
for any continuous bilinear map. Nevertheless, Proposition 4.3 gives a necessary
condition for a multilinear map to be integral, and so it provides an easy way to
see when a multilinear map is not integral.
Example 4.4. Let (rn) be a bounded, orthonormal sequence (with respect to the
usual scalar product) in C([0,1]) and let T:C([0,1]) ×C([0,1]) −→ `2be defined
by
T(f, g) = Z1
0
frng1
n∞
n=1
Then T(clearly weakly compact) is not integral. In fact, if gndenotes the function
which is equal to 1 at 1
n, 0 in [0,1
2n] and [ 3
2n,1] and linear elsewhere, the sequence

10 FERNANDO BOMBAL AND IGNACIO VILLANUEVA
(gn) converges pointwise to 0 and so is weakly null in C([0,1]). But T(rn, gn) = en,
the usual `2-basis, and so hT(rn, gn), eni= 1 for every n.
Let us state a definition: given Banach spaces E1, . . . , Ek, X , a multilinear op-
erator T∈ Lk(E1, . . . , Ek;X) is called regular if every one if the linear operators
Ti∈ L(Ei;Lk−1(E1,[i]
. . ., Ek;X)) associated to it are weakly compact (see [1] and
[10] for some properties of these operators). In case T∈ Lk(C(K1), . . . , C (Kk); X),
it follows from [3] that its representing polymeasure γis uniform if and only if Tis
regular.
Proposition 4.5. Let T:C(K1)× · · · × C(Kk)−→ Xbe an integral multi-
linear operator with representing polymeasure γ, and suppose that its extension
T:C(K1×,· · · ,×Kk)−→ Xis weakly compact. Then γis uniform, and therefore
Tis regular.
Proof. Let us prove, for instance, that γis uniform in the first variable. Ac-
cording to [3, Theorem 2.4], it suffices to prove that the corresponding operator
T1:C(K1)−→ Lk−1(C(K2), . . . C (Kk); X) is weakly compact or, equivalently, it
maps weakly null sequences into norm null sequences ([5, Corollary VI.2.17]). Let
(fn
1)⊂C(K1) be a weakly null sequence. We have to prove that kT1(fn
1)k → 0 when
n→ ∞. If not, there would be an  > 0 and a subsequence (denoted in the same
way) such that kT1(fn
1)k>  for every n. Then we could produce fn
j∈C(Kj) (2 ≤
j≤k), kfn
jk ≤ 1, such that kT1(fn
1)(fn
2, . . . , f n
k)k=kT(fn
1, . . . , f n
k)k>  for ev-
ery n. But (fn
1·fn
2· · · fn
k)⊂C(K1×,· · · ,×Kk) converges weakly to 0. Hence,
from the aforementioned property of weakly compact operators on C(K)-spaces,
kT(fn
1· · · fn
k)k=kT(fn
1, . . . , f n
k)ktends to 0 as ntends to ∞, which is a contradic-
tion. 
Remark 4.6.Note that if Xis reflexive, in particular if X=K, then it follows
from the above result that integral operators are regular.
We do not know if the converse of the Proposition 4.5 is true. In any case,
if T:C(K1)× · · · × C(Kk)→Xis integral and regular and, for instance, we
denote by T1:C(K1)−→ Lk−1(C(K2), . . . , C(Kk); X) the associated linear map,
it is easily checked that T(ϕ1) is integral for any ϕ1∈C(K1), and its representing
measure takes also values in the space of integral (k−1)-linear operators. Thus, it
can be considered as a measure m1: Σ1−→ L(C(K2× · · · × Kk); X). Reasoning
in a similar way as in [3, Theorem 2.4], we can prove that m1coincides with the
representing measure of the operator ˆ
T:C(K1, C(K2× · · · × Kk)) −→ Xgiven by
the Dinculeanu-Singer Theorem (see, e.g. [5, p. 182]), and the weak compactness
of Tis clearly equivalent to that of ˆ
T.
In the linear case, an operator T:C(K)−→ Xis weakly compact if and only if
its representing measure µtakes values in X, if and only if µis countably additive.
This is not longer true in the multilinear case, where the role of weakly compact
operators seems to be played by the so called completely continuous multilinear
maps (see [17]). In the case of integral multilinear maps one could conjecture that
the weak compactness and the behaviour of the representing polymeasure of T
should be analogous to that of the extended linear operator. This is not true, as
the following example shows:
Example 4.7. Let us consider `∞=C(βN). Let q:`∞→`2be a linear, con-
tinuous and onto map ([15, Remark 2.f.12]), and let us take a bounded sequence

INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES 11
(an)⊂`∞such that q(an) = en(the canonical basis of `2) for any n. Suppose
kank ≤ C. Then (en⊗an) is a basic sequence in `∞ˆ
⊗`∞≈C(βN, `∞) ([13,
Proposition 3.15]), equivalent to the canonical basis of co, since
k
n
X
i=1
λiei⊗aik= sup
kx∗k≤1
k
n
X
i=1
λix∗(ai)eik∞≤Ck(λi)k∞.
Moreover, if ϕn:= e∗
n⊗q∗(e∗
n)∈`∗
∞⊗`∗
∞⊂(`∞ˆ
⊗`∞)∗we have kϕnk ≤ kqkfor all
nand so, as ϕn(a⊗b)→0 when ntends to ∞, it turns out that (ϕn) is a weak∗
null sequence. Hence P(u) := P∞
n=1 ϕn(u)en⊗anis a continuous projection from
`∞ˆ
⊗`∞onto the closed subspace (isomorphic to c0) spanned by {en⊗an:n∈N}.
Consequently,
ˆ
T:`∞ˆ
⊗`∞−→ c0
defined as ˆ
T(u) := (ϕn(u)) ∈c0, is linear, continuous and onto. In particular,
ˆ
Tis not weakly compact. By construction, the corresponding bilinear map T:
`∞×`∞−→ c0is integral. Also, since (ϕn(x⊗y))∞
n∈`2for x, y ∈`∞and
kT(x, y)k2≤ kqkkxk∞kyk∞, it follows that Tfactors continuously through `2and
consequently it is weakly compact. In particular, the representing bimeasure γof T
takes values in c0, ([2, Corollary 2.2]), but the extended measure µthat represents
T:C(βN, `∞)→c0does not.
The next proposition characterizes when Tis weakly compact in terms of the
representing polymeasure of T:
Proposition 4.8. Let T:C(K1)× · · · × C(Kk)−→ Xbe an integral multilinear
operator with representing polymeasure γ, and let µbe the representing measure of
its extension T:C(K1× · · · × Kk)−→ X. The following assertions are equivalent:
a) Tis weakly compact.
b) µtakes values in X.
c) µis countably additive.
d) γm(see Section 2) takes values in Xand is strongly additive.
Proof. The equivalences between (a), (b) and (c) are well known (see [5, Theo-
rem VI.2.5]), and obviously they imply (d). Finally, since µis a w∗-countably
additive extension of γm, (d) implies that γmis weakly countably additive (and
strongly additive). The Hahn-Kluvanek extension Theorem ([5, Theorem I.5.2])
provides an (unique) X-valued countably additive extension of γmto Σ := Bo(K1)⊗
· · · ⊗Bo(Kk), which clearly coincides with µ. Obviously, every function in C(K1)⊗
· · · ⊗ C(Kk) is Σ-measurable. Hence, by density, every continuous function on
K1× · · · × Kkis Σ-measurable. Urysohn’s lemma proves that every closed, Fσset
belongs to Σ, and so is sent by µto X. A well known result of Grothendieck ([12,
Th´eor`eme 6]) proves that µsends any Borel subset of K1× · ·· × Kkto X.
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Departamento de An´
alisis Matem´
atico, Facultad de Matem´
aticas, Universidad Com-
plutense de Madrid, Madrid 28040
E-mail address:bombal@eucmax.sim.ucm.es, ignacio villanueva@mat.ucm.es