Unconditionally Converging Multilinear Operators

Abstract

We introduce a notion of unconditionally converging multilinear operator which allows to extend many of the results of the linear case to the multilinear case. We prove several characterizations of these multilinear operators (one of which seems to be new also in the linear case), which allow to considerably simplify the work with this kind of operators

Full text

UNCONDITIONALLY CONVERGING MULTILINEAR
OPERATORS
FERNANDO BOMBAL, MAITE FERN ´
ANDEZ-UNZUETA, AND IGNACIO VILLANUEVA
Abstract. We introduce a notion of unconditionally converging multilinear
operator which allows to extend many of the results of the linear case to
the multilinear case. We prove several characterizations of these multilinear
operators (one of which seems to be new also in the linear case), which allow
to considerably simplify the work with this kind of operators.
Keywords and phrases: Unconditionally converging, multilinear operators, weakly
unconditionally Cauchy series.
Unconditionally converging operators were introduced by Pelczynski in [13] and
since then, they have been extensively studied. In [9] a notion of unconditionally
converging polynomial is introduced, but it seems to be too general, since the co-
incidence of this class with other classes of polynomials impose severe restrictions
on the space (see [9, Theorem 10], e.g.). In [5] the author gives a new definition of
unconditionally converging polynomial which allows to extend many of the results
of unconditional converging operators (see [6]). In this note, we extend this defi-
nition to multilinear operators, proving some properties and characterizations, and
studying the relationship existing with other definitions in the literature.
Let Ebe a Banach space and E∗its dual. We recall that a formal series
P∞
n=1 xn⊂Eis said to be weakly unconditionally Cauchy if for every x∗∈E∗,
the series P∞
n=1 |x∗(xn)|converges (for equivalent definitions and basic properties
of this kind of series we refer the reader to [4]).
Definition 1. Let E1, . . . , Ek, X be Banach spaces. A multilinear operator T∈
Lk(E1, . . . , Ek;X)is said to be unconditionally converging, and we will write
T∈ Lk
uc(E1, . . . , Ek;X)if, for every weakly unconditionally Cauchy (w.u.C.) series
Pxn
i⊂Ei, with i∈ {1, . . . , k}, the sequence (T(sm
1, . . . , sm
k))mconverges in norm,
where sm
i=Pm
n=1 xn
i.
Analogously, a polynomial P∈ P(kE;X)is said to be unconditionally con-
verging, and we will write P∈ Puc (kE;X)if, for every weakly unconditionally
converging series Pxn⊂E, the sequence (P(sm))mconverges in norm, where
sm=Pm
n=1 xn([5]).
Recall that there is a canonical isomorphism between
Lk(E1, . . . , Ek;X) and L(E1ˆ
N· · · ˆ
NEk;X), where E1ˆ
N· · · ˆ
NEkdenotes projec-
tive tensor product. A continuous multilinear map T∈ Lk(E1, . . . , Ek;X) is said
to be compact (resp. weakly compact) if the corresponding linear operator on the
1991 Mathematics Subject Classification. 46B28.
Key words and phrases. unconditionally converging,multilinear operators.
First and third authors are partially supported by DGICYT grant PB97-0240.
Second author is partially supported by Conacyt grant I 29875-E.
1

2 FERNANDO BOMBAL, MAITE FERN´
ANDEZ-UNZUETA, AND IGNACIO VILLANUEVA
tensor product is compact (resp. weakly compact); equivalently, if Tmaps the
product of the corresponding unit balls into a relatively compact (resp., weakly
compact) subset of X.
Using the polarization formula, (see [12, pg. 6]) it is easy to check that a poly-
nomial is unconditionally converging if and only if so is its associated symmetric
multilinear generator.
In the linear case, a direct consequence of the Orlicz-Pettis theorem is that the
unconditionally converging operators are precisely those which transform w.u.C
series into weakly convergent ones. We shall show that this result extends to the
multilinear case. The proof is a suitable modification of the corresponding result
given in [6] for polynomials. We shall need the following extension of [3, Lemma
7.4]:
Lemma 2. Let E1, . . . , Ekbe Banach spaces, of which all but at most one have
the Dunford-Pettis property. Let {xn
i}∞
n=1,{yn
i}∞
n=1 be weakly Cauchy sequences
in Ei,(1 ≤i≤k)such that {xn
i−yn
i}∞
n=1 is weakly null (1 ≤i≤k)). Then
{xn
1⊗ · ·· ⊗ xn
k−yn
1⊗ · ·· ⊗ yn
k}∞
n=1 is a weak null sequence in E1ˆ
⊗ · · · ˆ
⊗Ek.
Proof. We have
xn
1⊗ · ·· ⊗ xn
k−yn
1⊗ · ·· ⊗ yn
k= (xn
1−yn
1)⊗xn
2⊗ · ·· ⊗ xn
k+
+yn
1⊗(xn
2−yn
2)⊗ · ·· ⊗ xn
k+. . . +yn
1⊗ · ·· ⊗ yn
k−1⊗(xn
k−yn
k).
Every member of the right side has k−1 factors which are weak Cauchy sequences,
and the other factor is a weak null sequence. Hence, by the arguments of [3, Lemma
7.4] each summand converges weakly to zero in the tensor product. 
Theorem 3. Let E1, . . . , Ek, X be Banach spaces. A multilinear operator T∈
Lk(E1, . . . , E;X)is unconditionally converging if and only if, for every weakly un-
conditionally Cauchy series Pxn
i⊂Ei, with i∈ {1, . . . , k}, the sequence (T(Pm
n=1 xn
1,
. . . , Pm
n=1 xn
k))mconverges weakly in X.
Proof. Suppose Tsatisfy the weaker hypothesis, but it is not unconditionally con-
verging. Then, there are w.u.C. series Pxn
iin Ei(1 ≤i≤k),  > 0 and a pair of
subsequences (nj),(mj) with nj> mj, such that
kT(snj
1, . . . , snj
k)−T(smj
1, . . . , smj
k)k ≥ , (∗)
where sm
i=Pm
n=1 xn
i. Let
S:co× · ·· × co→E1× · · · × Ek
be the continuous k−linear map defined by
S(Xan
1en, . . . Xan
ken) = (S1(Xan
1en), . . . Sk(Xan
ken)) :=
:= (Xan
1xn
1, . . . Xan
kxn
k),
where (en) is the usual basis in co, and put V:= T◦S. With this notation, we
have
T(snj
1, . . . , snj
k)−T(smj
1, . . . , smj
k) = T(snj
1−smj
1, snj
2, . . . , snj
k)+
+T(smj
1, snj
2−smj
2, snj
3, . . . , snj
k) + . . . +T(smj
1, . . . , smj
k−1, snj
k−smj
k)
=V(zj−wj, zj, . . . , zj) + . . . +V(wj, . . . , wj, zj−wj),

UNCONDITIONALLY CONVERGING MULTILINEAR OPERATORS 3
with zj:= Pnj
k=1 ek, wj:= Pmj
k=1 ek∈co. Then (zj),(wj) are weakly Cauchy
sequences, and (zj−wj) converges weakly to zero. Hence, by Lemma 2 for instance,
each member of the right side of the above sum converges weakly to zero. If we were
able to prove that they converge in norm, we should get a contradiction to (*) and
the proof would be over. But Pelczynski showed in [14] that every weakly compact
multilinear operator on a product of spaces with the Dunford-Pettis property (like
co), transforms weakly Cauchy sequences into a norm convergent one. Therefore,
it suffices to prove the following
Claim: Vis weakly compact.
In fact, let (vn
1), . . . (vn
k) be ksequences in the unit ball of co. Passing to some
subsequences if necessary, we can suppose that all of them are weakly Cauchy. Prop-
erty (u) of co(see [11, 1.c.1]) guarantees the existence of w.u.C. series Pwn
i,1≤
i≤k, such that the sequences (σm
i−vm
i) are weakly null for i= 1, . . . , k (where
σm
i=Pm
n=1 wn
i). By Lemma 2 (σm
1⊗ · · · ⊗ σm
k−vm
1⊗ · · · ⊗ vm
k) is weakly null and
so
V(σm
1, . . . , σm
k)−V(vm
1, . . . , vm
k)w
→0.
But, by hypothesis, V(σm
1, . . . , σm
k) = T(Pm
n=1 S1(wn
1), . . . Pm
n=1 Sk(wn
k)) converges
weakly to some x∈X(since PSi(wn
i) are w.u.C. series in Ei), and thus V(vm
1, . . . , vm
k)
also converges weakly to x, which proves the claim. 
We remark that the above proof can be repeated word for word to show that
whenever E1, . . . Ekare Banach spaces with the Dunford-Pettis property, the prop-
erty (u) of Pelczynski and without copies of `1, every k−linear unconditionally
converging operator on E1×. . . ×Ekis weakly compact. When all the spaces
coincide with co, we can say more:
Corollary 4. Every k−linear unconditionally converging operator on co×. . . ×co
is compact.
Proof. As we have mentioned before, every T∈ Lk
uc(co, . . . , co;X) is weakly com-
pact. By induction on k, using the facts that every operator from cointo c∗
o(≈`1)
is compact and that c∗
ohas the approximation property, it can be easily proved that
the dual of coˆ
⊗ · · · ˆ
⊗cois the injective tensor product of kspaces `1, and hence a
Schur space, since this property is preserved by taking injective tensor products (see
[15, Proposition 1.3], e.g.). If ˜
T∈ L(coˆ
⊗ · · · ˆ
⊗co;X) is the operator corresponding
to T, it is weakly compact and, by Gantmacher’s theorem, so is its transpose ˜
T∗.
But the range of this operator is a Schur space. Therefore, ˜
T∗and hence Tare
compact, by Schauder’s theorem. 
The above result is also true if instead of cowe take any C(Ω) space, with
Ω a compact scattered Hausdorff topological space. The proof is the same, once
we know that every T∈ Lk
uc(C(Ω1), . . . , C (Ωk); X)is weakly compact, which was
proved in [10].
Also embedded in the proof of Theorem 3 is the result that the sequence {T(Pm
n=1 xn
1,
. . . , Pm
n=1 xn
k)}mis weakly Cauchy for every T∈ Lk(E1, . . . , Ek;X)and w.u.C. se-
ries Pxn
i⊂Ei,1≤i≤k. In fact, with the above notation,
T(
m
X
n=1
xn
1, . . . ,
m
X
n=1
xn
k) = V(
m
X
n=1
en, . . .
m
X
n=1
en)

4 FERNANDO BOMBAL, MAITE FERN´
ANDEZ-UNZUETA, AND IGNACIO VILLANUEVA
and {Pm
n=1 en⊗ · · · ⊗ Pm
n=1 en}mis weakly Cauchy by Lemma 2. An immediate
consequence of this is the following corollary:
Corollary 5. i) Every weakly compact multilinear operator is unconditionally con-
verging.
ii) If Xis weakly sequentially complete, every T∈ Lk(E1, . . . , En;X)is uncon-
ditionally converging.
Later we shall give more conditions under which every multilinear operator is
unconditionally converging.
In [2, Theorem 2.3], a result about completely continuous multilinear operators
is proved which turns out to be very useful in working with this kind of mapping.
Now we shall prove an analogous result for unconditionally converging operators,
which allows to mimic for this class of operators many results and techniques of
completely continuous mappings, as can be seen for example in [10].
We will state first an auxiliary definition.
Definition 6. Let E1, . . . , Ek, X be Banach spaces. A multilinear operator T∈
Lk(E1, . . . , Ek;X)is said to be unconditionally continuous, if, for every weakly
unconditionally Cauchy series Pn∈Nxn
i⊂Eiwith i= 1, . . . , k such that
sn
i=
n
X
m=1
xm
i
ω
→xi∈Ei
the following holds
lim
n→∞ kT(sn
1, . . . , sn
k)−T(x1, . . . , xk)k= 0 .
The above definition is clearly easier to handle than that of unconditionally
converging operators. Our aim is to prove that both classes of operators coincide.
In one direction the proof of this is based on the proof of [2, Lemma 2.4 and
Theorem 2.3].
Theorem 7. Let E1, . . . , Ek, X be Banach spaces and let
T∈ Lk(E1, . . . , Ek;X). Then the following are equivalent:
i) Tis unconditionally converging.
ii) Tis such that if, for every j= 1, . . . , k,P∞
n=1 xn
j⊂Ejis a weakly uncon-
ditionally Cauchy series and there exists i∈ {1, . . . , k}such that P∞
n=1 xn
iweakly
converges to zero, then
lim
m→∞ kT(sm
1, . . . , sm
k)k= 0 , where sm
j=
m
X
n=1
xn
j.
iii) Tis unconditionally continuous.
Proof. i) ⇒ii): We will do the proof assuming without loss of generality that in
ii), i= 1. Let Tand Pnxn
1,. . . , Pnxn
kbe as in the hypothesis. If the result is not
true, then there exist an  > 0 and an increasing sequence of indexes J= (m(l))l
such that, for every l∈N,


T(sm(l)
1, . . . , sm(l)
k)

>  .
On the other hand, it is clear that, if we fix m∈N, the operator Tm=Tsm
2,...,sm
k∈
L(E1;X) defined as
Tm(x) = T(x, sm
2, . . . , sm
k)

UNCONDITIONALLY CONVERGING MULTILINEAR OPERATORS 5
is unconditionally converging, and therefore there exists n(m)> m such that for
every r≥n(m),
kTm(sr
1)k<
2.
We can assume that if j∈J, then n(j)∈J. Then, for every j∈N,


T(sm(l)
1, . . . , sm(l)
k)−T(sn(m(l))
1, sm(l)
2, . . . , sm(l)
k)

>
2
Let p(0) = 1, p(1) = n(m(1)) and, if l > 1, p(l) = n(p(l−1)). We then have that,
for every l≥1,


T(sp(l)
1, . . . , sp(l)
k)−T(sp(l+1)
1, sp(l)
2, . . . , sp(l)
k)

>
2.
Let us now define
yl
1=






p(j+1)
X
n=p(j)+1
xn
1if l= 2j+ 1
0 if l= 2j
and for i > 1
yl
i=






0 if l= 2j+ 1
p(j)
X
n=p(j−1)+1
xn
iif l= 2j
(with p(r) = 0 if r < 0.)
Then it is clear that Pnyn
i⊂Eiis a w.u.C. series for each i∈ {1, . . . , k}and,
if we call σm
i=Pm
n=1 yn
i, we get that
σ2j+1
1=sp(j+1)
1, σ2j
1=σ2j−1
1=sp(j)
1
and, if i > 1,
σ2j+1
i=σ2j
i=sp(j)
i
Therefore, for every j∈N,


T(σ2j+1
1, . . . , σ2j+1
k)−T(σ2j
1, . . . , σ2j
k)

=
=

T(sp(j+1)
1, sp(j)
2, . . . , sp(j)
k)−T(sp(j)
1, sp(j)
2, . . . , sp(j)
k)

>
2
a contradiction to the fact that Tis unconditionally converging.
The proof that ii) ⇒iii) is easy considering that
kT(sn
1, . . . , sn
k)−T(x1, . . . , xk)k ≤ kT(sn
1−x1, sn
2, . . . , sn
k)k+
+kT(x1, sn
2−x2, x3. . . , xk)k+· · · +kT(x1, . . . , xk−1, sn
k−xk)k.
iii) ⇒ii): This proof is an adaptation of the proof of [2, Lemma 2.4]; we will do it
by induction on k. If k= 1 the result is clear. Let us suppose it true for k−1. Now
we will consider the case of k-linear operators: if the result is false, for 1 ≤j≤k
there exist weakly unconditionally Cauchy series P∞
n=1 xn
j⊂Ejwhere we suppose
that P∞
n=1 xn
1weakly converges to 0, such that (T(Pm
n=1 xn
1, . . . , Pm
n=1 xn
k)m∈Ndoes
not converge to 0. Then there exist  > 0 and an infinite subset J⊂Nsuch that
for every j∈J,
kT(
j
X
n=1
xn
1, . . . ,
j
X
n=1
xn
k)k> 

6 FERNANDO BOMBAL, MAITE FERN´
ANDEZ-UNZUETA, AND IGNACIO VILLANUEVA
We will consider J={j(1) < j(2) <· · · < j(n)<· · · } Now let us fix m∈N. The
function
Tsj(m)
k
:E1× · ·· × Ek−17→ X
defined as
Tsj(m)
1(x1, . . . , xk−1) = T(x1, . . . , xk−1,
j(m)
X
n=1
xn
k)
is clearly unconditionally continuous. Then the induction hypothesis assures that
there exist an index p(m)∈Nsuch that
kTsj(m)
1(
l
X
n=1
xn
1, . . . ,
l
X
n=1
xn
k−1)k< /2 for every l > p(m)
Clearly we can choose j(m)< p(m)< p(m+ 1) and p(m)∈Jfor every m∈N. In
particular for every m∈Nwe get
/2≤ kT(
p(m)
X
n=1
xn
1, . . . ,
p(m)
X
n=1
xn
k−1,
p(m)
X
n=1
xn
k)k−
−kT(
p(m)
X
n=1
xn
1, . . . ,
p(m)
X
n=1
xn
k−1,
j(m)
X
n=1
xn
k)k=
=kT(
p(m)
X
n=1
xn
1, . . . ,
p(m)
X
n=1
xn
k−1,
p(m)
X
n=1
xn
k−
j(m)
X
n=1
xn
k)k
Let us consider now the series
ym
i=
p(m+1)
X
n=p(m)+1
xn
ifor i= 1 . . . k −1
and
ym
k=
p(m+1)
X
n=p(m)+1
xn
k−
j(m+1)
X
n=j(m)+1
xn
kfor i=k
These kseries are weakly unconditionally Cauchy, they have the property that at
least two of them (the first and the last) weakly sum to 0 and they verify that





T m
X
n=1
yn
1, . . . ,
m
X
n=1
yn
k!




≥/2
Repeating the reasoning we obtain k weakly unconditionally Cauchy series P∞
n=1 zn
i
such that all of them weakly sum to 0 and





T j
X
n=1
zn
1, . . . ,
j
X
n=1
zn
k!




≥/(2k−1)
which is a contradiction.
ii) ⇒i) Let Pxn
1⊂E1, . . . , Pxn
k⊂Ekbe weakly unconditionally Cauchy series
and let (p(r))rand (q(r))rbe two increasing sequences of indexes with p(0) = q(0) =
0. Then
kT(sp(r)
1, . . . , sp(r)
k)−T(sq(r)
1, . . . , sq(r)
k)k=
=kT(sp(r)
1−sq(r)
1, sp(r)
2, . . . , sp(r)
k)k+kT(sq(r)
1, sp(r)
2−sq(r)
2, sp(r)
3, . . . , sp(r)
k)k+

UNCONDITIONALLY CONVERGING MULTILINEAR OPERATORS 7
+· · · +kT(sq(r)
1, . . . , sq(r)
k−1, sp(r)
k−sq(r)
k)k
Let us see that each of these terms converges to zero when rgrows to infinity.
We will prove it only for the second term, since the others are treated similarly.
Let us consider the following series:
ym
1=
p(m)
X
n=p(m−1)+1
xn
1,
ym
2=
p(m)
X
n=p(m−1)+1
xn
2−
q(m)
X
n=q(m−1)+1
xn
2and
ym
i=
p(m)
X
n=p(m−1)+1
xn
ifor i= 3, . . . , k .
Clearly, for every i= 1, . . . , k,Pyn
iis a weakly unconditionally Cauchy series and
besides Pyn
2weakly converges to zero. Therefore the hypothesis proves that this
term converges to zero, which finishes the proof. 
We want to point out that this result seems to be new also in the linear case.
The authors have verified that several proofs of results concerning unconditionally
converging linear operators can be made much simpler by using it.
Of course the result implies the existence of weakly unconditionally Cauchy series
that weakly converge to zero but do not converge weakly unconditionally. A simple
and not new example of one such series is Pxn⊂c0where
x1=e1
xn=en−en−1if n > 1.
As we mentioned at the beginning, a different definition of unconditionally con-
verging polynomials which also extends the linear one was given in [9]. The natural
extension to multilinear operators would be the following:
Definition 8. Let E1, . . . , Ek, X be Banach spaces. A multilinear operator T∈
Lk(E1, . . . , Ek;X)will be called weakly unconditionally converging, and we will
write T∈ Lk
wuc(E1, . . . , Ek;X), if for every weakly unconditionally Cauchy series
Pxn
i⊂Ei, with i∈ {1, . . . , k}, the series PT(xn
1, . . . , xn
k)converges in norm.
This definition gives rise to a strictly wider class, as the next two propositions
show:
Proposition 9. Every unconditionally converging multilinear operator is weak un-
conditionally converging.
Proof. Let E1, . . . , Ek, X and Pxn
ibe as in the above definition and let T∈
Lk
uc(E1, . . . , Ek;X). The k−Rademacher generalized functions {sn(t)}∞
n=1 ([1]) sat-
isfy the following orthogonality properties:
Z1
0
si1(t)· · · sik(t)dt =1 if i1=. . . = 1k
0 otherwise.

8 FERNANDO BOMBAL, MAITE FERN´
ANDEZ-UNZUETA, AND IGNACIO VILLANUEVA
Then
m
X
n=1
T(xn
1, . . . , xn
k) = R1
0Pm
i1···ik=1 si1(t)· · · sik(t)T(xi1
1, . . . , xik
k)dt
=R1
0T(Pm
i1=1 si1(t)xi1
1, . . . , Pm
ik=1 sik(t)xik
k)dt.
For every t∈[0,1] the series Psn(t)xn
i(1 ≤i≤k) are weakly unconditionally
Cauchy, and therefore the sequence {TPm
i1=1 si1(t)xi1
1, . . . , Pm
ik=1 sik(t)xik
k}con-
verges in norm. Hence, P∞
n=1 T(xn
1, . . . , xn
k) converges in norm, that is, Tis weakly
unconditionally converging. 
However, not every weak unconditionally converging operator is unconditionally
converging. In fact, we have:
Proposition 10. Let k > 1and let E1, . . . Ekbe Banach spaces. The following
assertions are equivalent:
i) No E1, . . . , Ekcontains copies of co.
ii) Lk
uc(E1, . . . , Ek;X) = Lk(E1, . . . , Ek;X), for every Banach space X.
iii) Lk
uc(E1, . . . , Ek;X) = Lk
wuc(E1, . . . , Ek:X), for every Banach space X.
Proof. If no Eicontains copies of co, every w.u.C. series is converges in norm, and
so (i) implies (ii).
Obviously, (ii) implies (iii).
Finally, let us prove that (iii) implies (i): Suppose for instance that E1contains
a copy of coand let (xn
1) be a sequence equivalent to the usual cobasis. Choose
norm one elements x∗
i∈E∗
i(2 ≤i≤k) and define
T:E1× · ·· × Ek→E1
by the formula T(z1, . . . , zk) := x∗
2(z2)· · · x∗
k(zk)z1. If Pyn
i⊂Ei(1 ≤i≤k) are
w.u.C. series, we have
∞
X
n=1
kT(yn
1, . . . yn
k)k ≤ (sup
n
kyn
1k · · · kyn
k−1k)(
∞
X
n=1
|x∗
k(yn
k)|)<∞
which proves that Tis weakly unconditionally converging. However, if zi∈Ei(2 ≤
i≤k) are chosen in such a way that x∗
i(zi) = 1 for i= 2, . . . k, then
T(
m
X
n=1
xn
1, z2, . . . , zk) =
m
X
n=1
xn
1
which does not converge in norm. Therefore, Tis not unconditionally converging.

The following two propositions were told to us by J. Gutirrez, to whom thanks
are given. Proposition 11 was proved in [7] under the stronger hypothesis that X∗
contains no copy of `1. Proposition 12 is an application of Theorem 7, and it refines
[8, Theorem 3].
Proposition 11. If Xdoes not contain a copy of c0, then for every Banach space
E1, . . . Ek, every multilinear operator from E1, . . . Ekinto Xis unconditionally con-
verging.

UNCONDITIONALLY CONVERGING MULTILINEAR OPERATORS 9
Proof. Let T∈ Lk(E1, . . . , Ek;X) and let Pnxn
1⊂E1,. . . , Pnxn
k⊂Ekbe weakly
unconditionally Cauchy series. Let us define
S∈ Lk(c0;E1× · ·· × Ek) as in the proof of 3, and put Q:= T◦S.
From [8, Theorem 6] (or from [16]) it follows that Qsends weakly Cauchy se-
quences in E1×,· · · ,×Ekinto norm converging ones, and therefore it is uncondi-
tionally converging. So, the sequence
Q(
m
X
n=1
en, . . . ,
m
X
n=1
en) = T(
m
X
n=1
xn
1, . . . ,
m
X
n=1
xn
k)
converges, which implies that Tis unconditionally converging. 
Proposition 12. Let E1, . . . Ekand Xbe Banach spaces such that L(Ei;X) =
Luc(Ei;X)for each i∈ {1, . . . , k}. Then it is also true that Lk(E1, . . . Ek;X) =
Lk
uc(E1, . . . Ek;X).
Proof. Suppose first that one of the Ei’s contains a complemented copy of c0. Then
Xdoes not contain a copy of c0and Proposition 11 gives us the result. Now, if
none of the Ei’s contains a complemented copy of c0, then, by [8, Lemmas 1 and
2], we have that L(Ei;c(X)) = Luc(Ei;c(X)) (where c(X) stands for the Banach
space of all X-valued convergent sequences, endowed with the sup norm), and we
proceed by induction on k. Suppose the result to be true for k−1 multilinear
mappings, and let us consider T∈ Lk(E1, . . . , Ek;X) and weakly unconditionally
Cauchy series Pnxn
1⊂E1,. . . , Pnxn
k⊂Eksuch that one of them, say the first
one, weakly converges to zero. Let us define
S:E17→ c(X)
by
S(x1) = (T(x1, sm
2, . . . , sm
k))m
where sm
i=Pm
n=1 xn
i. From the induction hypothesis it is clear that Sis well
defined. Since Sis an unconditionally converging operator, we get that
kT(sm
1, . . . , sm
k)k ≤ kS(sm
1)k → 0
which, by Theorem7, implies that Tis unconditionally converging. 
References
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10 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ-UNZUETA, AND IGNACIO VILLANUEVA
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Proc. Amer. Math. Soc.
Departamento de Anlisis Matemtico, Facultad de Matemticas, Universidad Complutense
de Madrid, MADRID 28040
E-mail address:bombal@eucmax.sim.ucm.es
C.I.M.A.T., A.P. 402, 36000, Guanajuato, Gto., Mxico
E-mail address:fernan@cimat.mx
Departamento de Anlisis Matemtico, Facultad de Matemticas, Universidad Complutense
de Madrid, MADRID 28040
E-mail address:ignacio villanueva@mat.ucm.es