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SOME CLASSES OF MULTILINEAR OPERATORS ON C(K)
SPACES
FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
Abstract. We obtain a classification of the projective tensor product
of C(K) spaces according to the fact that none, exactly one or more
than one of the factors contain copies of `1, in terms of the behaviour of
certain classes of multilinear operators on the product of the spaces or
the verification of certain Banach space properties on the corresponding
tensor product. The main tool is an improvement of some results of
G. Emmanuele and Hensgen about the Reciprocal Dunford-Pettis and
Pelczynski’s V properties on the projective tensor product of Banach
spaces. We also study the relationship between several classes of multi-
linear operators and the corresponding linear associated operators.
1. Introduction
In the past years much research has been done in the theory of multi-
linear operators and polynomials between Banach spaces. In particular,
different classes of multilinear operators or polynomials have been defined
which extended the corresponding notions for linear operators, and the re-
lations between some of these classes have been studied. If E, F and Xare
Banach spaces and T:E×F−→ Xis a bilinear operator, it is well known
that there exists only one linear operator ˆ
T:Eˆ
⊗πF−→ Xcanonically
associated to T, where Eˆ
⊗πFis the projective tensor product of Eand
F. In Section 2 we improve some results of G. Emmanuele and Hensgen
to establish, under suitable conditions, some non trivial relationships be-
tween several classes of bilinear operators. In Section 3 we use the results
of Section 2 to obtain a classification of the projective tensor product of
several C(K) spaces, according to the fact that none, exactly one o more
than one of the factors contain copies of `1, in terms of the behaviour of
certain classes of multilinear operators on the product of the spaces or the
verification of certain Banach space properties on the corresponding tensor
product. Finally, in Section 4 we study the relation between ˆ
Tbelonging
1991 Mathematics Subject Classification. 46B25, 46B28.
Key words and phrases. Tensor products, C(K) spaces.
First and third authors are partially supported by DGICYT grant PB97-0240. Second
author partially supported by Conacyt Grant J32150-E.
1
2 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
to certain operator ideals and Tbelonging to certain classes of multilinear
operators.
The notations and terminology used along the paper will be the standard
in Banach space theory, as for instance in [9]. However, before going any
further, we shall clear out some terminology: Lk(E1. . . , Ek;X) will be the
Banach space of all the continuous k-linear mappings from E1× · · · × Ek
into Xand Lk
wc(E1. . . , Ek;X) will be the closed subspace of it formed by
the weakly compact multilinear operators. When X=Kor k= 1, we
omit them. We write K(E;X) for the space of compact operators from
Einto X. As usual, E1ˆ
⊗π· · · ˆ
⊗πEkstands for the (complete) projective
tensor product of the Banach spaces E1, . . . , Ek. It T∈ Lk(E1. . . , Ek;X)
we denote by ˆ
T:E1ˆ
⊗π· · · ˆ
⊗πEk→Xits linearization.
We say that T∈ Lk(E1, . . . , Ek;X) is completely continuous, and we
write T∈ Lk
cc(E1, . . . , Ek;X), if, given weak Cauchy sequences (xn
i)n∈
N
⊂
Ei(1 ≤i≤k), the sequence (T(xn
1, . . . , xn
k))nis norm convergent in X.
This definition may be adapted to polynomials in an obvious way. The
space of completely continuous polynomials is denoted by Pcc(kE;X). By
the polarization formula [17, Theorem 1.10], a polynomial is completely
continuous if and only if so is its associated symmetric multilinear operator.
If X=K, i.e., if Tis a multilinear form, we will use the notation weakly
sequentially continuous instead of completely continuous.
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),
We shall say that Tis regular if all the maps Ti,1≤i≤kdefined above,
are weakly compact.
Recall that Ehas the Dunford-Pettis property (DPP, for short) if, for ev-
ery X,Lwc(E;X)⊆ Lcc (E;X). Examples of spaces with the DPP are C(K)
and L1(µ) spaces. Ehas the reciprocal Dunford-Pettis property (RDPP, for
short) if, for every X,Lcc(E;X)⊆ Lwc (E;X). The spaces containing no
copy of `1, and C(K) spaces have the RDPP. Both properties were intro-
duced in [14].
A formal series Pxnin a Banach space Eis weakly unconditionally
Cauchy (w.u.C., for short) if there is C > 0 such that, for any finite subset
∆ of Nand any signs ±, we have kPn∈∆±xnk ≤ C. For other equivalent
definitions, see [8, Theorem V.6]. The series Pxnis unconditionally con-
vergent if every subseries is norm convergent. Other equivalent definitions
may be seen in [9, Theorem 1.9].
SOME CLASSES OF MULTILINEAR OPERATORS ON C(K) SPACES 3
A linear operator between Banach spaces is unconditionally converging
if it takes w.u.C. series into unconditionally convergent series. A Banach
space Eis said to have PeÃlczy´nski’s property (V) if every unconditionally
converging linear operator on Eis weakly compact. This property was in-
troduced in [18], where it is shown that C(K) spaces have property (V), and
that the dual of a space with property (V) is weakly sequentially complete.
Following [12], we say that T∈ Lk(E1, . . . , Ek;X) is unconditionally
converging if, given w.u.C. series Pn∈
N
xn
iin Ei(1 ≤i≤k), the sequence
(T(sm
1, . . . , sm
k))m
is norm convergent in X, where sm
i=Pm
n=1 xn
i. This definition may be
adapted to polynomials in an obvious way. Since a linear operator fails to
be unconditionally converging if and only if it preserves a copy of c0[8,
Exercise V.8], it is clear that the definition of unconditionally converging
k-linear operators agrees for k= 1 with that of unconditionally converging
linear operators.
By the polarization formula, a polynomial is unconditionally converging
if and only if so is its associated symmetric multilinear operator.
Since BEˆ
⊗πF=coe(BE⊗BF), it follows that Tis (weakly) compact if
and only in ˆ
Tis (weakly) compact.
2. Some properties of Bilinear operators
In [10] and [11], the following results are proved:
Theorem 2.1. Let Ebe a Banach space not containing `1and Fa Banach
space with the RDPP. If L(E;F∗) = K(E;F∗), then Eˆ
⊗πFhas the RDPP.
Theorem 2.2. Let E, F be Banach spaces with the RDPP such that E∗and
F∗are weakly sequentially complete. If L(E;F∗) = K(E;F∗), then Eˆ
⊗πF
has the RDPP.
Theorem 2.3. Let E, F be Banach spaces with Property (V) such that
L(E;F∗) = K(E;F∗). Then Eˆ
⊗πFhas property (V).
Taking advantage of the ideas in those papers, we prove now a strength-
ening of these results. First we will need some definitions and lemmas.
Definition 2.4. Let Ebe a Banach space. A set M⊂E∗is an L-set
(respectively a V-set), if for every weakly null sequence (xn)⊂E(resp.
every w.u.C. series Pnxn⊂E), we have
lim
n→∞ sup{|x∗(xn)|:x∗∈M}= 0.
The following result is well known.
4 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
Proposition 2.5. Let Ebe a Banach space. Then:
(a) Ehas RDPP if and only if every L-set is relatively weakly compact
([16]. see also [2]).
(b) Ehas property (V) if and only if every V-set is relatively weakly
compact ([18]).
We will also need some results concerning the Aron-Berner extension of
a multilinear operator: if T:E×F−→ Xis a bilinear operator, then we
can define its Aron-Berner extension,
AB(T) : E∗∗ ×F∗∗ −→ X∗∗
by
AB(T)(z1, z2) = lim
αlim
βT(xα, yβ),
where (xα)⊂Eis a net weak-star converging to z1and (yβ)⊂Fis a
net weak-star converging to z2. Related to this extension we will use the
following results from [15].
Lemma 2.6. Let E, F be Banach spaces with the RDPP, and Xany Banach
space. If T:E×F−→ Xis a completely continuous bilinear operator, then
its Aron-Berner extension AB(T)takes values in X.
Lemma 2.7. Let E, F be Banach spaces such that their duals E∗and F∗
have the Dunford-Pettis Property and such that L(E;F∗) = Lwc(E;F∗). For
any Banach space X, if T:E×F−→ Xis a bilinear operator such that its
Aron-Berner extension AB(T)is X-valued, then AB(T) : E∗∗ ×F∗∗ −→ X
is completely continuous.
Lemma 2.8. Let E, F be Banach spaces with property V, and Xany Banach
space. If T:E×F−→ Xis an unconditionally converging bilinear operator,
then its Aron-Berner extension AB(T)takes values in X.
Lemma 2.9. Let E, F be Banach spaces such that L(E;F∗) = Lwc(E;F∗).
For any Banach space X, if T:E×F−→ Xis a bilinear operator such that
its Aron-Berner extension AB(T)is X-valued, then AB(T) : E∗∗ ×F∗∗ −→
Xis unconditionally converging.
Now we can prove the following.
Proposition 2.10. Let Ebe a Banach space not containing `1and Fa
Banach space with the RDPP. Assume further that L(E;F∗) = K(E;F∗)
and that E∗and F∗have the Dunford-Pettis Property. For every Banach
space X, if T:E×F−→ Xis a completely continuous bilinear operator,
then Tis weakly compact.
SOME CLASSES OF MULTILINEAR OPERATORS ON C(K) SPACES 5
Proof. Let Tbe as in the hypothesis and let ˆ
T:Eˆ
⊗πF−→ Xbe the
operator canonically associated to T. Since Tis weakly compact if and
only if ˆ
Tis weakly compact, it suffices to prove that ˆ
T∗is weakly compact.
Let then M=ˆ
T∗(BX∗)⊂(Eˆ
⊗πF)∗=K(E;F∗). Let (hn)n⊂Mand let
(ϕn)n⊂BX∗be such that ˆ
T∗(ϕn) = hnfor every n∈N. Define Hby
H= span[hn(x) : x∈E, n ∈N]. Then His a closed subspace of F∗and H
is separable, because, for every n∈N,hn:E−→ F∗is compact. Let now
Y⊂Fbe a countable norming set of Hand let y∈Y.
Claim 1: The set {h∗
n(y); n∈N} ⊂ E∗is an L-set.
Proof of the claim: Let (xm)m⊂Ebe a weakly converging to 0 sequence.
We have
h∗
n(y)(xm) = hn(xm)(y) = ˆ
T∗(ϕn)(xm⊗y) = hˆ
T(xm⊗y), ϕni=hT(xm, y), ϕni.
Therefore
lim
m→∞ sup
n∈
N
|h∗
n(y)(xm)| ≤ lim
m→∞ kT(xm, y)k= 0,
and the claim is proved.
So {h∗
n(y); n∈N} ⊂ E∗is relatively weakly compact and therefore we can
suppose (using the fact that Yis countable and considering subsequences
if necessary) that, for every y∈Y, (h∗
n(y))nis a weakly Cauchy sequence.
Let now x∗∗ ∈E∗∗.
Claim 2: The set {h∗∗
n(x∗∗); n∈N} ⊂ F∗is an L-set.
Proof of the claim: If we think of hn∈ K(E;F∗) as a bilinear form, hn:
E×F−→ K, it is clear that h∗∗
n(x∗∗)(y) = AB(hn)(x∗∗, y), where AB(hn)
denotes any of the two Aron-Berner extensions of hn. Let then (ym)m⊂F
be a weakly converging to 0 sequence. Then
h∗∗
n(x∗∗)(ym) = AB(ˆ
T∗(ϕn)(x∗∗, ym).
Let us see now that AB(ˆ
T∗(ϕn)(x∗∗, ym) = hAB(T)(x∗∗, ym), ϕni: let (xα)α⊂
Ebe a bounded net weak-star convergent to x∗∗. Then
AB(ˆ
T∗(ϕn)(x∗∗, ym) = lim
α
ˆ
T∗(ϕn)(xα, ym) =
= lim
αhˆ
T(xα⊗ym), ϕni= lim
αhT(xα, ym), ϕni=hAB(T)(x∗∗ , ym), ϕni.
Therefore,
lim
m→∞ sup
n∈
N
|h∗∗
n(x∗∗)(ym)| ≤ lim
m→∞ kAB(T)(x∗∗ , ym)k= 0
The last limit is 0 because AB(T) is completely continuous, as follows from
Lemmas 2.6 and 2.7. So, the claim is proved.
Now we can proceed as in [10] to obtain h∈ K(E;F∗) such that hnweakly
converges to h, which finishes the proof. ¤
6 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
Corollary 2.11. Let E, F be Banach spaces with the RDPP such that E∗
and F∗are weakly sequentially complete and have the Dunford-Pettis Prop-
erty. Assume further that L(E;F∗) = K(E;F∗). For every Banach space
X, if T:E×F−→ Xis a completely continuous bilinear operator, then T
is weakly compact.
Proof. The beginning of the proof runs as in Proposition 2.10 to prove that
(hn)nis weakly Cauchy. To finish it, we must reason as in the proof of
Theorem 2.2 (see [10, Cor. 4]). ¤
Since compact operators are completely continuous, Proposition 4.1 below
implies that our results are indeed a strengthening (under the additional
hypothesis that E∗and F∗have the DP property) of Theorems 2.1 and 2.2.
For instance, note that, since c0ˆ
⊗π`∞does not have the DP property ([6]),
there are completely continuous bilinear operators defined on c0×`∞such
that their linear associated operator defined on c0ˆ
⊗π`∞is not completely
continuous.
We have a similar result for unconditionally converging bilinear operators.
This time we do not need additional hypothesis on Eand F.
Proposition 2.12. Let E, F be Banach spaces with Property (V). Assume
further that L(E;F∗) = K(E;F∗). For every Banach space X, if T:E×
F−→ Xis an unconditionally converging bilinear operator, then Tis weakly
compact.
Proof. As in the proof of Proposition 2.10, it suffices to prove that ˆ
T∗(BX∗) =
M⊂(Eˆ
⊗πF)∗=K(E;F∗) is relatively weakly compact. Let then hn,ϕn,
Hand Ybe as in the proof of Proposition 2.10.
Claim 1: The set {h∗
n(y); n∈N} ⊂ E∗is a V-set.
Proof of the claim: Let Pmxm⊂Ebe a w.u.C. series. As in the proof
of Proposition 2.10,
h∗
n(y)(xm) = hn(xm)(y) = hT(xm, y), ϕni.
It is very easy to see that Tis separately unconditionally converging, so
lim
m→∞ sup
n∈
N
|h∗
n(y)(xm)| ≤ lim
m→∞ kT(xm, y)k= 0,
and the claim is proved
So {h∗
n(y); n∈N} ⊂ E∗is relatively weakly compact and again as in the
proof of Proposition 2.10 we can suppose that, for every y∈Y, (h∗
n(y))n⊂
E∗is a weakly Cauchy sequence. Let now x∗∗ ∈E∗∗ .
Claim 2: The set {h∗∗
n(x∗∗); n∈N} ⊂ F∗is a V-set.
Proof of the claim: Let us observe that it follows from Lemmas 2.8 and 2.9
that AB(T) is unconditionally converging, hence separately unconditionally
SOME CLASSES OF MULTILINEAR OPERATORS ON C(K) SPACES 7
converging. So, proceeding as in the proof of Proposition 2.10 we get
lim
m→∞ sup
n∈
N
|h∗∗
n(x∗∗)(ym)| ≤ lim
m→∞ kAB(T)(x∗∗ , ym)k= 0,
and the claim is proved.
Now we can proceed as in [11] to obtain h∈ K(E;F∗) such that hnweakly
converges to h, which finishes the proof. ¤
From Theorem 4.2 below it follows that Proposition 2.12 is strictly stronger
than Theorem 2.3.
3. The projective tensor product of C(K)spaces
This section was the original motivation of this paper. We apply now the
results of the previous sections to obtain a classification of the projective
tensor products of C(K) spaces in terms of some of the classical Banach
space properties.
It is known that the projective tensor product of Banach spaces is associa-
tive, that is, if E, F, G are Banach spaces, then Eˆ
⊗πFˆ
⊗πG=Eˆ
⊗π(Fˆ
⊗πG) =
(Eˆ
⊗πF)ˆ
⊗πG. We will make frequent use of this fact.
Recall that a compact Hausdorff space Kis said to be scattered (or dis-
persed) if it does not contain any non void perfect set. In [19] it is proved,
among other interesting results, that Kis scattered if and only if C(K)
contains no copy of `1. In this case, C(K)∗can be identified with `1(Γ) for
some Γ and, consequently, it is a Schur space.
Theorem 3.1. Let k≥2and K1, . . . , Kkbe infinite compact Hausdorff
spaces. Then, the following assertions are equivalent:
(a1) For every i∈ {1, . . . , k},Kiis scattered.
(b1)C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)has properties DP, RDP and V.
(c1)C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)does not contain any isomorphic copy of `1.
(d1) For any Banach space X, and any k-linear operator T:C(K1)×
· · · × C(Kk)−→ Xthe following are equivalent:
(1) Tis completely continuous.
(2) Tis unconditionally converging.
(3) Tis weakly compact.
(4) Tis regular.
(5) Tis compact.
Proof. We will first prove that (a1) implies all of the others. By a standard
argument (see, for instance, the proof of [5, Theorem 3.1]), it can be proved
that
(C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk))∗=C(K1)∗ˇ
⊗²· · · ˇ
⊗²C(Kk)∗=
=K(C(K1); (C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk))∗)
8 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
Hence, (C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)∗is a Schur space, and so, by a well known
result of Pethe and Thakare, X:= C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk) has the DPP and
contains no copy of `1. Also, from the associativity of projective tensor
products and Theorem 2.3 it follows that Xhas property V, and hence it
also has the RDPP. So (b1) and (c1) hold.
Let us check (d1): Compact multilinear operators are always weakly com-
pact, and weakly compact multilinear operators are completely continuous
on a product of spaces with the DP property. On C(K) spaces, multilinear
unconditionally converging and completely continuous operators, coincide
([15]). Since no C(Ki) (1 ≤i≤k) contains copies of `1by hypothesis,
every completely continuous multilinear operator on C(K1)× · · ·× C(Kk) is
compact. On C(K) spaces, every regular multilinear operator is completely
continuous ([5, Lemma 2.6]). Finally, reasoning as in [1] we can prove that,
under the assumption (a1), if Tis completely continuous, it is weakly con-
tinuous on bounded sets ([1, Proposition 2.12]), hence regular ([1, Theorem
2.9]).
For the converse implications, let us notice that one and only one of the
conditions (a1), or (a2), (a3) (in Theorems 3.3 and 3.4 below) hold. Then,
by exclusion, it is enough to prove that conditions (ai) (i= 1,2,3) imply all
the others in Theorems 3.1, 3.3 and 3.4. ¤
Thanks are given to Joaqu´ın Guti´errez for his help on shortening the proof
of (a1)⇒(d1).
Corollary 3.2. Let Kbe an infinite compact Hausdorff space and 2≤k∈
N. Then, the following assertions are equivalent:
(a) Kis scattered.
(b) ˆ
⊗k
π,sC(K)has properties DP, RDP and V.
(c) ˆ
⊗k
π,sC(K)does not contain any isomorphic copy of `1.
(d) Any k-homogeneous unconditionally converging polynomial on C(K)
is weakly compact.
Proof. Reasoning as in the last part of the proof of Theorem 3.1, it suffices
to prove that condition (a) implies all of the others in this Corollary and in
Corollary 3.5. Hence, suppose (a) holds. Since ˆ
⊗k
π,sC(K) is complemented
in ˆ
⊗k
πC(K), (b) and (c) follow. If a polynomial P:C(K)−→ Xis un-
conditionally converging, then its associated symmetric multilinear form T
is also unconditionally converging. By Theorem 3.1, Tis weakly compact,
hence Pis weakly compact. ¤
Theorem 3.3. Let K1, . . . , Kkbe compact Hausdorff spaces. Then the fol-
lowing assertions are equivalent
SOME CLASSES OF MULTILINEAR OPERATORS ON C(K) SPACES 9
(a2) There exists precisely one i∈ {1, . . . , k}such that Kiis not scattered
(i.e., C(Ki)⊃`1).
(b2)C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)has properties RDP and V, but it does not
have the DP property.
(c2)C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)contains `1, but not complemented.
(d2) For any Banach space Xand any k-linear operator T:C(K1)×
· · ·×C(Kk)−→ X,Tis unconditionally converging (equivalently completely
continuous) if and only if Tis weakly compact, but there are weakly compact
multilinear operators on C(K1)×· · · ×C(Kk)which are neither compact nor
regular.
Proof. As mentioned before, we only need to show that (a2) implies all of
the others. Let us prove (b2): The statement about the DP property can
be seen in [6]. As for properties V and RDP, we will do it by induction
on k. For k= 2 the result follows from Theorems 2.1 and 2.3. Let us
suppose it true for k−1, and let us suppose that C(Kk)⊃`1. From
the induction hypothesis it follows that C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk) has prop-
erty V, hence it can not contain complemented copies of `1. Therefore,
(C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk))∗does not contain copies of c0. Then, every opera-
tor from C(K1) into (C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk))∗is compact. Now we use the
associativity of the projective tensor product and Theorem 2.3.
Clearly C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk) contains a copy of `1. But since it has
property V, none of such copies can be complemented. So, (b2) implies (c2).
Let us now see that (a2) implies (d2). Let us first see that, under such
assumption, unconditionally converging multilinear operators on C(K1)×
· · · × C(Kk) are weakly compact. The proof is a refinement of the proof
of Proposition 2.12. We apply induction on k. For k= 2 the result has
already been proved. Let us suppose it true for k−1, and let T:C(K1)×
· · · × C(Kk)−→ Xbe an unconditionally converging multilinear operator
and let ˆ
Tits linearization. We define
S:C(K1)×(C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk)) −→ X
by
S(f1, y) := ˆ
T((f1⊗y).
Clearly, Sis bilinear and continuous, with kSk=kˆ
Tk=kTk. Let
ˆ
S:C(K1)ˆ
⊗πC(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk)−→ X
be the linear operator associated to S. Clearly, we just have to check that
ˆ
S∗:X∗−→ K(C(K1); (C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk))∗) is weakly compact. As
before, let M=ˆ
S∗(BX∗), let (ϕn)n⊂BX∗and let hn=ˆ
S∗(ϕn). We just
need to extract a weakly converging subsequence from (hn)n. Let H=
span[hn(f1) : f1∈C(K1), n ∈N]. Then His a separable closed subspace
10 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
of (C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk))∗As before, let Y⊂C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk) be a
countable norming set of Hand let y∈Y.
Let us prove that Sy=S(·, y) : C(K1)−→ Xis unconditionally con-
verging. Since Tis separately unconditionally converging, this is clear when
y=f2⊗· · ·⊗fk, and it follows readily for y=Pn
i=1 fi
2⊗· · ·⊗fi
k. For the gen-
eral case it suffices to take into account the density of C(K2)⊗ · · · ⊗ C(Kk)
is C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk) and the fact that the canonical continuous linear
map
C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk)3y7→ Sy∈ L(C(K1), X)
takes values in the closed subspace of the unconditionally converging oper-
ators when y∈C(K2)⊗ · · · ⊗ C(Kk).
Using this, we can reason as in the proof of Proposition 2.12 to establish
that the set {h∗
n(y); n∈N} ⊂ C(K1)∗is a V-set.
So {h∗
n(y); n∈N} ⊂ C(K1)∗is relatively weakly compact and we can
suppose that, for every y∈Y, (h∗
n(y))n⊂C(K1)∗is a weakly Cauchy
sequence. Let now z∈C(K1)∗∗.
Claim : The set {h∗∗
n(z); n∈N} ⊂ (C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk))∗is a V-set.
Proof of the claim: Let Pnyn⊂C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk) be a w.u.C series.
As in the proof of Proposition 2.12, we get that
|h∗∗
n(z)(ym)| ≤ kAB(ˆ
S)(z, ym)k.
Tis unconditionally converging, hence so is AB(T) (Lemma 2.9). There-
fore AB(T)z:C(K2)×. . . ×C(Kk)−→ Xdefined by
AB(T)z(f2, . . . , fk) = AB(T)(z, f2, . . . , fk)
is unconditionally converging. Let ˆ
AB(T)z:C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk)−→ X
be the linear operator associated to it. By the induction hypothesis, ˆ
AB(T)z
is weakly compact, hence unconditionally converging. Clearly we have that,
for every y∈C(K2)ˆ
⊗π· · · ˆ
⊗πC(Kk), AB(S)(z, y) = ˆ
AB(T)z(y) and so the
claim follows.
Now we can again proceed as in [11] to finish the proof that uncondition-
ally converging multilinear operators are weakly compact.
For a weakly compact, neither regular nor compact multilinear operator
on C(K1)×. . . ×C(Kk) we proceed similarly to the proof of the main
result of [6]: suppose that C(K1)⊃`1. Then, there exists a surjective
operator q:C(K1)−→ `2([9, Corollary 4.16]). Let (xn
2)⊂C(K2) and
(µn
2)⊂BC(K2)∗be two sequences such that (xn
2) converges weakly to 0 and
so that µn
2(xn
2) = 1 for every n∈N. Let us choose norm one elements
µi∈C(Ki)∗,x0
i∈C(Ki) such that µi(x0
i) = 1 (3 ≤i≤k). Then we can
SOME CLASSES OF MULTILINEAR OPERATORS ON C(K) SPACES 11
consider the multilinear operator
T:C(K1)×. . . ×C(Kk)−→ `2
defined by
T(x1, . . . , xk) = Ãq(x1)nµn
2(x2)
k
Y
i=3
µi(xi)!n
Tis clearly weakly compact. Let (xn
1)⊂ kqkBC(K1)be a sequence such that
q(xn
1) = en. The sequence (xn
1⊗xn
2⊗x0
3⊗· · ·⊗ x0
k)n⊂C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)
converges weakly to 0, since (xn
1⊗xn
2)nconverges weakly to 0 in
C(K1)ˆ
⊗πC(K2) ([6, Lemma 2.1]). However,
T(xn
1, xn
2, x0
3, . . . , x0
k) = 1,
for every n. So, ˆ
Tis not completely continuous. Hence, ˆ
T, and consequently
T, can not be compact.
Moreover, the operator T1:C(K2)−→ Lk−1(C(K1), C(K3), . . . , C(Kk); `2)
associated to Tis not completely continuous, because
kT1(xn
2)k ≥ kqkkT1(xn
2)(xn
1, x0
3, . . . , x0
k)k=kqk.
So, T1is not weakly compact, i.e., Tis not regular. ¤
Finally, we consider the remaining possibility:
Theorem 3.4. Let K1, . . . , Kkbe infinite compact Hausdorff spaces. Then,
the following assertions are equivalent:
(a3) At least two of the spaces K1, . . . , Kkare not scattered.
(b3)C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)does not have any of the properties DP, RDP
and V.
(c3)C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk)contains a complemented copy of `1.
(d3) There exists a Banach space Xand an unconditionally converging,
multilinear operator T:C(K1)× · · · × C(Kk)−→ Xwhich is not weakly
compact.
Proof. We just show that (a3) implies all of the others. [13, Proposition
13] states that, if E⊃`1, then the space of two homogeneous polynomials
P(2E) contains a copy of `∞. That proof can be easily modified to show
that, if both Eand Fcontain copies of `1, then L2(E , F )⊃`∞. These facts
imply that, in that case, Eˆ
⊗π,sEand Eˆ
⊗πFcontain complemented copies
of `1. To finish the proof of (c3) we just need to observe that C(Ki)ˆ
⊗πC(Kj)
is complemented in C(K1)ˆ
⊗π··· ˆ
⊗πC(Kk).
Let us suppose that (c3) (and (a3)) hold. Since `1is Schur and not re-
flexive, the projection π:C(K1)ˆ
⊗π··· ˆ
⊗πC(Kk)−→ `1is an example of
a completely continuous (hence unconditionally converging) operator not
12 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
weakly compact. So, C(K1)ˆ
⊗π· · · ˆ
⊗πC(Kk) has neither property V nor
RDPP. The statement about the DP property can be seen in [6].
The multilinear operator ˜π:C(K1)× · · · × C(Kk)−→ `1associated to π
above, proves (d3). ¤
Corollary 3.5. Let Kbe an infinite compact Hausdorff space and 2≤k∈
N. Then the following assertions are equivalent:
(a) Kis not scattered.
(b) ˆ
⊗k
π,sC(K)does not have properties DP, RDPP and V.
(c) ˆ
⊗k
π,sC(K)contains a complemented copy of `1.
(d) There exists a k-homogeneous unconditionally converging polynomial
on C(K)which is not weakly compact.
Proof. As we already said in Corollary 3.2, we only have to show that (a)
implies all of the others. As above, (c) follows from [13, Proposition 13] and
the fact that ˆ
⊗2
π,sC(K) is complemented in ˆ
⊗k
π,sC(K).
Let us prove (b): The statement about the DP property can again be
seen in [6]. As before, the projection π:ˆ
⊗k
π,sC(K)−→ `1is a completely
continuous and unconditionally converging operator not weakly compact.
The multilinear symmetric operator ˜
π:C(K)×(k)
· · · ×C(K)−→ `1asso-
ciated to πimmediately above proves (d). ¤
4. Relationships between some classes of multilinear
operators and their linearization
Let us start this section with a simple and essentially known result which
relates the complete continuity of a multilinear operator Tand its lineariza-
tion ˆ
T:
Proposition 4.1. Let E1, . . . , Ekbe Banach spaces. Then the following
assertions are equivalent:
(a) For every Banach space X, if ˆ
T:E1ˆ
⊗π· · · ˆ
⊗πEk−→ Xis completely
continuous then its multilinear associated operator T:E1× · · · × Ek−→ X
is completely continuous.
(b) Every multilinear form ϕ∈ Lk(E1, . . . , Ek)is weakly sequentially con-
tinuous.
(c) If (xn
i)n⊂Ei(1≤i≤k) is a weakly Cauchy sequence, then (xn
1⊗
· · · ⊗ xn
k)n⊂E1ˆ
⊗π· · · ˆ
⊗πEkis weakly Cauchy.
Moreover, if k= 2,
(d) L(E1;E∗
2) = Lcc(E1;E∗
2)
implies all of the others.
Proof. The equivalence between (a), (b) and (c) is very easy and can be left
to the reader. Under the hypothesis (d), if (xn
1)⊂E1is a weakly Cauchy
SOME CLASSES OF MULTILINEAR OPERATORS ON C(K) SPACES 13
sequence and (xn
2)⊂E2is bounded, then xn
1⊗xn
2⊂E1ˆ
⊗πE2is weakly
Cauchy, which is (c). ¤
So we see that if ˆ
Tis completely continuous, Tdoes not need to be
completely continuous. Conversely if Tis completely continuous, ˆ
Tneed
not be completely continuous, even if Eand Fhave DP (see [6]).
We study now the relation between Tbeing an unconditionally converg-
ing multilinear operator and ˆ
Tbeing an unconditionally converging linear
operator. From the previous section it follows that, for every Banach space
Xand k∈N,T∈ Lk(c0;X) is unconditionally converging if and only if
ˆ
T∈ L(c0ˆ
⊗π
(k)
· · · ˆ
⊗πc0;X) is unconditionally converging (if and only if both
of them are weakly compact).
As a consequence we have
Theorem 4.2. Let E1, . . . , Ekand Xbe Banach spaces. If the operator
ˆ
T:E1ˆ
⊗π· · · ˆ
⊗πEk−→ Xis unconditionally converging, then T:E1× · · · ×
Ek−→ Xis also unconditionally converging.
Proof. Let ˆ
T:Eˆ
⊗π· · · ˆ
⊗πEk−→ Xbe unconditionally converging. If Tis
not unconditionally converging, then there exist w.u.C series Pnxn
1⊂E1,
. . . , Pnxn
k⊂Eksuch that (T(Pm
n=1 xn
1, . . . , Pm
n=1 xn
k))mis not a Cauchy
sequence. Let ii:c0−→ Eibe defined by ii(en) = xn
i(1 ≤i≤k). Then,
the multilinear operator
V:c0×(k)
· · · ×c0−→ X
defined by
V(y1, . . . , yk) = T(i1(y1), . . . , ik(yk))
is not unconditionally converging. Hence, ˆ
Vis not unconditionally converg-
ing. On the other hand, it is easy to check that ˆ
V=ˆ
T◦(i1⊗ · · · ⊗ ik) which
is unconditionally converging by the hypothesis. This contradiction finishes
the proof. ¤
The converse of Theorem 4.2 is not true, as the following example shows.
Example 4.3. In [7], the authors provide an example of a Banach space X
with the RNP (hence it does not contain c0) such that Xˆ
⊗πXcontains c0.
Let us consider the operator
γ:X×X−→ Xˆ
⊗πX
defined by
γ(x, y) = x⊗y.
14 FERNANDO BOMBAL, MAITE FERN ´
ANDEZ, AND IGNACIO VILLANUEVA
Since X6⊃ c0,γis unconditionally converging (see [4, Proposition 2.10]),
but ˆγ, which is the identity on Xˆ
⊗πX, is not unconditionally converging,
since it fixes a copy of c0.
We have not been able to find a less exotic example. Let us observe
that one can prove that every bilinear operator T:`∞×`∞−→ c0is
unconditionally converging, so any not unconditionally converging operator
(for example any projection) P:`∞ˆ
⊗π`∞−→ c0would provide a more
familiar counterexample. We have not been able to find such object, in
particular we do not know if `∞ˆ
⊗π`∞contains complemented copies of c0.
From Theorem 4.2 it follows easily that, if every unconditionally converg-
ing bilinear operator on E×Fis weakly compact, then Eˆ
⊗πFhas property
V. We do not believe the converse to be true, but we do not have a coun-
terexample. For a wide variety of spaces the converse is true. For C(K)
spaces, this follows from Section 3, but more generally we have
Proposition 4.4. Let Eand Fbe Banach spaces such that Eˆ
⊗πFhas
property V. Assume further that at least one of the following conditions
holds:
(1) E∗or F∗has the metric approximation property
(2) Eor Fhas an unconditional compact expansion of the identity
(3) E∗or F∗has the compact approximation property and is a subspace
of a Banach space Zpossessing an unconditional compact expansion
of the identity
Then every unconditionally converging bilinear operator on E×Fis
weakly compact.
Proof. Clearly, both Eand Fhave property V. Moreover, [11, Theorem
7] states that, in the hypothesis, L(E;F∗) = K(E;F∗) (or L(F;E∗) =
K(F;E∗)). Now, Proposition 2.12 applies. ¤
For `pspaces we can precise this last result a little more
Proposition 4.5. Let 1< pi<∞. Then the following are equivalent:
(a) `p1ˆ
⊗π· · · ˆ
⊗π`pkhas property V
(b) L(`p1ˆ
⊗π· · · ˆ
⊗π`pk−1;`qk) = K(`p1ˆ
⊗π··· ˆ
⊗π`pk−1;`qk), where `∗
pk=`qk.
(c) `p1ˆ
⊗π··· ˆ
⊗π`pkis reflexive
(d) Pk
i=1
1
pi<1
(e) Every unconditionally converging multilinear operator on `p1×· · ·× `pk
is weakly compact
Proof. If (a) holds, then (b) follows from [11, Theorem 7]. The equivalence
between (b), (c) and (d) can be seen in [3, Section 4].
SOME CLASSES OF MULTILINEAR OPERATORS ON C(K) SPACES 15
Let us observe that, in general, if E1, . . . , Ekare reflexive spaces, then
E1ˆ
⊗π· · · ˆ
⊗πEkis reflexive if and only if every unconditionally converging
multilinear operator defined on E1×· · ·× Ekis weakly compact. For the non
trivial implication of this statement, it suffices to realize that the multilinear
operator
T:E1× · · · × Ek−→ E1ˆ
⊗π· · · ˆ
⊗πEk
given by
T(x1, . . . xk) = x1⊗ · · · ⊗ xk
is unconditionally converging (this can be seen, for instance, applying [4,
Proposition 2.10] as in Example 4.3). Therefore, (c) and (e) are equivalent.
We already mentioned before that (e) implies (a) always, i.e., not only for
`pspaces. ¤
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Departamento de An´
alisis Matem´
atico, Facultad de Matem´
aticas, Uni-
versidad Complutense de Madrid, Madrid 28040
E-mail address:bombal@eucmax.sim.ucm.es
CIMAT, A.P. 402, Guanajuato, Gto. 36000, M´
exico
E-mail address:maite@cimat.mx
Departamento de An´
alisis Matem´
atico, Facultad de Matem´
aticas, Uni-
versidad Complutense de Madrid, Madrid 28040
E-mail address:ignacio villanueva@mat.ucm.es
