CUBO A Mathematical Journal
Vol.14, No¯ 01, (21–27). March 2012

Remarks on cotype and absolutely summing multilinear
operators

A. Thiago Bernardino

UFRN/CERES,

Centro de Ensino Superior do Seridó,

Rua Joaquim Gregório, S/N, 59300-000, Caicó- RN, Brazil,

email: thiagobernardino@yahoo.com.br

ABSTRACT

In this short note we present some new results concerning cotype and absolutely sum-

ming multilinear operators, extending recent results from different authors.

RESUMEN

En esta nota presentamos nuevos resultados sobre cotipo y suma absoluta de operadores

multilineales, extendiendo resultados recientes de diferentes autores.

Keywords and Phrases: Absolutely p-summing multilinear operators, cotype.

2010 AMS Mathematics Subject Classification: 46G25, 47B10.



22 A. Thiago Bernardino CUBO
14, 1 (2012)

1 Introduction

In this note the letters X1, ..., Xn, X, Y will denote Banach spaces over the scalar field K = R or C.

From now on the space of all continuous n-linear operators from X1 × · · · × Xn to Y will be
denoted by L(X1, ..., Xn; Y). If 1 ≤ s < ∞, the symbol s∗ represents the conjugate of s. It will be
convenient to adopt that s/∞ = 0 for any s > 0. For 1 ≤ q < ∞, we denote by `wq (X) the set
{(xj)

∞
j=1 ⊂ X : supϕ∈BX∗

∑
j
|ϕ(xj)|

q
< ∞}.

If 0 < p, q1, ..., qn < ∞ and
1

p
≤

1

q1
+ · · · +

1

qn
,

a multilinear operator T ∈ L(X1, ..., Xn; Y) is absolutely (p; q1, ..., qn)-summing if

(T(x
(1)
j

, ..., x
(n)
j

))∞j=1 ∈ `p(Y)
for every (x

(k)
j

)∞j=1 ∈ `wqk(Xk), k = 1, ..., n. In this case we write T ∈ Πnp,q1,...,qn (X1, ..., Xn; Y). If
q1 = · · · = qn = q, we write Πnp,q (X1, ..., Xn; Y) instead of Πnp,q,...,q (X1, ..., Xn; Y) . For details on
the linear theory we refer to the excellent monograph [9] and for the multilinear theory we refer to

[1, 7, 14] and references therein.

This paper deals with the connection between cotype and absolutely summing multilinear

operators; this line of investigation begins with [4] and was followed by several recent papers (we

refer, for example, to [5, 6, 8, 11, 12, 13, 15, 16] and for a full panorama we mention [14]). The

following result appears in [10, Theorem 3 and Remark 2] and [16, Corollary 4.6] (see also [5,

Theorem 3.8 (ii)] for a particular case):

Theorem 1.1 (Inclusion Theorem). Let X1, ..., Xn be Banach spaces with cotype s and n ≥ 2 be
a positive integer:

(i) If s = 2, then

Πnq;q(X1, ..., Xn; Y) j Π
n
p;p(X1, ..., Xn; Y)

holds true for 1 ≤ p ≤ q ≤ 2.

(ii) If s > 2, then

Πnq;q(X1, ..., Xn; Y) j Π
n
p;p(X1, ..., Xn; Y)

holds true for 1 ≤ p ≤ q < s∗ < 2.

As a consequence of results from [3] one can easily prove the following generalization of this

result (see [2] for details):

Theorem 1.2. If X1 has cotype 2 and 1 ≤ p ≤ s ≤ 2, then

Πns;s,q,...,q(X1, ..., Xn; Y) j Π
n
p;p,q,....,q(X1, ..., Xn; Y)



CUBO
14, 1 (2012)

Remarks on cotype absolutely summing multilinear operators 23

for all X2, ..., Xn,Y and all q ≥ 1. In particular

Πns;s(X1, ..., Xn; Y) j Π
n
p;p,s,....,s(X1, ..., Xn; Y) j Π

n
p;p(X1, ..., Xn; Y). (1.1)

A similar result, mutatis mutandis, holds if Xj (instead of X1) has cotype 2.

In this note we remark that analogous results hold for other situations in which the spaces

involved may have different cotypes and no space may have necessarily cotype 2.

2 Results

The following proposition can be found in [5]:

Proposition 2.1. Let 1 ≤ p1, ..., pn, p, q1, ..., qn, q ≤ ∞ such that 1/t ≤ ∑nj=1 1/tj for t ∈ {p, q} .
Let 0 < θ < 1 and define

1

r
=

1 − θ

p
+

θ

q
and

1

rj
=

1 − θ

pj
+

θ

qj
for all j = 1, ..., n

and let T ∈ L (X1, ..., Xn; Y) . Then

T ∈ Πnp;p1,...,pn ∩ Π
n
q;q1,...,qn

implies T ∈ Πnr;r1,...,rn,

provided that for each j = 1, ..., n, one of the following conditions holds:

(i) Xj is an L∞-space;
(ii) Xj is of cotype 2 and 1 ≤ pj, qj ≤ 2;

(iii) Xj is of finite cotype sj > 2 and 1 ≤ pj, qj < s∗j ;

(iv) pj = qj = rj.

Next lemma appears in [13, Theorem 3.1] without proof. We present a proof for the sake of

completeness:

Lemma 2.2. Let s > 0. Suppose that Xj has cotype sj for all j = 1, ..., n and at least one of the

sj is finite. If
1

s
≤

1

s1
+ . . . +

1

sn
,

then

L (X1, ..., Xn; Y) = Πns;b1,...,bn (X1, ..., Xn; Y)

for

bj = 1 if sj < ∞ and bj = ∞ if sj = ∞.



24 A. Thiago Bernardino CUBO
14, 1 (2012)

Proof. Let j1, ..., jk ∈ {1, ..., n} , k ≤ n such that sj1, ..., sjk are finite and sj = ∞ if j 6= j1, ..., jk.
If

(
x
(jl)
i

)∞
i=1

∈ `w1 (Xjl) and (x
(j)
i

)∞i=1 ∈ `∞(Xj), j 6= jl, l = 1, ..., k, using Generalized Hölder
Inequality, we obtain( ∞∑

i=1

∥∥∥T(x(1)i , ..., x(n)i )∥∥∥s
) 1

s

≤ ‖T‖
( ∞∑

i=1

(∥∥∥x(1)i ∥∥∥ · · · ∥∥∥x(n)i ∥∥∥)s
) 1

s

≤ C ‖T‖
( ∞∑

i=1

∥∥∥x(j1)i ∥∥∥sj1
)1/sj1

· · ·
( ∞∑

i=1

(∥∥∥x(jk)i ∥∥∥)sjk
)1/sjk

where C is such that
n∏

j=1,j6=j1,...,jn

∥∥∥x(j)i ∥∥∥ ≤ C
for all i. Since Xj has cotype sj, for j1, ..., jk, we have( ∞∑

i=1

∥∥∥T(x(1)i , ..., x(n)i )∥∥∥s
) 1

s

≤ C ‖T‖
( ∞∑

i=1

∥∥∥x(j1)i ∥∥∥sj1
)1/sj1

· · ·
( ∞∑

i=1

(∥∥∥x(jk)i ∥∥∥)sjk
)1/sjk

= C ‖T‖
k∏

t=1

( ∞∑
i=1

∥∥∥idXj
t

(
x
(jt)
i

)∥∥∥sjt )1/sjt < ∞
and the result follows.

The main result of this note is the following Theorem. At first glance it seems to have too

restrictive assumptions, but Corollary 2.4 and Example 2.5 will illustrate its usefulness:

Theorem 2.3. Let k, n be natural numbers, n ≥ k ≥ 2 and Xk+1, ..., Xn, Y be arbitrary Banach
spaces. If Xj has finite cotype sj ≥ 2 for j = 1, ..., k, then

Πnp;p1,...,pk,q,...,q (X1, ..., Xn; Y) j Π
n
r;r1,...,rk,q,...,q

(X1, ..., Xn; Y)

for any (q, θ) ∈ [1, ∞] × (0, 1),
1 ≤ pj ≤ 2 (when sj = 2),
1 ≤ pj < s∗j (when sj > 2)

and s ∈ [1, ∞) so that
1

s
≤

1

s1
+ · · · +

1

sk
,

1

r
=

1 − θ

s
+

θ

p
,

1

rj
=

1 − θ

1
+

θ

pj
,

for all j = 1, ..., k.



CUBO
14, 1 (2012)

Remarks on cotype absolutely summing multilinear operators 25

Proof. Let T ∈ Πnp;p1,...,pk,q,...,q (X1, ..., Xn; Y) . By the previous lemma,

L (X1, ..., Xn; Y) = Πns;1,...,1,∞,...,∞ (X1, ..., Xn; Y) ,
where 1 is repeated k times. A fortiori, we have

L (X1, ..., Xn; Y) = Πns;1,...,1,q,...,q (X1, ..., Xn; Y) .

So,

T ∈ Πns;1,...,1,q,...,q (X1, ..., Xn; Y) ∩ Π
n
p;p1,...,pk,q,...,q

(X1, ..., Xn; Y) .

From Proposition 2.1 we get

T ∈ Πnr;r1,...,rk,q,...,q (X1, ..., Xn; Y) .

Corollary 2.4. Let k, n be natural numbers, n ≥ k ≥ 2, Xk+1, ..., Xn, Y be arbitrary Banach spaces
and q ∈ [1, ∞). If Xj has finite cotype sj ≥ 2, j = 1, ..., k and 1 ≤ 1/s1 + · · · + 1/sk, then

Πnp;p1,...,pk,q,...,q (X1, ..., Xn; Y) j Π
n
r;r1,...,rk,q,...,q

(X1, ..., Xn; Y) ,

where pj = p and rj = r for all j = 1, ..., k, for all r so that

1 ≤ r < p < min s∗j if sj 6= 2 for some j = 1, ..., k,
1 ≤ r < p ≤ 2 if sj = 2 for all j = 1, ..., k.

In particular

Πnp;p (X1, ..., Xn; Y) j Π
n
r;r (X1, ..., Xn; Y)

for all r so that

1 ≤ r < p < min s∗j if sj 6= 2 for some j = 1, ..., k,
1 ≤ r < p ≤ 2 if sj = 2 for all j = 1, ..., k.

Proof. Since 1 ≤ 1/s1 + · · · + 1/sk, we can use s = 1 in the previous theorem. Since p = pi and
r = ri for all i = 1, ..., k and s = 1, we conclude that

Πnp;p,...,p,q,...,q (X1, ..., Xn; Y) j Π
n
r;r,...,r,q,...,q (X1, ..., Xn; Y) .

In fact, for any 1 ≤ r < p there is a θ ∈ (0, 1) so that

1

r
=

1 − θ

1
+

θ

p

and since p = pi and r = ri, the same θ ∈ (0, 1) satisfies

1

ri
=

1 − θ

1
+

θ

pi
.



26 A. Thiago Bernardino CUBO
14, 1 (2012)

Choosing q = p, since r < p we have

Πnp;p (X1, ..., Xn; Y) j Π
n
r;r,...,r,p,...,p (X1, ..., Xn; Y) j Π

n
r;r (X1, ..., Xn; Y) .

Example 2.5. Let X4, ..., Xn, Y be arbitrary Banach spaces. Then

Πnp;p,p,p,q,...,q (`3, `3, `3, X4, ..., Xn; Y) j Π
n
r;r,r,r,q,...,q (`3, `3, `3, X4, , , ., Xn; Y)

for all q ∈ [1, ∞) and 1 ≤ r < p < 3∗. In particular
Πnp;p (`3, `3, `3, X4, ..., Xn; Y) j Πnr;r (`3, `3, `3, X4, , , ., Xn; Y)

for all 1 ≤ r < p < 3∗.

Received: January 2011. Revised: February 2011.

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Remarks on cotype absolutely summing multilinear operators 27

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