summing sequences

“garcia de galdeano”

garcía de galdeano

seminariomatemáticon. 3

PRE-PUBLICACIONES delseminario matematico 2002

J. L. ArreguiO. Blasco

(p;q) - SUMMING SEQUENCES

Universidad de Zaragoza

(p, q)-SUMMING SEQUENCES.

Jose Luis Arregui and Oscar Blasco

§1 Introduction.

Given a real or complex Banach space X and 1 ≤ p ≤ ∞, we denote by `p(X) and`wp (X) the Banach spaces of sequences in X with norms ‖(xn)‖`p(X) = ‖(‖xn‖)‖`p

and ‖(xj)‖`wp (X) = supx∗∈BX∗ ‖(x∗xj)‖`p respectively.

It is well known (see page 50, [DJT]) that `wp (X) = `p(X) for some 1 ≤ p <

∞ if and only if X is finite dimensional, and of course `w∞(X) = `∞(X). We

also have natural isometric isomorphisms `wp (X) ' L(`p′ ,X) (1 < p ≤ ∞) and

`w1 (X) ' L(c0,X) by associating to each operator u the sequence (xj) ⊂ X given

by the image of the canonical basis xj = u(ej). In this sense, the space K(c0,X) ofcompact operators corresponds with the sequences (xj) such that the series

∑j xj

converges unconditionally. For any infinite dimensional Banach space X we havethat `1(X) is strictly contained in K(c0,X), because there exist unconditionallyconvergent series which are not absolutely convergent, as the Dvoretzky-Rogers’theorem asserts (see [DR], [D] or [DJT, 1.2]).

An operator u ∈ L(X,Y ) is absolutely summing if for every unconditionallyconvergent series

∑xj in X it holds that

∑uxj is absolutely convergent in Y . The

study of these operators began with [G]. In spite of the difference between K(c0, X)and L(c0,X), absolutely summing operators coincide with 1-summing operators.For 1 ≤ p < ∞, an operator u: X → Y is p-summing (see [Pi]) if it maps sequences(xj) ∈ `w

p (X) into sequences (uxj) ∈ `p(Y ). Equivalently, if there exists a constantC such that

( n∑

j=1

‖uxj‖p)1/p ≤ C sup

x∗∈BX∗

( n∑

j=1

|x∗xj |p)1/p

for any finite family x1, x2, . . . xn of vectors in X.The least of such constants is the p-summing norm of u, denoted by πp(u). The

space Πp(X,Y ) of all p-summing operators from X to Y is then a Banach spacefor 1 ≤ p < ∞.

Grothendieck’s theorem, in this setting, says that, for any measure space (Ω, µ)and any Hilbert space H , L(L1(µ),H) = Π1(L1(µ),H). Therefore there existsa constant KG, called the Grothendieck constant, such that π1(u) ≤ KG‖u‖ forany bounded u:L1(µ) → H and which is the least among all constants with thisproperty.

This formulation of Grothendieck’s theorem and the meaning of KG was givenby Pietsch ([P1]) and Lindenstrauss and PeÃlczynski ([LP]), by changing from the

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2 J.L. ARREGUI, O. BLASCO

original tensor norms context to the Pietsch’s operator ideals setting (see [P2] and[DF, 9.1]).

Given, now, 1 ≤ p ≤ q < ∞, the space Πp,q(X, Y ) of (p,q)-summing operatorsis formed by those mapping sequences in `w

q (X) into sequences in `p(Y ), in otherwords u ∈ Πp,q if there exists C such that

‖(uxj)‖`p(Y ) ≤ C‖(xj)‖`wq (X)

for any finite family of vectors xj in X, and the least of such C is the (p, q)-summingnorm of u, denoted by πp,q(u). Obviously, (p, p)-summing is just the same as p-summing.

Despite the identity is not p-summing for any Banach space with infinite di-mension, it can be (p, q)-summing if p 6= q. For some spaces the identity is (2, 1)-summing, which means that if

∑j xj is an unconditionally convergent series then

(xj) ∈ `2(X). These spaces are said to have Orlicz property, because Orlicz (see[O])was the first to notice that this is the case for X = Lp[0, 1] with 1 ≤ p ≤ 2.After the work by Maurey and Pisier (see [MP]) it became clear that they actuallyhave a property, known as cotype 2, stronger than the Orlicz property.

For each 1 ≤ p ≤ ∞, Radp(X) will be taken as the closure in Lp([0, 1],X) of theset of functions of the form

∑nj=1 rjxj , xj ∈ X, where (rj)j∈N are the Rademacher

functions on [0, 1] defined by rj(t) = sig(sin 2jπt).In fact Rad∞(X) can be described as the set of compact operators K(c0,X),

or also the space of unconditonally convergent series, since for any finite family(xj)j≤n we have

∥∥ ∑

j

rjxj

∥∥∞ ∼ ‖(xj)‖`w

1 (X) =∥∥T : ej 7→ xj

∥∥L(`n

∞,X).

Making use of the Kahane’s inequalities (see [DJT], page 211) it follows thatRadp(X) coincide up to equivalent norms for all p < ∞. The space will be denotedthen Rad(X), and we shall use the L1-norm throughout the paper. It is alsoknown that Rad(X) can be identified with the space of the almost unconditionallysummable sequences (xj), corresponding to functions given by t 7→

∑∞j=1 rj(t)xj ,

where the series converges for almost every t with respect to Lebesgue measure.Finally, we recall the fundamentals on type and cotype. For 1 ≤ p ≤ 2 (re-

spect. q ≥ 2), a Banach space X is said to have (Rademacher) type p (respect.(Rademacher) cotype q) if `p(X) ⊂ Rad(X) (respect. Rad(X) ⊂ `q(X)). In otherwords if there exists a constant C such that

∫ 1

0||

n∑

j=1

xjrj(t)||dt ≤ C( n∑

j=1

‖xj‖p)1/p

(respect.( n∑

j=1

‖xj‖q)1/q ≤ C

∫ 1

0||

n∑

j=1

xjrj(t)||dt),

for any finite family x1, x2, . . . xn of vectors in X.The basic theory of p-summing and (p, q)-summing operators, type and cotye

can be found, for example, in the books [DJT], [DF], [J], [TJ] [Pi] or [W].

(p, q)-SUMMING SEQUENCES 3

Let X and Y be two real or complex Banach spaces and let E(X) and F (Y ) betwo Banach spaces whose elements are defined by sequences of vectors in X andY (containing any eventually null sequence in X or Y ). A sequence of operators(un) ∈ L(X,Y ) is called a multiplier sequence from E(X) to F (Y ) if there existsa constant C > 0 such that

∥∥(ujxj)nj=1

∥∥F (Y ) ≤ C

∥∥(xj)nj=1

∥∥E(X)

for all finite families x1, . . . , xn in X .The set of all of multiplier sequences is denoted by (E(X), F (Y )).In this article we consider the case of the classical sequence spaces E(X) = `w

p (X)and F (Y ) = `p(Y ). Note that the constant sequences uj = u for all j ∈ N belongingto (`w

q (X), `p(Y )) corresponds to u being an operator in Πp,q(X,Y ). Also the caseuj = λj .u ∈ (`w

q (X), `1(Y )) for all (λj) ∈ `p′ , where (1/p)+(1/p′) = 1, correspondsto u ∈ Πp,q(X,Y ). These facts suggest the use of the notation `πp,q(X,Y ) insteadof (`w

q (X), `p(Y )) and `πp(X,Y ) for q = p.A sequence (uj)j∈N of operators in L(X,Y ) is a (p, q)-summing multiplier, in

short (uj) ∈ `πp,q(X,Y ), if there exists a constant C > 0 such that, for any finitecollection of vectors x1, x2, . . . xn in X , it holds that

( n∑

j=1

‖ujxj‖p)1/p

≤ C sup( n∑

j=1

|x∗xj |q)1/q

; x∗ ∈ BX∗

.

The reader is referred to [AF] for the particular case p = q, X = Y and uj =αjIdX . A scalar sequence (αj) is there defined to be a p-summing multiplier ifuj = αjIdX belongs to `πp(X,Y ).

Actually, in this article we shall consider mainly the case Y = K, what leadsus to define a new family of spaces of vector valued sequences, not only for dualspaces, that will be called spaces of (p, q)-summing sequences in X.

For any Banach space X , we define the space `πp,q (X) as the set of all sequences(xj) in X such that there exists a constant C > 0 for which

( n∑

j=1

|x∗jxj |p

)1/p

≤ C sup( n∑

j=1

|x∗jx|q

)1/q

; x ∈ BX

for any finite collection of vectors x∗1, . . . , x

∗n in X∗. This will be shown to be

equivalent to have that (xj), considered as operators in L(X∗, K) , is a (p, q)-summing multiplier.

Our main objectives thereafter is to study these spaces and to be able to describesome classical aspects of the theory of geometry of Banach spaces and operatorideals, from this point of view.

The paper is divided into three sections. In the first one we introduce the spacesof (p, q)-summing multipliers and (p, q)-summing sequences in X. They make sensefor any 1 ≥ p, q, but the cases 1 ≤ p < q and p ≥ q are different in kind. Thecase p = 1 ≤ q is shown to correspond with integral operators from `q to X , and inparticular we can identify the (1, q)-summing sequences in (L1(µ)). We also showthat the space of operators L(X, `q(Y )) is isometrically isomorphic to the space ofmultipliers (`πp,q(X), `q(Y )) for any p ≤ q.


In Section 3 we try to recover some results from the finite dimensional setting inthe infinite dimensional situation, at least under certain assumptions on the spaceX. For instance, in the case X = K, we have that `πp,q = `r where (1/r) = ((1/p)−(1/q))+. This, in particular, implies that `πp,q = `πr,s whenever (1/p) − (1/q) =(1/r) − (1/s). We are able so show, among other things, that if X∗ has cotype 2then `π1(X) = `π2(X) and `π1,q(X) = `πr,2(X) for 1/r = (1/q′) + (1/2), or that`πp,1(X) = `∞(X) is equivalent to X being finite dimensional for p < 2, to X∗

having Orlicz property for p = 2 and to X∗ having cotype p for p > 2. We finallyobserve that `w

p (X) ⊂ `πp,1(X), and use it to find the values or p, q and r wherethe canonical basis (ej) belongs to `πp,q(`r).

In the final section we point out an equivalent formulation to Grothendieck’stheorem in terms of these spaces, by showing that the bilinear map defined by((x∗

j ), (xk)) → ((x∗jxk)k)j maps `∞(X∗) × `∞(X) into `π1(`∞) for Banach spaces

isomorphic to Hilbert spaces . We also present a lemma, that together with a resultby Pisier, allows to conclude that the spaces where `π1,2(X) = Rad(X) are thosesatisfying Grothendieck’s theorem and having cotype 2.

Notation is fairly standard. We follow the usual terms L(X,Y ) for the spaceof bounded linear operators between Banach spaces, BX and SX for the unit balland sphere in X, X ∼ Y if two Banach spaces are isomorphic and X ' Y if theyare isometrically isomorphic. We write the action of an operator or functional on xmerely as ux and x∗x, though we prefer to use x∗(x) or 〈x∗, x〉 if we think it helps,and we use the tensor form for expressing finite rank operators: (x∗ ⊗y)x = x∗(x)y.Finally (ej) is the canonical basis of the sequence spaces `p and c0, p′ denotes theconjugate exponent of p, x+ = maxx, 0 and K denotes R or C if no difference isrelevant.

§2 (p, q)-summing multipliers and sequences.

Let us start off by mentioning the different behaviour when using a single op-erator or a sequence of them. Let u 6= 0 be a bounded linear operator betweentwo Banach spaces X and Y . If u maps sequences (xj) ∈ `q(X) into sequences(uxj) ∈ `p(Y ) then necessarily q ≤ p ( for q > p one can take xj = (1/j)1/px wherex /∈ Ker(u) to get a contradiction). Nevertheless for sequences of operators we getthe following result:

Proposition 2.1. Let X and Y be Banach spaces, and 1 ≤ p, q ≤ ∞. For 1/r =((1/p) − (1/q))+ we have that

(`q(X), `p(Y )) = `r(L(X, Y )).

Proof. Any multiplier sequence (uj) must be in `∞(L(X,Y )), as we see by takingsequences in X of the form (0, . . . , 0, xj , 0, 0, . . . ). If q ≤ p it is plain that theconverse is true.

Let q > p and 1/p = (1/r) + (1/q). By Holder’s inequality,

(∑

j

‖ujxj‖p)1/p ≤ (∑

j

‖uj‖p‖xj‖p)1/p ≤ (∑

j

‖uj‖r)1/r(∑

j

‖xj‖q)1/q.

Conversely, given n we note that the `r/p-norm of (‖uj‖p)nj=1 equals, by duality, the

norm of (λj‖uj‖p) in `1 for some 0 ≤ λj such that∑

j λq/pj = 1. Let βj = λ

1/pj and


xj ∈ SX such that ‖ujxj‖ is arbitrarily close to ‖uj‖; then (∑

j ‖uj(βjxj)‖p)1/p

approximates (∑

j ‖uj‖pβpj )1/p, and hence

( n∑

j=1

‖uj‖r)1/r is bounded by a constant

independent of n. ¤

The study of multiplier sequences between `wq (X) and `p(Y ) is far more com-

plicated. For the reason explained in the introduction, we find more convenient tochange the notation from (`w

q (X), `p(Y )) to the following:

Definition 2.2. Let X and Y be Banach spaces, and let p, q ≥ 1. A sequence(uj)j∈N of operators in L(X, Y ) is a (p, q)-summing multiplier if there exists aconstant C > 0 such that, for any finite collection of vectors x1, x2, . . . xn in X , itholds that

( n∑

j=1

‖ujxj‖p)1/p

≤ C sup( n∑

j=1

|x∗xj |q)1/q

; x∗ ∈ BX∗

.

We use `πp,q(X,Y ) to denote the set of (p, q)-summing multipliers, and πp,q[uj ] isthe least constant C for which (uj) verifies the inequality in the definition. In orderto avoid ambiguities, sometimes we shall use πp,q[uj ;X,Y ]. Of course if p = q wesimply say that the sequence (uj) is a p-summing multiplier and write `πp(X,Y ),πp[uj ;X,Y ].

Remark 2.3. The obvious modifications for p = ∞ or q = ∞ make sense, butthen `πp,∞(X, Y ) = `p(L(X, Y )) and `π∞,q(X,Y ) = `∞(L(X,Y )).

Remark 2.4. The example before Proposition 2.1 shows that Πp,q(X,Y ) = 0for p < q. Let us observe that c00(L(X,Y )) ⊂ `πp,q(X, Y ) for any 1 ≤ p, q, wherec00(L(X,Y )) stands for all sequences of operators with a finite number of non-zeroelements.

Indeed, if uj = 0 for all j > N then πp,q[uj ] ≤ N1/p maxj≤N

||uj ||.

Remark 2.5. For any Banach space X, and the usual identification between Xand L(K,X), it follows from Proposition 2.1 that `πp,q (K,X) = `r(X), where 1/r =((1/p) − (1/q))+.

Remark 2.6. For any couple of Banach spaces X and Y , 1 ≤ p, q < ∞ anduj ∈ L(X, Y ), we clearly have

(uj) ∈ `πp,q(X,Y ) if and only if (λjuj) ∈ `π1,q (X,Y ) for all (λj) ∈ `p′ .

Moreoverπp,q[uj ;X,Y ] = supπ1,q[λjuj ;X,Y ] : ||λj ||`p′ = 1.

Let us mention that the characterization of the absolutely summing operators interms of unconditional series can be generalized as follows, with the same standardproof:


Proposition 2.7. A sequence (uj) is in `πp,1(X,Y ) if and only if it holds that forany unconditionally convergent series

∑xj in X we have (ujxj)j ∈ `p(Y ) .

The subspace of L(`wq (X), `p(Y )) formed by (p, q)-summing sequences of opera-

tors is closed, and then the summing norm πp,q is complete:

Proposition 2.8. For any X and Y Banach spaces, and for any 1 ≤ p, q < ∞,(`πp,q(X,Y ), πp,q) is a Banach space.

Let us now concentrate in the case Y = K. Being `πp,q (X, K) a space of sequencesin X∗, it seems natural, somehow, to denote it in the form E(X∗). For any Banachspace X we can give sense to `πp,q(X) in such a way that `πp,q(X∗) = `πp,q (X, K):

Proposition 2.9. For any sequence (x∗j ) in X∗, the following are equivalent:

(i) (x∗j ) ∈ `πp,q(X, K), with πp,q [x∗

j ] ≤ C.(ii) There exists C > 0 such that

( n∑

j=1

|x∗∗j x∗

j |p)1/p

≤ C sup( n∑

j=1

|x∗∗j x∗|q

)1/q

; x∗ ∈ BX∗

for every x∗∗1 , . . . , x∗∗

n in X∗∗.

Proof. It is only that (i) implies (ii) which deserves a proof, and it is a consequenceof the principle of local reflexivity, in its weak form (see [DF]): Let E and F thesubspaces of X∗∗ and X∗ generated by x∗∗

1 , . . . , x∗∗n and by x∗

1, . . . , x∗n respectively.

Given ε > 0, there exists a linear operator u:E → X such that ‖u‖ < 1 + ε andx∗(ux∗∗) = x∗∗x∗ for any x∗∗ ∈ E and x∗ ∈ F . Therefore

( n∑

j=1

|x∗∗j x∗

j |p)1/p =

( n∑

j=1

|x∗j (ux∗∗

j )|p)1/p ≤ C sup

x∗∈BX∗

( n∑

j=1

|x∗(ux∗∗j )|q

)1/q.

The last term equals the norm of the linear operator uv: `nq′ → X, where v: `n

q′ →E is given by ej 7→ x∗∗

j , and the norm of v equals supx∗∈BX∗

( ∑nj=1 |x∗∗

j x∗|q)1/q. Since

‖uv‖ ≤ (1 + ε)‖v‖, we obtain that

( n∑

j=1

|x∗∗j x∗

j |p)1/p ≤ C(1 + ε) sup

x∗∈BX∗

( n∑

j=1

|x∗∗j x∗|q

)1/q

for any ε > 0. ¤

Definition 2.10. For any Banach space X, we define the space `πp,q(X) as the setof all sequences (xj) in X such that there exists a constant C > 0 for which

( n∑

j=1

|x∗jxj |p

)1/p

≤ C sup( n∑

j=1

|x∗jx|q

)1/q

; x ∈ BX

for any finite collection of vectors x∗1, . . . , x

∗n in X∗.

Remark 2.11. Alternatively we could have defined

`πp,q(X) = `πp,q(X∗, K) ∩ `∞(X).

Remark 2.12. Note that if X, Y are Banach spaces and uj ∈ L(X, Y ) then

πp,q[uj ] = sup‖y∗

j ‖=1πp,q [y∗

j uj ;X∗] .


Lemma 2.13. Let X be a Banach space, 1 ≤ p, q ≤ ∞, (αj) ⊆ K and x ∈ X:Then

πp,q[αjx] = ‖(αj)‖r‖x‖

where 1/r = ((1/p) − (1/q))+.

Proof. Given x∗1, . . . , x

∗n ∈ X∗ we see that

‖(αjx∗jx)‖p ≤ ‖(αj)‖r ‖(x∗

jx)‖q

≤ ‖(αj)‖r ‖x‖ ‖(x∗j )‖`w

q (X∗) ,

and thus we obtain an inequality.To get the converse we note that, choosing x∗ ∈ SX∗ such that x∗x = ‖x‖, we

may take a sequence (βj) of scalars so that, if x∗j = βjx

∗, the value of

‖(αjx∗jx)‖p = ‖(αjβj)‖p‖x‖

is arbitrarily close to ‖x‖ ‖(αj)‖r ‖(βj)‖q , and then we just need to observe that‖(βj)‖q = ‖(x∗

j )‖`wq (X∗). ¤

Note in particular that any non trivial constant sequence is in `πp,q(X) if andonly if p ≥ q.

Proposition 2.14. Let X be a Banach space, 1 ≤ p, q and (1/r) = ((1/p) −(1/q))+. Then

`p(X) ⊂ `πp,q(X) ⊂ `r(X), and also

`r⊗X ⊂ `πp,q(X),

with continuous inclusions of norm 1.

Proof. The first one is a direct consequence of `wq (X∗) ⊂ `∞(X∗), and we get

the second by `q(X∗) ⊂ `wq (X∗) and Proposition 2.1. The third one follows from

Lemma 2.13. ¤

Remark 2.15. `πp,q(X) ⊂ c0(X) if and only if p < q.

In order to see when the second embedding in the previous proposition becomesan identity, we need the following lemma:

Lemma 2.16. Let X be a Banach space, 1 ≤ t ≤ s < ∞ and 1/r = (1/t) − (1/s).If x∗

1, x∗2, ..., x

∗n ∈ X∗ then

(n∑

j=1

‖x∗j‖s)1/s = sup(

n∑

j=1

|x∗jxj |t)1/t : ‖(xj)‖`r(X) = 1.

Proof. For t = 1 this is just the duality `s(X∗) = (`s′(X))∗.The general case follows from using that

(n∑

j=1

|x∗jxj |t)1/t = sup

n∑

j=1

|αjx∗jxj | :

n∑

j=1

|αj |t′= 1.


Let us observe now that `s′(X) = `t′`r(X) (which follows simply by writing xn =‖xn‖s′/t′

yn where yn = ‖xn‖−s′/t′xn for xn 6= 0 and yn = 0 for xn = 0).

Therefore

sup(n∑

j=1

|x∗jxj |t)1/t : ‖(xj)‖`r(X) = 1

= supn∑

j=1

|αjx∗jxj | :

n∑

j=1

|αj |t′= 1, ‖(xj)‖`r(X) = 1

= supn∑

j=1

|x∗jyj | :

n∑

j=1

‖yj‖s′= 1 = (

n∑

j=1

‖x∗j‖s)1/s. ¤

Corollary 2.17. Let X be a Banach space. The following are equivalent.(a) X is finite dimensional.(b) `πp,q(X) = `r(X) for all 1 ≤ p ≤ q < ∞ and 1/r = (1/p) − (1/q).(c) `πp,q(X) = `r(X) for some 1 ≤ p ≤ q < ∞ and 1/r = (1/p) − (1/q).

Proof. The only thing to be shown is that (c) implies (a). Note that using Lemma2.16 one has

(n∑

k=1

||x∗k||q)1/q = sup(

n∑

k=1

|x∗kxk|p)1/p :

n∑

k=1

||xk||r = 1.

Therefore assuming `r(X) ⊂ `πp,q(X) we shall have `wq (X∗) = `q(X∗), what gives

that X∗ is finite dimensional. ¤

We’ll see later that there are infinite dimensional spaces X such that `πp,q (X) =`∞(X) with p > q.

We remark now another difference between the cases p < q and p ≥ q: note firstthat, in general, the πp,q-norm of any sequence is independent from any reorderingof its terms:

Proposition 2.18. Let (xj) a bounded sequence in any Banach space X, and let1 ≤ p, q ≤ ∞. Then

πp,q[xσ(j)] = πp,q[xj ]

for any bijection σ: N → N

The proof follows from the definition and the fact that the p-norm and the weakq-norm are reordering invariant.

When p ≥ q we can say more:

Proposition 2.19. Let (xj) a bounded sequence in any Banach space X, and let1 ≤ q ≤ p < ∞. Then

πp,q[xσ(j)] ≤ πp,q[xj ]


for any map σ: N → N.

Proof. Given x∗1, x∗

2, . . . , x∗n ∈ X∗ we have

( ∑

j

|x∗jxσ(j)|p

)1/p =( ∑

k

(∑

σ(j)=k

|x∗jxk|p)

)1/p ≤( ∑

k

(∑

σ(j)=k

|x∗jxk|q)p/q

)1/p

=( ∑

k

∣∣(∑

σ(j)=k

αjx∗j )xk

∣∣p))1/p (where (αj)σ(j)=k ∈ B`q′ for each k)

=( ∑

k

|y∗kxk|p

)1/p (with y∗k =

∑

σ(j)=k

αjx∗j ∈ X∗)

≤ πp,q[xj ] ‖(y∗k)‖`w

q (X∗) = πp,q[xj ] sup‖(βk)‖q′ ≤1

∥∥∑

k

βky∗k

∥∥

= πp,q[xj ] sup‖(βk)‖q′≤1

∥∥∑

j

αjβσ(j)x∗j

∥∥ ≤ πp,q[xj ] sup‖(γj)‖q′ ≤1

∥∥∑

k

γjx∗j

∥∥

= πp,q[xj ] ‖(x∗j )‖`w

q (X∗) . ¤

The result does not hold if 1 ≤ p < q: take σ a constant map.Proposition 2.19 implies that all (p, q)-sequences verify something apparently

stronger than the condition in Definition 2.10:

Corollary 2.20. For any Banach space X and p ≥ q ≥ 1, a sequence (xj) ⊂ X is(p, q)-summing if and only if there exists a constant C such that

( n∑

k=1

supj

|x∗kxj |p

)1/p ≤ C supx∈BX

( n∑

k=1

|x∗kx|q

)1/q

for any x∗1, . . . , x

∗n ∈ X∗, and the least such C is πp,q[xj ].

Recall the following definition: an operator u ∈ L(X,Y ) is p-integral if there

exists a measure µ such that the composition Xu→ Y

jY−→ Y ∗∗ factorizes as Xβ→

L∞(µ)ip−→ Lp(µ) α→ Y ∗∗ for some bounded operators α and β (ip and jY are the

respective inclusions).The p-integral norm of u is the infimum of all the possible values of ‖α‖‖β‖ in

the previous expression. The set of p-integral operators (a Banach operator ideal)is denoted by Ip(X,Y ). For p = 1 it is denoted simply by I(X,Y ), the space ofintegral operators.

Any p-integral operator u is also p-summing, and πp(u) is not greater than thep-integral norm, but the converse is not true in general. Basic results on p-integraloperators can be seen in [DJT].

We’ll make use of the following fact: u:X → Y is integral if and only if thereexists a constant C > 0 such that

| tr(uv)| ≤ C‖v‖

for any finite rank linear operator v: Y → X , and the least such C is the integralnorm of u.

This makes easy to characterize the (1, q)-sequences in terms of integral opera-tors:


Theorem 2.21. Let X be any Banach space, and let 1 ≤ q < ∞. Then a sequence(xj) ⊂ X is (1, q)-summing if and only if it defines an integral operator u: `q → Xby uej = xj, and the integral norm of u is then π1,q [xj ].

Proof. Let u an integral operator `q → X with uej = xj for all j, and let C itsintegral norm. Given x∗

1, . . . x∗n ∈ X∗, let v =

∑nj=1 x∗

j ⊗λjej , where λj = sig(x∗jxj).

Thenn∑

j=1

|x∗jxj | =

n∑

j=1

λjx∗jxj = tr(uv) ,

so∑n

j=1 |x∗jxj | ≤ C‖v‖, and ‖v‖ is just ‖(x∗

j )‖`wq (X∗). Therefore π1,q [xj ] ≤ C.

Conversely, let (xj) ∈ `π1,q(X). Then (xj) ∈ `q′(X), so u: ej 7→ xj definesa bounded operator in L(`q, X). Now, if v =

∑nj=1 x∗

j ⊗ ξj with ξj = (ξjk)k ∈`q then, for v∗

k =∑

j ξjkx∗j ∈ X∗, it turns out that | tr(uv)| =

∑k |v∗

kxk| ≤π1,q [xk]‖(v∗

k)‖`wq (X∗) and ‖(v∗

k)‖`wq (X∗) = ‖v‖, giving that the integral norm of u

is bounded by π1,q [xj ] . ¤

Corollary 2.22. If a bounded sequence (xj) defines an integral operator `1 → X,then any sequence in the set xj defines an integral operator as well.

As an application of Theorem 2.21, we can identify the sequences in `π1,q(L1(µ)),for any σ-finite space µ:

For any Banach lattice X , an operator u: X → L1(µ) is integral if and only if∫ (sup

x∈BX

|ux|)dµ < ∞, its value being the integral norm of u (see [DJT, Th. 5.19]).

When applied to X = `q, theorem 2.21 gives the following:

Theorem 2.23. Let 1 ≤ q < ∞, and let µ a σ-finite measure. Then (fj) ∈`π1,q(L1(µ)) if and only if

∫‖(fj(w))‖`q′ dµ(w) < ∞ ,

and then the integral equals π1,q[fj ].

Proof. Just note that sup‖(λj)‖q=1

∣∣ ∑

j

λjfj(w)∣∣ = ‖(fj(w))‖q′ for any w in the mea-

sure space.

Remark 2.24. When 1 < q < ∞ we thus have an isometry between `π1,q(L1(µ))and L1(µ, `q′).

Of course, for q = ∞ the result keeps true, since `π1,∞(L1(µ)) = `1(L1(µ)) 'L1(µ, `1).

As for q = 1, recall that we can have∫

supj

|fj(w)|dµ(w) < ∞ with w 7→ (fj(w))

not being a measurable function. For example, for the Rademacher functions rj in([0, 1], dt

)we have that (rj(t)), t ∈ [0, 1] = −1, 1N is not esentially separable

and then the sequence does not define a function in L1(dt, `∞). Anyway (rj) ∈`π1

(L1[0, 1]

), as Theorem 2.23 gives the following for q = 1:


Corollary 2.25. Let µ a σ-finite measure. Then (fj) ∈ `π1(L1(µ)) if and only ifthere exists another function f ∈ L1(µ) such that, for every j, |fj | ≤ f µ-a.e.

By Remark 2.12 we can also identify the sequences of `π1,q (c0, Y ) in the followingway:

Corollary 2.26. Let (uj) be a sequence of bounded operators in L(c0, Y ), such

that uj(ek) = ykj. Then π1,q[uj ] = sup‖y∗

j ‖≤1

∑k

( ∑j |y∗

j ykj |q′)1/q′

for q > 1 and

π1[uj ] = sup‖y∗

j ‖≤1

∑

k

supj

|y∗j ykj |.

Another consequence of the interpretation of π1-sequences as integral operatorsis the following:

Corollary 2.27. Let X be a Banach space and let (xj) be a bounded sequence inX. Then (xj) ∈ `π1(X) if and only if there exist a Banach space Y , a sequence(y∗

j ) ∈ `∞(Y ∗) and u ∈ Π1(X∗, Y ) such that xj = y∗j u ∈ X∗∗ for each j.

Proof. Let us assume that such u and (y∗j ) do exist. The constant sequence (uj = u)

is a multiplier from `w1 (X∗) to `1(Y ), and (y∗

j ) ∈ (`1(Y ), `1), so the composition(xj) = (y∗

j u) belongs to (`w1 (X∗), `1) = `π1(X∗∗).

Conversely, if (xj) ∈ `π1(X) then Theorem 2.21 says that v: `1 → X givenby vej = xj is an integral operator, and in particular v∗ is absolutely summing(v∗ is integral if v is so, and integral operators with values in `∞ are absolutelysumming). Then we can take Y = `∞, u = v∗ and (y∗

j ) = (ej) in `1 ⊂ (`∞)∗. Sinceej(v∗x∗) = x∗(vej) = x∗xj for any x∗ ∈ X∗ and each j, the result follows. ¤

We finish this section with a result on multipliers in(`πp,q(X), `p(Y )

), showing

that these spaces coincide for any 1 ≤ p ≤ q:

Theorem 2.28. Let X, Y be Banach spaces and 1 ≤ p ≤ q. Then(`πp,q(X), `p(Y )

)' L(X, `q(Y )).

The isometry is given by mapping (uj) ∈(`πp,q(X), `p(Y )

)to the bounded linear

operator U : X → `q(Y ) defined by U(x) = (ujx).

Proof. Assume first that (uj) ∈(`πp,q(X), `p(Y )

); let r such that 1/p = (1/r) +

(1/q). Given x ∈ X, we may write

‖(ujx)‖q = ‖(ujαjx)‖p

for some numbers (αj) such that ‖(αj)‖r = 1. Now the assumption and Lemma2.13 give

‖(ujx)‖q ≤ ||(uj)||(`πp,q (X),`p(Y ))πp,q[αjx] = ||(uj)||(`πp,q (X),`p(Y ))‖x‖.

Conversely, let us assume that U : X → `q(Y ) is a bounded linear operator andU(x) = (uj(x))j∈N. If πp,q[xj ] ≤ 1 then

‖(ujxj)‖p = ‖(y∗j (ujxj))‖p = ‖((u∗

jy∗j )xj)‖p


for some y∗j ∈ Y ∗ with ‖y∗

j ‖ = 1, so

‖(ujxj)‖p ≤ πp,q [xj ]‖(u∗jy

∗j )‖`w

q (X∗) ≤ sup‖x‖≤1

‖((u∗jy

∗j )x)‖q

= sup‖x‖≤1

‖(y∗j (ujx))‖q ≤ sup

‖x‖≤1‖(ujx)‖q = ‖U‖L(X,`q(Y )). ¤

Corollary 2.29. Let X and Y be two Banach spaces, let q ∈ [1, ∞] and µ a σ-finitemeasure. Let (vj) ⊂ L(X,L1(µ)) be a sequence such that

∥∥(∑

j |vjx|q′)1/q′∥∥

1 ≤C1‖x‖ for all x ∈ X (‖(supj |vjx|)‖1 ≤ C1‖x‖ if q = 1), and let (uj) ⊂ L(L1(µ), Y )

be a sequence such that( ∑

j ‖ujf‖q)1/q ≤ C2‖f‖ for any f ∈ L1(µ) (‖uj‖ ≤ C2

for all j if q = ∞). Then there exists another constant C (depending on C1 andC2) such that

∑j ‖ujvjx‖ ≤ C‖x‖ for all x ∈ X.

Proof. By Theorems 2.23 and 2.28 we have that

X(vj(·))−→ `π1,q(L1)

(uj)−→ `1(Y )

is a composition of bounded operators, where (uj) means the operator defined bythe sequence as a multiplier. ¤

§3 Inclusions among the spaces `πp,q(X).

Let us point out first some elementary embeddings among these spaces.

Proposition 3.1. Let 1 ≤ r, s < ∞, 1 ≤ p1 ≤ p2, 1 ≤ q1 ≤ q2 and 1 ≤ p ≤ q.Then

`πp1,s(X,Y ) ⊆ `πp2,s(X,Y ) ,

`πr,q2(X,Y ) ⊆ `πr,q1

(X,Y )

`πp(X,Y ) ⊆ `πq (X,Y )

with continuous inclusions of norm 1.In particular, for 1 ≤ p, q < ∞,

`π1,q (X) ⊂ `π1(X) ⊂ `πp(X) ⊂ `πp,1(X).

Proof. The proofs of the two first embeddings are straighforward.To see the last one, take (uk) ∈ `πp(X, Y ), (xk) ∈ `w

q (X) and (λk) ∈ `r where(1/r) + (1/q) = (1/p). Then

(n∑

k=1

|λkuk(xk)|p)1/p ≤ πp[uj ]‖(λjxj)‖`wp (X) ≤ πp[uj ]‖(xj)‖`w

q (X)‖(λj)‖`r .

Taking the supremum over the unit ball of `r we get the result. ¤

We can actually get a general formulation which cover all the cases above andmany more ones.


Theorem 3.2. Let X a be Banach space, 1 ≤ p ≤ r, 1 ≤ q, s and (1/q) + (1/r) ≤(1/p) + (1/s). Then `πp,q (X) ⊆ `πr,s(X) , with continuous inclusion of norm 1.

Proof. The case s ≤ q follows from the norm 1 inclusions `ws (X∗) ⊆ `w

q (X∗) and`p(X) ⊆ `r(X). For q < s, r = ∞ or s = ∞ Proposition 2.1 and Remark 2.3 giveus the result. So we assume that q < s and r, s < ∞. Then 1 < r/p, s/q < ∞; leta and b their conjugate numbers, that is 1 = (1/a) + (p/r) = (1/b) + (q/s).

If πp,q[xj ] ≤ C, for any finite set of vectors x∗j in X∗ we have, for appropiate

scalars αj ≥ 0 such that∑

αaj = 1, that

(∑

j

|x∗jxj |r

)1/r =( ∑

j

|x∗j (α

1/pj xj)|p

)1/p ≤ C sup‖x∗‖≤1

( ∑

j

αq/pj |x∗xj |q

)1/q.

From our assumptions we have that ap ≤ bq, so that∑

j

αqp b

j ≤ 1, and for any x∗

we get, by Holder’s inequality,( ∑

j αq/pj |x∗xj |q

)1/q ≤( ∑

j |x∗xj |s)1/s

. This showsthat πr,s[xj ] ≤ C. ¤

Note that, in the scalar-valued case, for (1/p) − (1/q) = (1/r) − (1/s) we have

(`p, `q) = (`r , `s).

To find cases where `πp,q(X) = `πr,s(X) for (1/q) + (1/r) = (1/p) + (1/s) weneed the following lemma:

Lemma 3.3. Let X be a Banach space and 1 < r < ∞. Then `w1 (X) = `r`

wr′(X)

if and only if L(c0, X) = Πr(c0,X).

Proof. Assume `w1 (X) = `r`

wr′(X) and take u ∈ L(c0,X). According to the identi-

fication with `w1 (X) we have that the sequence xn = u(en) = αnx′

n where αn ∈ `r

and x′n ∈ `w

r′(X). This allows to factorize u = w.v where v ∈ L(c0, `r) is given byv(en) = αnen and w ∈ L(`r,X) is given by w(en) = x′

n. It is not difficult to show(see [DJT], page 41) that v ∈ Πr(c0, `r), and then u ∈ Πr(c0,X).

Conversely, assume L(c0,X) = Πr(c0,X) and let us take (xn) ∈ `w1 (X). Consider

now the operator u : c0 → X defined by u(en) = xn. From the assumptionu ∈ Πr(c0, X). Now, since (en) ∈ `w

1 (c0) and u ∈ Πr(c0,X) then (see [DJT] page53) u(en) = αnx′

n where αn ∈ `r and x′n ∈ `w

r′(X). ¤

Proposition 3.4. Let X be a Banach space such that L(c0,X∗) = Πs′(c0,X

∗) forsome 1 < s < ∞ . Then `πr,s(X) ⊆ `πp,q(X) for 1 ≤ p, q, r, s < ∞ such that(1/p) − (1/q) = (1/r) − (1/s).

Proof. Let us take (xn) ∈ `πr,s(X) and (x∗n) ∈ `w

q (X∗). To show that (x∗nxn) ∈ `p,

it suffices to see that for any (αn) ∈ `q′ we get (αnx∗nxn) ∈ `u where (1/p) +

(1/q′) = 1/u. Given now a sequence (αn) ∈ `q′ we have that (αnx∗n) ∈ `w

1 (X∗).Using Lemma 3.3 we have that there exists (βn) ∈ `s′ and (y∗

n) ∈ `ws (X∗) so

that αnx∗n = βny∗

n. Therefore (αnx∗n) = (βny∗

nxn) ∈ `s′`r = `u because 1/u =(1/p) + (1/q′) = (1/s′) + (1/r). ¤

Combining Theorem 3.2 and Proposition 3.4 we get the following result:


Theorem 3.5. Let X be a Banach space such that L(c0, X∗) = Πs′(c0,X

∗) forsome 1 < s < ∞. Then `πr,s(X) = `πp,q(X) for 1 ≤ p, q, r, s < ∞ such that1 ≤ p ≤ r and (1/p) − (1/q) = (1/r) − (1/s).

Proposition 3.6.(a) If X has cotype 2 then `w

1 (X) = `2`w2 (X).

(b) If X has cotype q > 2 then `w1 (X) = `r`

wr′(X) for any r > q.

Proof. Use Lemma 3.3 and the fact that L(c0, Y ) = Π2(c0, Y ) for any Y of cotype2 and L(c0, Y ) = Πr(c0, Y ) for any Y of cotype q > 2 and r > q (see Theorem11.14, [DJT]). ¤

Remark 3.7. Let X be any space with GL-property (see Page 350, [DJT] fordefinition and results). Then X has cotype 2 if and only if `w

1 (X) = `2`w2 (X).

Use Lemma 3.3 and the fact that, in the setting of spaces with the GL-property,L(c0,X) = Π2(c0,X) if an only if X is of cotype 2, (see page 352, [DJT]).

Remark 3.8. Let X be a G.T. space, i.e. X satisfies the Grothendieck theoremL(X, `2) = Π1(X, `2). Then `w

1 (X) = `2`w2 (X).

Indeed, if u ∈ L(c0,X) then u∗ ∈ L(X∗, `1). Now GT property on X gives thatu∗ ∈ Π2(X∗, `1) (see [Pi], page 71 ) which implies that u∗ factors through a Hilbertspace, and so u does. Therefore u ∈ Π2(c0,X).

Corollary 3.9. If X∗ has cotype 2 then `πp,q (X) = `πr,2(X) for any p ≤ r and1/q = (1/p) − (1/r) + (1/2).

In particular `π1(X) = `π2(X) and `π1,q(X) = `πr,2(X) for 1/r = (1/q′)+ (1/2).

Corollary 3.10. If X∗ has cotype q0 > 2 then `πp,q(X) = `πr,s(X) for any p ≤ r,s < q′

0 and (1/p) − (1/q) = (1/r) − (1/s).In particular `πp(X) = `π1(X) for any 1 ≤ p < q′

0 and `π1,q (X) = `πr,s(X) fors < q′

0 and 1/r = (1/q′) + (1/s).

Let us now see a connection between the (p, q)-summing sequences and someproperties in Banach space theory.

Proposition 3.11. Let X be a Banach space and 1 ≤ q ≤ p < ∞ and r ≥ p′. Thefollowing are equivalent:

(i) idX∗ is (p, q)-summing.(ii) `r(X) ⊆ `πs,q (X) for any 1 ≤ s ≤ r such that 1/s = (1/r) + (1/p).Moreover, πp,q(idX∗) = supπs,q [xj ] : ‖(xj)‖`r(X) = 1.

Proof. Assume first that the identity in X∗ is (p, q)-summing. If r and s are asstated, (xj) ∈ B`r(X) and x∗

1, . . . , x∗n ∈ X∗ we see that

( ∑

j

|x∗jxj |s

)1/s ≤( ∑

j

‖x∗j‖p

)1/p ≤ πp,q(idX∗) ‖(x∗j )‖`w

q (X∗).

Conversely, we assume now that `r(X) ⊆ `πs,q (X) and x∗1, . . . , x

∗n in X∗. From

Lemma 2.16 we have

( ∑

j

‖x∗j‖p

)1/p = sup(n∑

k=1

|x∗kxk|s)1/s;

n∑

k=1

‖xk‖r = 1.


Then (xj) is of norm 1 in `r(X), and if C is the norm of the inclusion of `r(X) in`πs,q (X) we have

( ∑

j

|x∗jxj |s

)1/s ≤ C‖(x∗j )‖`w

q (X∗). This yields( ∑

j

‖x∗j‖p

)1/p ≤

C‖(x∗j )‖`w

q (X∗). ¤

Some particularly interesting cases are given in the following corollaries.

Corollary 3.12. Let X be a Banach space and 1 ≤ p. The following are equivalent:(a) idX∗ is (p, 1)-summing.(b) `∞(X) = `πp,1(X).(c) `p′(X) ⊂ `π1(X).

Corollary 3.13. Let X be a Banach space. The following are equivalent.(a) X is finite dimensional.(b) `πp,q(X) = `∞(X) for some p ≥ q with (1/q) − (1/p) < 1/2.(c) `s(X) ⊂ `πp,q (X) for some 1 ≤ p ≤ q and p < s < r with (1/s)−(1/r) < 1/2.(d) `πp,1(X) = `∞(X) for some (or for all) 1 ≤ p < 2.(e) `p′(X) ⊆ `π1(X) for some (or for all) 1 ≤ p < 2.

Proof. To see that (b) implies (a) use the fact that idX∗ ∈ Πp,q(X∗,X∗) for (1/q)−(1/p) < 1/2. This gives that X∗ is finite dimensional (see [DJT] page 199).

To see that (c) implies (a), from Proposition 3.11 idX∗ ∈ Πq1,q(X∗,X∗) where(1/s)+(1/q1) = (1/p), what again allows to get the result because (1/q)− (1/q1) <1/2.

(d) is the particular case of (b) for q = 1.(e) is equivalent to (d) from Corollary 3.12. ¤

As applications of Corollary 3.12 we have:

Corollary 3.14. Let X be a Banach space. The following are equivalent:(a) `π2,1(X) = `∞(X).(b) `2(X) ⊆ `π1(X).(c) X∗ has Orlicz property.

Corollary 3.15. Let X be a Banach space and p > 2. The following are equivalent:(a) `πp,1(X) = `∞(X).(b) `p′(X) ⊆ `π1(X).(c) X∗ has cotype p.

Proof. It follows from Corollary 3.12 and the deep result, due to M. Talagrand (see[T1]), which asserts that, for 2 < q < ∞, the identity in X is (q, 1)-summing if andonly if X has cotype q. ¤

Remark 3.16. For p > 1 and 1 ≤ q < ∞, in general `πp,q (X) 6= Ip(`q,X).Indeed, recalling that I2(X,Y ) = Π2(X,Y ) for every couple of spaces X and Y

(see corollary 5.9 in [DJT]), we deduce that `π2,1(`∞) 6= I2(`1, `∞): By Corollary3.14 we have that `π2,1(`∞) = `∞(`∞) ' L(`1, `∞), but L(`1, `∞) does not coincidewith Π2(`1, `∞) because Π2(`1, `∞) is the same as Π1(`1, `∞) (for `1 is of cotype 2,and Corollary 11.16 in [DJT] applies) and on the other hand Π1(`1, `∞) 6= L(`1, `∞):the operator given by x ∈ `1 7→ (

∑nj=1 xj)n ∈ `∞ is not absolutely summing (see

[W], exercise III.F.3).


Proposition 3.17. Let E be a subspace of X. Then `πp,q(E) ⊆ `∞(E) ∩ `πp,q(X),but equality does not hold in general.

Proof. The embedding is straightforward.Let us show that for p = q = 1 there exists E such that `π1(E) 6= `∞(E)∩`π1(X):Let us take E such that `2(E) 6⊆ `π1(E) (for instance E = `1). Since E is a

subspace of X = `∞(Γ) for Γ = BE∗ and (`∞(Γ))∗ = (`1(Γ))∗∗ is of cotype 2 ,then `2(E) ⊆ `∞(E) ∩ `π1(X). Therefore `∞(E) ∩ `π1(X) does not coincide with`π1(E). ¤

We have seen that `πp,1 (X) coincides with `∞(X) in various situations, but letus get that it always contains `w

p (X):

Theorem 3.18. Let X be a Banach space, p > q and 1/s′ = (1/q) − (1/p). Then`ws (X) ⊂ `πp,q (X) with inclusion of norm 1.

Proof. For any finite family of vectors (xj)1≤j≤n in X and (x∗j )1≤j≤n in X∗, since

1/p′ > 1/q′ and 1/p′ = (1/s′) + (1/q′), we can write

(∑

j

|x∗jxj |p)1/p = sup

‖(αj)‖p′=1|∑

j

αjx∗jxj |

= sup‖(βj)‖s′ =1

sup‖(λj)‖q′ =1

|∑

j

βjλjx∗jxj |

= sup‖(βj)‖s′ =1


|∫ 1

0〈∑

j

rj(t)λjx∗j ,

∑

k

rk(t)βkxk〉dt|

≤ sup‖(βj)‖s′ =1


supt∈[0,1]

‖∑

j

rj(t)λjx∗j‖X∗‖

∑

k

rk(t)βkxk‖X

≤ ‖(x∗j )‖`w

q (X∗)‖(xj)‖`ws (X). ¤

Corollary 3.19. Let X be a Banach space. Then `wp (X) ⊂ `πp,1(X) with inclusion

of norm 1.

As a consequence, next we study whether the sequence given by the canonicalbasis (ej) belongs to `πp,q(`r), depending on the values of p, q and r.

Proposition 3.20. For any p ≥ 1 we have (ej) ∈ `πp,1(`p′), with πp,1[ej ; `p′ ] = 1.

Proof. Note that for p ≥ 2 this is follows from Corollaries 3.14 and 3.15, because(`p′)∗ = `p has cotype p.

For 1 ≤ p < 2 it follows from the fact (ej) ∈ `wp (`p′) and Corollary 3.19. ¤

Theorem 3.21. (ej) ∈ `πp,q(`r) if and only if it holds that p = ∞ or 1/r ≤(1/q) − (1/p). Moreover, in these cases πp,q [ej ] = 1.

Proof. For p < q we have that `πp,q (`r) ⊂ `( 1p − 1

q )−1(`r). Hence if we assume that

(ej) ∈ `πp,q (`r) then q ≤ p. As the norm of the inclusion `nq′ → `n

r′ is n( 1q − 1

r )+ , wesee that

( n∑

j=1

|〈ej , ej〉|p)1/p = n

1p ≤ πp,q[ej ]n( 1

q − 1r )+ ,

which leads to p = ∞ or q < r with 0 ≤ 1/q − 1/p − 1/r.


Conversely, if p = ∞ then (ej) ∈ `∞(`r) = `π∞,q (`r). And if 1/q − 1/p − 1/r ≥ 0then, by Proposition 3.20 and Theorem 3.2, we obtain

(ej) ∈ `πr′,1(`r) ⊆ `πp,q(`r).

The inclusion above is of norm 1, and then πp,q[ej ] = 1 when bounded. ¤

Note that, as a consequence, we get the well-known result asserting that id: `p →`q is integral if and only if p = 1 and q = ∞, according to Theorem 2.21.

§4 (p, q)-summing sequences and Grothendieck’s Theorem.

Theorem 4.1. Let X be a Banach space. Then `π1,2(X) ⊂ Rad(X) ⊂ `π1(X).

Proof. Let us take a finite family of vectors (xj)1≤j≤n in X. Using that L1([0, 1], X)isometrically embedds into the dual of L∞([0, 1],X∗), we have∫ 1

0

∥∥n∑

k=1

xkrk(t)∥∥dt = sup

‖g‖L∞([0,1],X∗)=1

∣∣n∑

k=1

< xk

∫ 1

0g(t)rk(t)dt >

∣∣

≤ π1,2[xj ] sup‖g‖L∞([0,1],X∗)=1

sup‖(αk)‖2=1

∥∥n∑

k=1

αk

∫ 1

0g(t)rk(t)dt

∥∥

= π1,2[xj ] sup‖g‖L∞([0,1],X∗)=1

sup‖(αk)‖2=1

∥∥∫ 1

0(

n∑

k=1

αkrk(t))g(t)dt∥∥

= π1,2[xj ] sup‖(αk)‖2=1

∫ 1

0

∣∣n∑

k=1

αkrk(t)∣∣dt

≤ π1,2[xj ].

On the other hand, for any finite family of vectors (xj)1≤j≤n in X and (x∗j )1≤j≤n

in X∗ we can writen∑

j=1

|x∗jxj | ∼ sup

εk=±1

∣∣n∑

j=1

< x∗j , εjxj >

∣∣

= supεk=±1

∣∣∫ 1

0<

n∑

j=1

εjx∗jrj(t),

n∑

j=1

xjrj(t) > dt∣∣

≤∥∥

n∑

j=1

xjrj

∥∥Rad(X)‖(x∗

j )‖`w1 (X∗)

This gives the other inclusion. ¤Remarks 4.2. Let X be a Banach space and 1 < p ≤ 2. It follows from Theorem4.1 that if X has type p then `p(X) ⊂ `π1(X). But this is also a consequence ofCorollary 3.15 and the fact that if X has type p then X∗ has cotype p′.

If Rad(X) ⊂ `π1,p(X) then X has cotype p′.

Recall that a Banach space X (see Remark 3.8) is a G.T. space if it verifiesGrothendieck’s theorem in the sense that Π1(X,H) = L(X,H) for every Hilbertspace H). G. Pisier gave a characterization of G.T. spaces of coytpe 2 (see theorem6.6 and corollary 6.7 in [Pi]) as those satisfying something equivalent to Rad(X) ⊂`π1,2(X). Then, combining this with Theorem 4.1, we can state the following:


Theorem 4.3. Rad(X) = `π1,2(X) if and only if X is a G.T. space of cotype 2.

Remark 4.4. Using Kintchine’s inequalities we have that L1(µ, `2) = Rad(L1(µ)).Then Theorem 2.23, that is `π1,2(L1(µ)) = L1(µ, `2), also follows from Theorem 4.3.

Grothendieck’s theorem has been stated in a lot of different ways. We shall giveyet another formulation of it in terms of the `πp,q spaces.

For any Banach space X, let us consider the bilinear map VX : `∞(X∗)×`∞(X) →`∞(`∞) given by VX

((x∗

j ), (xk))

=((x∗

jxk)k

)j. It’s clear that VX is bounded. For

some Banach spaces we can get that actually the bilinear map is bounded notonly with values in `∞(`∞) but in some smaller space. This is the case for `p if1 < p < ∞.

Theorem 4.5. Let X be a Banach space. Then L(`1,X) = Πp,q(`1,X) if and onlyif VX defines a bounded bilinear map VX : `∞(X∗) × `∞(X) → `πp,q(`∞).

Proof. Let (xj) ⊂ X and (x∗j ) ⊂ X∗ be such that ‖xj‖, ‖x∗

j‖ ≤ 1 for all j. Letu: `1 → X the continuous operator such that uej = xj for all j; clearly ‖u‖ ≤ 1.

By hypothesis we can take C (independently of (xj)) such that πp,q(u) ≤ C‖u‖ ≤C. That is,

‖(uyj)‖`p(X) ≤ C‖(yj)‖`wq (`1)

for any finite family (yj) ⊂ `1. Therefore if ξj = x∗j u for each j then

((〈ξj , ek〉)k)j = ((x∗j (uek))k)j = ((x∗

jxk)k)j = VX ((x∗j ), (xj)).

Therefore‖(〈ξj , yj〉)‖`p = ‖(〈x∗

j , uyj〉)‖`p ≤ ‖(uyj)‖`p(X),

and hence‖(〈ξj , yj〉)‖`p ≤ C‖(yj)‖`w

q (`1),

showing that πp,q[ξj ; `∞] ≤ C.Let us assume now that VX : `∞(X∗)×`∞(X) → `πp,q(`∞) is bounded with norm

C. Given u ∈ L(`1, X), for every finite family (yj) ∈ `1 we have that

‖(uyj)‖`p(X) = sup‖(〈x∗j , uyj〉)‖`p ; (x∗

j ) ⊂ BX∗≤ supπp,q[VX((x∗

j ), (uej)); `∞]; (x∗j ) ⊂ BX∗ ‖(yj)‖`w

q (`1)

≤ C‖u‖ ‖(yj)‖`wq (`1),

and then πp,q(u) ≤ C‖u‖. ¤

When applied to the Hilbert case we get the new formulation of Grothendieck’stheorem:

Corollary 4.6. If H is a Hilbert space, the bilinear form

VH : `∞(H) × `∞(H) → `π1(`∞)

is bounded, and its norm is Grothendieck constant KG.

Taking H = `2 (with no loss of generality) it is a particular case of the followingresult:


Corollary 4.7. If 1 ≤ p ≤ ∞ and 1/r = 1−|(1/p)− (1/2)|, then the bilinear form

V`p : `∞(`p′) × `∞(`p) → `πr,1(`∞)

is bounded, with ‖V`p‖ ≤ 2a2 K1−a

G , where a = |1 − 2/p|.Proof. Equivalently

πr,1(u) ≤ 2a2 K1−a

G ‖u‖for every operator u ∈ L(`1, `p), which is an extension, due to Kwapien, of Grothen-dieck’s theorem (see [K], and also [DF, 34.11] and [TJ]). ¤

Remark 4.8. Note in the previous result that 1 ≤ r ≤ 2. The case r = 2 is forp = 1 (or p = ∞). By Corollary 3.14 we know that `π2,1(`∞) = `∞(`∞), so thestatement is trivial in this case. However, Corollary 3.14 tells us that for r < 2 theinclusion `πr,1(`∞) ⊆ `∞(`∞) is proper.

References

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33–37.[P1] A. Pietsch, Absolut p-summeriende Abbildungen in normierten Raumen, Studia Math. 29

(1968), 275–326.[P2] A. Pietsch, Operator Ideals, North-Holland, 1980.[Pi] G. Pisier, Factorization of Operators and Geometry of Banach spaces, CBM 60, Amer.

Math. Soc., Providence R.I., 1985.[T1] M. Talagrand, Cotype of operators from C(K), Inventiones Math. 107 (1992), 1–40.[T2] M. Talagrand, Cotype and (q, 1)-summing norm in a Banach space, Inventiones Math.

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Jose Luis Arregui. Departamento de Matematicas, Universidad de Zaragoza,

50009 Zaragoza, Spain.

E-mail address : [email protected]

Oscar Blasco. Departamento de Analisis Matematico, Universidad de Valencia,

46100 Burjassot (Valencia), Spain.

E-mail address : [email protected]

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