(p;q) - SUMMING SEQUENCES
J. L. Arregui O. Blasco
n. 3
2002
“garcia de galdeano”
PRE-PUBLICACIONES del seminario matematico
seminario matemático
garcía de galdeano Universidad de Zaragoza
(p, q)-SUMMING SEQUENCES.
Jos´ e Luis Arregui and Oscar Blasco
§1 Introduction. Given a real or complex Banach space X and 1 ≤ p ≤ ∞, we denote by `p (X) and `w (X) the Banach spaces of sequences in X with norms k(xn )k`p (X) = k(kxn k)k`p p and k(xj )k`w = supx∗ ∈BX∗ k(x∗ xj )k`p respectively. p (X) It is well known (see page 50, [DJT]) that `w p (X) = `p (X) for some 1 ≤ p < ∞ if and only if X is finite dimensional, and of course `w ∞ (X) = `∞ (X). We 0 also have natural isometric isomorphisms `w (X) ' L(` , X) (1 < p ≤ ∞) and p p w `1 (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 ,P X) of compact operators corresponds with the sequences (xj ) such that the series j xj converges unconditionally. For any infinite dimensional Banach space X we have that `1 (X) is strictly contained in K(c0 , X), because there exist unconditionally convergent series which are not absolutely convergent, as the Dvoretzky-Rogers’ theorem asserts (see [DR], [D] or [DJT, 1.2]). An operator uP∈ L(X, Y ) is absolutely P summing if for every unconditionally convergent 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 constant C such that n n ¡X ¢ ¡X ¢1/p p 1/p kuxj k ≤ C sup |x∗ xj |p j=1
x∗ ∈BX ∗
j=1
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 space for 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 exists a constant KG , called the Grothendieck constant, such that π1 (u) ≤ KG kuk for any bounded u: L1 (µ) → H and which is the least among all constants with this property. This formulation of Grothendieck’s theorem and the meaning of KG was given by Pietsch ([P1]) and Lindenstrauss and PeÃlczy´ nski ([LP]), by changing from the Partially supported by D.G.E.S.I.C. PB98-1426
1
Typeset by AMS-TEX
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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 operators is formed by those mapping sequences in `w q (X) into sequences in `p (Y ), in other words u ∈ Πp,q if there exists C such that k(uxj )k`p (Y ) ≤ Ck(xj )k`w q (X) for any finite family of vectors xj in X, and the least of such C is the (p, q)-summing norm of u, denoted by πp,q (u). Obviously, (p, p)-summing is just the same as psumming. Despite the identity is not p-summing for any Banach space with infinite dimension, it can be (p, q)-summing P 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 actually have a property, known as cotype 2, stronger than the Orlicz property. For each 1 ≤ p ≤ ∞, RadP p (X) will be taken as the closure in Lp ([0, 1], X) of the n set of functions of the form j=1 rj xj , xj ∈ X, where (rj )j∈N are the Rademacher functions on [0, 1] defined by rj (t) = sig(sin 2j πt). 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 °X ° ° rj xj ° j
∞
° ° °T : ej 7→ xj ° n ∼ k(xj )k`w = . (X) 1 L(` ,X) ∞
Making use of the Kahane’s inequalities (see [DJT], page 211) it follows that Radp (X) coincide up to equivalent norms for all p < ∞. The space will be denoted then Rad(X), and we shall use the L1 -norm throughout the paper. It is also known that Rad(X) can be identified with the space of the almost unconditionally P∞ summable 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 (respect. 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 other words if there exists a constant C such that Z 1 X n n ¡X ¢1/p || xj rj (t)||dt ≤ C kxj kp 0
j=1
j=1
(respect. n ¡X j=1
kxj k
¢ q 1/q
≤C
Z
1 0
||
n X
xj rj (t)||dt),
j=1
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 ) be two Banach spaces whose elements are defined by sequences of vectors in X and Y (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 exists a constant C > 0 such that ° ° ° ° n ° °(uj xj )n ° ° j=1 F (Y ) ≤ C (xj )j=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 belonging to (`w q (X), `p (Y )) corresponds to u being an operator in Πp,q (X, Y ). Also the case 0 uj = λj .u ∈ (`w q (X), `1 (Y )) for all (λj ) ∈ `p0 , where (1/p) + (1/p ) = 1, corresponds to u ∈ Πp,q (X, Y ). These facts suggest the use of the notation `πp,q (X, Y ) instead of (`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 finite collection of vectors x1 , x2 , . . . xn in X, it holds that n ³X j=1
kuj xj k
p
´1/p
≤ C sup
n n³ X j=1
∗
|x xj |
q
´1/q
o
∗
; x ∈ BX ∗ .
The reader is referred to [AF] for the particular case p = q, X = Y and uj = αj IdX . A scalar sequence (αj ) is there defined to be a p-summing multiplier if uj = αj IdX belongs to `πp (X, Y ). Actually, in this article we shall consider mainly the case Y = K, what leads us to define a new family of spaces of vector valued sequences, not only for dual spaces, 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 ³X j=1
|x∗j xj |p
´1/p
≤ C sup
n n³ X j=1
|x∗j x|q
´1/q
; x ∈ BX
o
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 describe some classical aspects of the theory of geometry of Banach spaces and operator ideals, from this point of view. The paper is divided into three sections. In the first one we introduce the spaces of (p, q)-summing multipliers and (p, q)-summing sequences in X. They make sense for any 1 ≥ p, q, but the cases 1 ≤ p < q and p ≥ q are different in kind. The case p = 1 ≤ q is shown to correspond with integral operators from `q to X, and in particular we can identify the (1, q)-summing sequences in (L1 (µ)). We also show that the space of operators L(X, `q (Y )) is isometrically isomorphic to the space of multipliers (`πp,q (X), `q (Y )) for any p ≤ q.
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J.L. ARREGUI, O. BLASCO
In Section 3 we try to recover some results from the finite dimensional setting in the infinite dimensional situation, at least under certain assumptions on the space X. 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 2 then `π1 (X) = `π2 (X) and `π1,q (X) = `πr,2 (X) for 1/r = (1/q 0 ) + (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 finally observe that `w p (X) ⊂ `πp,1 (X), and use it to find the values or p, q and r where the canonical basis (ej ) belongs to `πp,q (`r ). In the final section we point out an equivalent formulation to Grothendieck’s theorem in terms of these spaces, by showing that the bilinear map defined by ((x∗j ), (xk )) → ((x∗j xk )k )j maps `∞ (X ∗ ) × `∞ (X) into `π1 (`∞ ) for Banach spaces isomorphic to Hilbert spaces . We also present a lemma, that together with a result by Pisier, allows to conclude that the spaces where `π1,2 (X) = Rad(X) are those satisfying Grothendieck’s theorem and having cotype 2. Notation is fairly standard. We follow the usual terms L(X, Y ) for the space of bounded linear operators between Banach spaces, BX and SX for the unit ball and sphere in X, X ∼ Y if two Banach spaces are isomorphic and X ' Y if they are isometrically isomorphic. We write the action of an operator or functional on x merely as ux and x∗ x, though we prefer to use x∗ (x) or hx∗ , xi 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 , p0 denotes the conjugate exponent of p, x+ = max{x, 0} and K denotes R or C if no difference is relevant. §2 (p, q)-summing multipliers and sequences. Let us start off by mentioning the different behaviour when using a single operator or a sequence of them. Let u 6= 0 be a bounded linear operator between two 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/p x where x∈ / Ker(u) to get a contradiction). Nevertheless for sequences of operators we get the 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 taking sequences in X of the form (0, . . . , 0, xj , 0, 0, . . . ). If q ≤ p it is plain that the converse is true. Let q > p and 1/p = (1/r) + (1/q). By H¨older’s inequality, X X X X ( kuj xj kp )1/p ≤ ( kuj kp kxj kp )1/p ≤ ( kuj kr )1/r ( kxj kq )1/q . j
j
j
j
Conversely, given n we note that the `r/p -norm of (kuj kp )nj=1 equals, by duality, the P q/p 1/p norm of (λj kuj kp ) in `1 for some 0 ≤ λj such that j λj = 1. Let βj = λj and
(p, q)-SUMMING SEQUENCES
5
P xj ∈ SX such that kuj xj k is arbitrarily close to kuj k; then ( j kuj (βj xj )kp )1/p n ¡X ¢1/r P p p 1/p approximates ( j kuj k βj ) , and hence kuj kr is bounded by a constant j=1
independent of n. ¤
The study of multiplier sequences between `w q (X) and `p (Y ) is far more complicated. For the reason explained in the introduction, we find more convenient to change 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 a constant C > 0 such that, for any finite collection of vectors x1 , x2 , . . . xn in X, it holds that n ³X j=1
kuj xj kp
´1/p
≤ C sup
n n³ X j=1
|x∗ xj |q
´1/q
o ; x∗ ∈ BX ∗ .
We use `πp,q (X, Y ) to denote the set of (p, q)-summing multipliers, and πp,q [uj ] is the least constant C for which (uj ) verifies the inequality in the definition. In order to avoid ambiguities, sometimes we shall use πp,q [uj ; X, Y ]. Of course if p = q we simply 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, but then `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 ) = {0} for p < q. Let us observe that c00 (L(X, Y )) ⊂ `πp,q (X, Y ) for any 1 ≤ p, q, where c00 (L(X, Y )) stands for all sequences of operators with a finite number of non-zero elements. Indeed, if uj = 0 for all j > N then πp,q [uj ] ≤ N 1/p max||uj ||. j≤N
Remark 2.5. For any Banach space X, and the usual identification between X and 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 < ∞ and uj ∈ L(X, Y ), we clearly have (uj ) ∈ `πp,q (X, Y ) if and only if (λj uj ) ∈ `π1,q (X, Y ) for all (λj ) ∈ `p0 . Moreover πp,q [uj ; X, Y ] = sup{π1,q [λj uj ; X, Y ] : ||λj ||`p0 = 1}. Let us mention that the characterization of the absolutely summing operators in terms of unconditional series can be generalized as follows, with the same standard proof:
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J.L. ARREGUI, O. BLASCO
Proposition 2.7. A sequence (uj ) is P in `πp,1 (X, Y ) if and only if it holds that for any unconditionally convergent series xj in X we have (uj xj )j ∈ `p (Y ) .
The subspace of L(`w q (X), `p (Y )) formed by (p, q)-summing sequences of operators 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 sequences in X ∗ , it seems natural, somehow, to denote it in the form E(X ∗ ). For any Banach space 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 n ³X ´1/p n³ X ´1/q o ∗ p ∗∗ ∗ q ∗ ∗ |x∗∗ x | ≤ C sup |x x | ; x ∈ B X j j j for every
j=1 ∗∗ x∗∗ 1 , . . . , xn
j=1
∗∗
in X .
Proof. It is only that (i) implies (ii) which deserves a proof, and it is a consequence of the principle of local reflexivity, in its weak form (see [DF]): Let E and F the ∗∗ ∗ ∗ subspaces of X ∗∗ and X ∗ generated by x∗∗ 1 , . . . , xn and by x1 , . . . , xn respectively. Given ε > 0, there exists a linear operator u: E → X such that kuk < 1 + ε and x∗ (ux∗∗ ) = x∗∗ x∗ for any x∗∗ ∈ E and x∗ ∈ F . Therefore n n n ¡X ¢ ¡X ¢ ¡X ¢ ∗∗ ∗ p 1/p ∗ ∗∗ p 1/p q 1/q |xj xj | = |xj (uxj )| ≤ C sup |x∗ (ux∗∗ . j )| j=1
x∗ ∈BX ∗
j=1
The last term equals the norm of the linear operator
E is given by ej 7→ x∗∗ j , and the norm of v equals
sup
j=1 uv: `nq0 → X, where v: `nq0 → ¡ Pn ¢ ∗∗ ∗ q 1/q |x x | . Since j j=1
x∗ ∈BX ∗
kuvk ≤ (1 + ε)kvk, we obtain that n n ¡X ¢ ¡X ¢ ∗ p 1/p ∗ q 1/q |x∗∗ x | ≤ C(1 + ε) sup |x∗∗ j j j x | x∗ ∈BX ∗
j=1
for any ε > 0.
j=1
¤
Definition 2.10. 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 n ³X ´1/p n³ X ´1/q o ∗ p |xj xj | ≤ C sup |x∗j x|q ; x ∈ BX j=1
for any finite collection of vectors
j=1 ∗ ∗ x1 , . . . , xn 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 πp,q [yj∗ uj ; X ∗ ] . kyj∗ k=1
(p, q)-SUMMING SEQUENCES
7
Lemma 2.13. Let X be a Banach space, 1 ≤ p, q ≤ ∞, (αj ) ⊆ K and x ∈ X: Then πp,q [αj x] = k(αj )kr kxk where 1/r = ((1/p) − (1/q))+ .
Proof. Given x∗1 , . . . , x∗n ∈ X ∗ we see that k(αj x∗j x)kp ≤ k(αj )kr k(x∗j x)kq
∗ , ≤ k(αj )kr kxk k(x∗j )k`w q (X )
and thus we obtain an inequality. To get the converse we note that, choosing x∗ ∈ SX ∗ such that x∗ x = kxk, we may take a sequence (βj ) of scalars so that, if x∗j = βj x∗ , the value of k(αj x∗j x)kp = k(αj βj )kp kxk is arbitrarily close to kxk k(αj )kr k(βj )kq , and then we just need to observe that ∗ . k(βj )kq = k(x∗j )k`w ¤ q (X ) Note in particular that any non trivial constant sequence is in `πp,q (X) if and only 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 ˆ ⊂ `πp,q (X), `r ⊗X with continuous inclusions of norm 1. ∗ ∗ Proof. The first one is a direct consequence of `w q (X ) ⊂ `∞ (X ), and we get ∗ w ∗ the second by `q (X ) ⊂ `q (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 becomes an 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 X j=1
kx∗j ks )1/s
n X = sup{( |x∗j xj |t )1/t : k(xj )k`r (X) = 1}. j=1
Proof. For t = 1 this is just the duality `s (X ∗ ) = (`s0 (X))∗ . The general case follows from using that (
n X j=1
|x∗j xj |t )1/t
= sup{
n X j=1
|αj x∗j xj |
:
n X j=1
0
|αj |t = 1}.
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J.L. ARREGUI, O. BLASCO
Let us observe now that `s0 (X) = `t0 `r (X) (which follows simply by writing xn = 0 0 0 0 kxn ks /t yn where yn = kxn k−s /t xn for xn 6= 0 and yn = 0 for xn = 0). Therefore sup{(
n X
|x∗j xj |t )1/t : k(xj )k`r (X) = 1}
j=1 n X
= sup{
= sup{
j=1 n X j=1
|αj x∗j xj | |x∗j yj |
:
:
n X
j=1 n X j=1
0
|αj |t = 1, k(xj )k`r (X) = 1}
n X kyj k = 1} = ( kx∗j ks )1/s . ¤ s0
j=1
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 Lemma 2.16 one has (
n X
k=1
||x∗k ||q )1/q
= sup{(
n X
k=1
|x∗k xk |p )1/p
:
n X
k=1
||xk ||r = 1}.
∗ ∗ Therefore assuming `r (X) ⊂ `πp,q (X) we shall have `w q (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 first that, in general, the πp,q -norm of any sequence is independent from any reordering of its terms: Proposition 2.18. Let (xj ) a bounded sequence in any Banach space X, and let 1 ≤ 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 weak q-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 let 1 ≤ q ≤ p < ∞. Then πp,q [xσ(j) ] ≤ πp,q [xj ]
(p, q)-SUMMING SEQUENCES
9
for any map σ: N → N.
Proof. Given x∗1 , x∗2 , . . . , x∗n ∈ X ∗ we have ¡X ∗ ¢1/p ¡ X X ¢1/p ¡ X X ¢1/p |xj xσ(j) |p = ( |x∗j xk |p ) ≤ ( |x∗j xk |q )p/q j
k
k
σ(j)=k
¯p ¢1/p ¡X¯ X ¯( = αj x∗j )xk ¯ ) k
=
¡X k
(where (αj )σ(j)=k ∈ B`q0 for each k)
σ(j)=k
|yk∗ xk |p
¢1/p
X
(with yk∗ =
sup k(βk )kq 0 ≤1
αj x∗j ∈ X ∗ )
σ(j)=k
∗ = πp,q [xj ] ≤ πp,q [xj ] k(yk∗ )k`w q (X )
= πp,q [xj ]
σ(j)=k
sup k(βk )kq0 ≤1
°X ° ° βk yk∗ ° k
°X ° ° αj βσ(j) x∗ ° ≤ πp,q [xj ] j
j
sup
k(γj )kq0 ≤1
∗ . = πp,q [xj ] k(x∗j )k`w ¤ q (X )
°X ° ° γj x∗ ° j
k
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 ¡X
k=1
sup |x∗k xj |p j
¢1/p
≤ C sup
x∈BX
n ¡X
k=1
|x∗k x|q
for any x∗1 , . . . , x∗n ∈ X ∗ , and the least such C is πp,q [xj ].
¢1/q
Recall the following definition: an operator u ∈ L(X, Y ) is p-integral if there u
jY
β
exists a measure µ such that the composition X → Y −→ Y ∗∗ factorizes as X → ip
α
L∞ (µ) −→ Lp (µ) → 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 kαkkβk 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 of integral operators. Any p-integral operator u is also p-summing, and πp (u) is not greater than the p-integral norm, but the converse is not true in general. Basic results on p-integral operators can be seen in [DJT]. We’ll make use of the following fact: u: X → Y is integral if and only if there exists a constant C > 0 such that | tr(uv)| ≤ Ckvk for any finite rank linear operator v: Y → X, and the least such C is the integral norm of u. This makes easy to characterize the (1, q)-sequences in terms of integral operators:
10
J.L. ARREGUI, O. BLASCO
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 → X by uej = xj , and the integral norm of u is then π1,q [xj ]. Proof. Let u an integral operator `q → X with Pn uej = xj for all j, and let C its integral norm. Given x∗1 , . . . x∗n ∈ X ∗ , let v = j=1 x∗j ⊗λj ej , where λj = sig(x∗j xj ). Then n n X X |x∗j xj | = λj x∗j xj = tr(uv) , j=1
j=1
Pn
∗ . Therefore π1,q [xj ] ≤ C. |x∗j xj | ≤ Ckvk, and kvk is just k(x∗j )k`w q (X ) Conversely, let (xj ) ∈ `π1,q (X). Then (xj ) ∈ `q 0 (X), so u: ej 7→ xj defines Pn ∗ a bounded operator in L(`q , X). Now, if v = j=1 xj ⊗ ξj with ξjP= (ξjk )k ∈ P ∗ ∗ ∗ `q then, for vk∗ = j ξjk xj ∈ X , it turns out that | tr(uv)| = k |vk xk | ≤ ∗ ∗ π1,q [xk ]k(vk∗ )k`w and k(vk∗ )k`w = kvk, giving that the integral norm of u q (X ) q (X ) is bounded by π1,q [xj ] . ¤
so
j=1
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: Z For any Banach lattice X, an operator u: X → L1 (µ) is integral if and only if ¡ ¢ sup |ux| dµ < ∞, its value being the integral norm of u (see [DJT, Th. 5.19]). x∈BX
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 Z
k(fj (w))k`q0 dµ(w) < ∞ ,
and then the integral equals π1,q [fj ]. ¯X ¯ Proof. Just note that sup ¯ λj fj (w)¯ = k(fj (w))kq 0 for any w in the measure space.
k(λj )kq =1
j
Remark 2.24. When 1 < q < ∞ we thus have an isometry between `π1,q (L1 (µ)) and L1 (µ, `q 0 ). Of course, for q = ∞ the result keeps true, since `π1,∞ (L1 (µ)) = `1 (L1 (µ)) ' L1 (µ, `1 ). Z As for q = 1, recall that we can have sup |fj (w)|dµ(w) < ∞ with w 7→ (fj (w)) j
not ¡ being¢ a measurable function. For example, for the NRademacher functions rj in [0, 1], dt we have that {(rj (t)), t ∈ [0, 1]} = {−1, 1} 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:
(p, q)-SUMMING SEQUENCES
11
Corollary 2.25. Let µ a σ-finite measure. Then (fj ) ∈ `π1 (L1 (µ)) if and only if there 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 following way: Corollary 2.26. Let (uj ) be a sequence of bounded operators in L(c0 , Y ), such ¢ 0 P ¡P ∗ q 0 1/q that uj (ek ) = ykj . Then π1,q [uj ] = sup for q > 1 and k j |yj ykj | π1 [uj ] = sup
X
kyj∗ k≤1 k
sup|yj∗ ykj |.
kyj∗ k≤1
j
Another consequence of the interpretation of π1 -sequences as integral operators is the following: Corollary 2.27. Let X be a Banach space and let (xj ) be a bounded sequence in X. Then (xj ) ∈ `π1 (X) if and only if there exist a Banach space Y , a sequence (yj∗ ) ∈ `∞ (Y ∗ ) and u ∈ Π1 (X ∗ , Y ) such that xj = yj∗ ◦ u ∈ X ∗∗ for each j. Proof. Let us assume that such u and (yj∗ ) do exist. The constant sequence (uj = u) ∗ ∗ is a multiplier from `w 1 (X ) to `1 (Y ), and (yj ) ∈ (`1 (Y ), `1 ), so the composition ∗ ∗∗ (xj ) = (yj∗ ◦ u) belongs to (`w 1 (X ), `1 ) = `π1 (X ). Conversely, if (xj ) ∈ `π1 (X) then Theorem 2.21 says that v: `1 → X given by 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 absolutely summing). Then we can take Y = `∞ , u = v ∗ and (yj∗ ) = (ej ) in `1 ⊂ (`∞ )∗ . Since ej (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) = (uj x). ¡ ¢ 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 k(uj x)kq = k(uj αj x)kp for some numbers (αj ) such that k(αj )kr = 1. Now the assumption and Lemma 2.13 give k(uj x)kq ≤ ||(uj )||(`πp,q (X),`p (Y )) πp,q [αj x] = ||(uj )||(`πp,q (X),`p (Y )) kxk. Conversely, let us assume that U : X → `q (Y ) is a bounded linear operator and U (x) = (uj (x))j∈N . If πp,q [xj ] ≤ 1 then k(uj xj )kp = k(yj∗ (uj xj ))kp = k((u∗j yj∗ )xj )kp
12
J.L. ARREGUI, O. BLASCO
for some yj∗ ∈ Y ∗ with kyj∗ k = 1, so ∗ ∗ ∗ ≤ sup k((u y )x)kq k(uj xj )kp ≤ πp,q [xj ]k(u∗j yj∗ )k`w j j q (X )
=
sup k(yj∗ (uj x))kq kxk≤1
kxk≤1
≤ sup k(uj x)kq = kUkL(X,`q (Y )) . ¤ kxk≤1
Corollary 2.29. Let X and Y be two Banach spaces, let q ∈ [1, ∞] and µ a σ-finite °P 0 0° ° measure. Let (vj ) ⊂ L(X, L1 (µ)) be a sequence such that ( j |vj x|q )1/q °1 ≤ C1 kxk for all x ∈ X (k(supj |vj x|)k1 ≤ C1 kxk if q = 1), and let (uj ) ⊂ L(L1 (µ), Y ) ¡P ¢ q 1/q be a sequence such that ≤ C2 kfk for any f ∈ L1 (µ) (kuj k ≤ C2 j kuj fk for all j if q =P∞). Then there exists another constant C (depending on C1 and C2 ) such that j kuj vj xk ≤ Ckxk for all x ∈ X. Proof. By Theorems 2.23 and 2.28 we have that (vj (·))
(uj )
X −→ `π1,q (L1 ) −→ `1 (Y ) is a composition of bounded operators, where (uj ) means the operator defined by the 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 X
k=1
|λk uk (xk )|p )1/p ≤ πp [uj ]k(λj xj )k`w ≤ πp [uj ]k(xj )k`w k(λj )k`r . p (X) q (X)
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 and many more ones.
(p, q)-SUMMING SEQUENCES
13
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. ∗ w ∗ Proof. The case s ≤ q follows from the norm 1 inclusions `w s (X ) ⊆ `q (X ) and `p (X) ⊆ `r (X). For q < s, r = ∞ or s = ∞ Proposition 2.1 and Remark 2.3 give us the result. So we assume that q < s and r, s < ∞. Then 1 < r/p, s/q < ∞; let a 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 P a scalars αj ≥ 0 such that αj = 1, that
¡X j
|x∗j xj |r
¢1/r
=
¡X j
1/p
|x∗j (αj xj )|p
¢1/p
≤ C sup
kx∗ k≤1
From our assumptions we have that ap ≤ bq, so that we get, by H¨older’s inequality, that πr,s [xj ] ≤ C. ¤
¡P
¢ q/p ∗ q 1/q j αj |x xj |
≤
¡X
X j
j
q
q/p
αj |x∗ xj |q
¢1/q
.
b
αjp ≤ 1, and for any x∗
¡P
j
|x∗ xj |s
¢1/s
. This shows
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) we need the following lemma: w Lemma 3.3. Let X be a Banach space and 1 < r < ∞. Then `w 1 (X) = `r `r 0 (X) if and only if L(c0 , X) = Πr (c0 , X). w Proof. Assume `w 1 (X) = `r `r0 (X) and take u ∈ L(c0 , X). According to the identi0 fication with `w 1 (X) we have that the sequence xn = u(en ) = αn xn where αn ∈ `r 0 w and xn ∈ `r0 (X). This allows to factorize u = w.v where v ∈ L(c0 , `r ) is given by v(en ) = αn en and w ∈ L(`r , X) is given by w(en ) = x0n . 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 ) ∈ `w 1 (X). Consider now the operator u : c0 → X defined by u(en ) = xn . From the assumption u ∈ Πr (c0 , X). Now, since (en ) ∈ `w 1 (c0 ) and u ∈ Πr (c0 , X) then (see [DJT] page 0 53) u(en ) = αn xn where αn ∈ `r and x0n ∈ `w r 0 (X). ¤
Proposition 3.4. Let X be a Banach space such that L(c0 , X ∗ ) = Πs0 (c0 , X ∗ ) for some 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 (xn xn ) ∈ `p , ∗ it suffices to see that for any (αn ) ∈ `q 0 we get (αn xn xn ) ∈ `u where (1/p) + ∗ (1/q0 ) = 1/u. Given now a sequence (αn ) ∈ `q 0 we have that (αn x∗n ) ∈ `w 1 (X ). ∗ Using Lemma 3.3 we have that there exists (βn ) ∈ `s0 and (yn∗ ) ∈ `w s (X ) so ∗ ∗ ∗ ∗ that αn xn = βn yn . Therefore (αn xn ) = (βn yn xn ) ∈ `s0 `r = `u because 1/u = (1/p) + (1/q 0 ) = (1/s0 ) + (1/r). ¤
Combining Theorem 3.2 and Proposition 3.4 we get the following result:
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J.L. ARREGUI, O. BLASCO
Theorem 3.5. Let X be a Banach space such that L(c0 , X ∗ ) = Πs0 (c0 , X ∗ ) for some 1 < s < ∞. Then `πr,s (X) = `πp,q (X) for 1 ≤ p, q, r, s < ∞ such that 1 ≤ p ≤ r and (1/p) − (1/q) = (1/r) − (1/s). Proposition 3.6. w (a) If X has cotype 2 then `w 1 (X) = `2 `2 (X). w (b) If X has cotype q > 2 then `w 1 (X) = `r `r 0 (X) for any r > q. Proof. Use Lemma 3.3 and the fact that L(c0 , Y ) = Π2 (c0 , Y ) for any Y of cotype 2 and L(c0 , Y ) = Πr (c0 , Y ) for any Y of cotype q > 2 and r > q (see Theorem 11.14, [DJT]). ¤ Remark 3.7. Let X be any space with GL-property (see Page 350, [DJT] for w definition and results). Then X has cotype 2 if and only if `w 1 (X) = `2 `2 (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 theorem w L(X, `2 ) = Π1 (X, `2 ). Then `w 1 (X) = `2 `2 (X). Indeed, if u ∈ L(c0 , X) then u∗ ∈ L(X ∗ , `1 ). Now GT property on X gives that ∗ u ∈ Π2 (X ∗ , `1 ) (see [Pi], page 71 ) which implies that u∗ factors through a Hilbert space, 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 and 1/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 0 ) + (1/2). Corollary 3.10. If X ∗ has cotype q0 > 2 then `πp,q (X) = `πr,s (X) for any p ≤ r, s < q00 and (1/p) − (1/q) = (1/r) − (1/s). In particular `πp (X) = `π1 (X) for any 1 ≤ p < q00 and `π1,q (X) = `πr,s (X) for s < q00 and 1/r = (1/q 0 ) + (1/s). Let us now see a connection between the (p, q)-summing sequences and some properties in Banach space theory. Proposition 3.11. Let X be a Banach space and 1 ≤ q ≤ p < ∞ and r ≥ p0 . The following 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 ∗ ) = sup{πs,q [xj ] : k(xj )k`r (X) = 1}.
Proof. Assume first that the identity in X ∗ is (p, q)-summing. If r and s are as stated, (xj ) ∈ B`r (X) and x∗1 , . . . , x∗n ∈ X ∗ we see that ¡ X ∗ s ¢1/s ¡ X ∗ p ¢1/p ∗ . |xj xj | ≤ kxj k ≤ πp,q (idX ∗ ) k(x∗j )k`w q (X ) j
j
Conversely, we assume now that `r (X) ⊆ `πs,q (X) and x∗1 , . . . , x∗n in X ∗ . From Lemma 2.16 we have ¡X j
¢1/p kx∗j kp
= sup{(
n X
k=1
|x∗k xk |s )1/s ;
n X
k=1
kxk kr = 1}.
(p, q)-SUMMING SEQUENCES
15
Then (xj ) is of norm 1 in `r (X), and if C is the norm of the inclusion of `r (X) in ¡ X ∗ s ¢1/s ¡ X ∗ p ¢1/p ∗ ) . This yields `πs,q (X) we have |xj xj | ≤ Ck(x∗j )k`w kxj k ≤ (X q Ck(x∗j )k`w ∗ . ¤ q (X )
j
j
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) `p0 (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) `p0 (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) `p0 (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 and only 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 Corollary 3.14 we have that `π2,1 (`∞ ) = `∞ (`∞ ) ' L(`1 , `∞ ), but L(`1 , `∞ ) does not coincide with Π2 (`1 , `∞ ) because Π2 (`1 , `∞ ) is the same as Π1 (`1 , `∞ ) (for `1 is of cotype 2, and Corollary 11.16 in [DJT] applies) Pnand on the other hand Π1 (`1 , `∞ ) 6= L(`1 , `∞ ): the operator given by x ∈ `1 7→ ( j=1 xj )n ∈ `∞ is not absolutely summing (see [W], exercise III.F.3).
16
J.L. ARREGUI, O. BLASCO
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 let us get that it always contains `w p (X): Theorem 3.18. Let X be a Banach space, p > q and 1/s0 = (1/q) − (1/p). Then `w s (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/p0 > 1/q 0 and 1/p0 = (1/s0 ) + (1/q 0 ), we can write (
X j
|x∗j xj |p )1/p = =
sup k(αj )kp0 =1
sup
|
X j
αj x∗j xj |
sup
k(βj )ks0 =1 k(λj )kq0 =1
=
sup
sup
k(βj )ks0 =1 k(λj )kq0 =1
≤
sup
sup
| |
X j
Z
βj λj x∗j xj |
1
0
h
X
sup k
k(βj )ks0 =1 k(λj )kq0 =1 t∈[0,1]
rj (t)λj x∗j ,
j
X j
X k
rk (t)βk xk idt|
rj (t)λj x∗j kX ∗ k
∗ k(xj )k`w (X) . ≤ k(x∗j )k`w ¤ q (X ) s
X k
rk (t)βk xk kX
Corollary 3.19. Let X be a Banach space. Then `w p (X) ⊂ `πp,1 (X) with inclusion of norm 1. As a consequence, next we study whether the sequence given by the canonical basis (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 (`p0 ), with πp,1 [ej ; `p0 ] = 1. Proof. Note that for p ≥ 2 this is follows from Corollaries 3.14 and 3.15, because (`p0 )∗ = `p has cotype p. For 1 ≤ p < 2 it follows from the fact (ej ) ∈ `w p (`p0 ) 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 ) ⊂ `( p1 − q1 )−1 (`r ). Hence if we assume that 1
1 +
(ej ) ∈ `πp,q (`r ) then q ≤ p. As the norm of the inclusion `nq0 → `nr0 is n( q − r ) , we see that n ¡X ¢1/p 1 1 1 + |hej , ej i|p = n p ≤ πp,q [ej ]n( q − r ) , j=1
which leads to p = ∞ or q < r with 0 ≤ 1/q − 1/p − 1/r.
(p, q)-SUMMING SEQUENCES
17
Conversely, if p = ∞ then (ej ) ∈ `∞ (`r ) = `π∞,q (`r ). And if 1/q − 1/p − 1/r ≥ 0 then, by Proposition 3.20 and Theorem 3.2, we obtain (ej ) ∈ `πr0 ,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 Z 1 X Z 1 n ° n ° ¯X ¯ ° ° ¯ xk rk (t) dt = sup < xk g(t)rk (t)dt > ¯ 0
kgkL∞ ([0,1],X ∗ ) =1 k=1
k=1
≤ π1,2 [xj ]
0
sup
sup
kgkL∞ ([0,1],X ∗ ) =1 k(αk )k2 =1
= π1,2 [xj ]
sup
sup
kgkL∞ ([0,1],X ∗ ) =1 k(αk )k2 =1
= π1,2 [xj ]
sup k(αk )k2 =1
Z
0
≤ π1,2 [xj ].
1
¯ ¯
n X
k=1
n °X ° αk k=1 Z ° 1
°
(
0
¯ αk rk (t)¯dt
Z
n X
k=1
1
0
° g(t)rk (t)dt°
° αk rk (t))g(t)dt°
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 write n n X ¯X ¯ |x∗ xj | ∼ sup ¯ < x∗ , εj xj > ¯ j
j=1
εk =±1
¯ = sup ¯ εk =±1
° ≤°
n X j=1
j
j=1
Z
0
1
<
n X j=1
εj x∗j rj (t),
n X j=1
° ∗ xj rj °Rad(X) k(x∗j )k`w 1 (X )
¯ xj rj (t) > dt¯
This gives the other inclusion. ¤
Remarks 4.2. Let X be a Banach space and 1 < p ≤ 2. It follows from Theorem 4.1 that if X has type p then `p (X) ⊂ `π1 (X). But this is also a consequence of Corollary 3.15 and the fact that if X has type p then X ∗ has cotype p0 . If Rad(X) ⊂ `π1,p (X) then X has cotype p0 . Recall that a Banach space X (see Remark 3.8) is a G.T. space if it verifies Grothendieck’s theorem in the sense that Π1 (X, H) = L(X, H) for every Hilbert space H). G. Pisier gave a characterization of G.T. spaces of coytpe 2 (see theorem 6.6 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:
18
J.L. ARREGUI, O. BLASCO
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 give yet 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 (xj ), (xk ) = (xj xk )k j . It’s clear that VX is bounded. For some Banach spaces we can get that actually the bilinear map is bounded not only with values in `∞ (`∞ ) but in some smaller space. This is the case for `p if 1 < p < ∞. Theorem 4.5. Let X be a Banach space. Then L(`1 , X) = Πp,q (`1 , X) if and only if VX defines a bounded bilinear map VX : `∞ (X ∗ ) × `∞ (X) → `πp,q (`∞ ). Proof. Let (xj ) ⊂ X and (x∗j ) ⊂ X ∗ be such that kxj k, kx∗j k ≤ 1 for all j. Let u: `1 → X the continuous operator such that uej = xj for all j; clearly kuk ≤ 1. By hypothesis we can take C (independently of (xj )) such that πp,q (u) ≤ Ckuk ≤ C. That is, k(uyj )k`p (X) ≤ Ck(yj )k`w q (`1 ) for any finite family (yj ) ⊂ `1 . Therefore if ξj = x∗j ◦ u for each j then ((hξj , ek i)k )j = ((x∗j (uek ))k )j = ((x∗j xk )k )j = VX ((x∗j ), (xj )). Therefore k(hξj , yj i)k`p = k(hx∗j , uyj i)k`p ≤ k(uyj )k`p (X) , and hence k(hξj , yj i)k`p ≤ Ck(yj )k`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 k(uyj )k`p (X) = sup{k(hx∗j , uyj i)k`p ; (x∗j ) ⊂ BX ∗ }
≤ sup{πp,q [VX ((x∗j ), (uej )); `∞ ]; (x∗j ) ⊂ BX ∗ } k(yj )k`w q (`1 ) ≤ Ckuk k(yj )k`w , q (`1 )
and then πp,q (u) ≤ Ckuk.
¤
When applied to the Hilbert case we get the new formulation of Grothendieck’s theorem: 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 following result:
(p, q)-SUMMING SEQUENCES
19
Corollary 4.7. If 1 ≤ p ≤ ∞ and 1/r = 1 − |(1/p) − (1/2)|, then the bilinear form V`p : `∞ (`p0 ) × `∞ (`p ) → `πr,1 (`∞ ) a
1−a is bounded, with kV`p k ≤ 2 2 KG , where a = |1 − 2/p|.
Proof. Equivalently
a
1−a πr,1 (u) ≤ 2 2 KG kuk
for every operator u ∈ L(`1 , `p ), which is an extension, due to Kwapie´ n, of Grothendieck’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 for p = 1 (or p = ∞). By Corollary 3.14 we know that `π2,1 (`∞ ) = `∞ (`∞ ), so the statement is trivial in this case. However, Corollary 3.14 tells us that for r < 2 the inclusion `πr,1 (`∞ ) ⊆ `∞ (`∞ ) is proper.
References [AF] [DF] [D] [DJT] [DR] [G] [J] [K] [LP] [MP] [O] [P1] [P2] [Pi] [T1] [T2] [TJ] [W]
S. Aywa, J.H. Fourie, On summing multipliers and applications (to appear). A. Defant, K. Floret, Tensor Norms and Operator Ideals, North-Holland, 1993. J. Diestel, Sequences and series in Banach spaces, Springer–Verlag, 1984. J. Diestel, H. Jarchow, A. Tonge, Absolutely summing operators, Cambridge University Press, 1995. A. Dvoretzky, C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. USA 36 (1950), 192–197. A. Grothendieck, R´ esum´ e de la th´ eorie m´ etrique des produits tensoriels topologiques, Bol. Soc. Mat. S˜ ao Paulo 8 (1953/1956), 1–79. G.J.O. Jameson, Summing and nuclear norms in Banach Space Theory, Cambridge University Press, 1987. S. Kwapie´ n, Some remarks on (p, q)-summing operators in `p -spaces, Studia Math. 29 (1968), 327–337. J. Lindenstrauss, A. PeÃlczy´ nski, Absolutely summing operators in Lp -spaces and their applications, Studia Math. 29 (1968), 275–326. B. Maurey, G. Pisier, S´ eries de variables al´ eatories vectorielles ind´ ependantes et propi´ et´ es g´ eom´ etriques des espaces de Banach, Studia Math. 58 (1976), 45–90. ¨ W. Orlicz, Uber unbedingte konvergenz in funktionen r¨ aumen (I), Studia. Math. 4 (1933), 33–37. A. Pietsch, Absolut p-summeriende Abbildungen in normierten R¨ aumen, Studia Math. 29 (1968), 275–326. A. Pietsch, Operator Ideals, North-Holland, 1980. G. Pisier, Factorization of Operators and Geometry of Banach spaces, CBM 60, Amer. Math. Soc., Providence R.I., 1985. M. Talagrand, Cotype of operators from C(K), Inventiones Math. 107 (1992), 1–40. M. Talagrand, Cotype and (q, 1)-summing norm in a Banach space, Inventiones Math. 110 (1992), 545–556. N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional Operator Ideals, Longman Scientific & Technical, 1989. P. Wojtaszczyk, Banach spaces for analysts, Cambridge University Press, 1991.
´ ticas, Universidad de Zaragoza, Jos ´ e Luis Arregui. Departamento de Matem a 50009 Zaragoza, Spain. E-mail address:
[email protected] ´ lisis Matem a ´ tico, Universidad de Valencia, Oscar Blasco. Departamento de Ana 46100 Burjassot (Valencia), Spain. E-mail address:
[email protected]
