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How many vectors are needed to compute (p,q)-summing norms?‡

arXiv:math/9302207v1 [math.FA] 4 Feb 1993How many vectors are needed to compute (p,q)-summing norms?‡Martin DefantMarius JungeAbstractWe show that for q < p there exists an α < ∞such thatπpq(T) ≤cpq π[nα]pq (T)for all T of rank n.Such a polynomial number is only possible if q = 2 or q < p. Furthermore, the growthrate is linear if q = 2 or 1q −1p > 12. Unless 1q −1p = 12 this is also a necessary condition.Based on similar ideas we prove that for q > 2 the Rademacher cotype constant of an-dimensional Banach space can be determined with essentially n(1 + ln n)cq manyvectors.IntroductionIn the local theory of Banach spaces the concept of summing operators is of special interest.The presented paper is concerned with the following problem raised up by T. Figiel.For given 1 ≤q ≤p ≤∞what is the best rate kn, such that(*)πpq(T)≤c πknpq (T)holds for all operators of rank n and some constant c?In [DJ] an observation of Figiel and Pelczynski was generalized in showingπpq(T) ≤3 π16npq (T)for all q, p and all operators T of rank n.This exponential growth can not be improvedin general.Figiel and Pelczynski also showed that there is an operator T : ℓ2n∞→ℓn2 (theRademacher projection) such that for all k ∈INπk1(T) ≤es1 + ln knπ1(T) .Recently, Johnson and Schechtman [JOS] discovered that for p = q and q ̸= 2 the rate can notbe polynomial.

More precisely, every sequence satisfying (*) growth faster than any polynomial,i.e.limn→∞kn n−t = ∞‡A preliminary version of this paper occurred as ”Absolutely summing norms with n vectors”[DJ2]1

for all 0 < t < ∞. This phenomenon is related with the fact that Lp spaces don’t have thepolynomial approximation property, which was proved by Bourgain.By far the nicest and most important result is Tomczak-Jaegermann’s inequality, namely(1)π2(T)≤√2 πn2 (T)for all T of rank n.In [DJ] a certain type of quotient formula was used to generalize Tomczak-Jaegermann’s in-equality:πp2(T) ≤√2 πnp2(T)for all T of rank n.K¨onig and Tzafriri showed that for all 2 < p < ∞(2)πp1(T)≤cpπnp1(T)for all T of rank n.In contrast to the case p = q we can show that for q < p the (p,q)-summing norm can bewell-estimated by a polynomial number of vectors.Theorem 1 Let 1 ≤q ≤p, r ≤∞with 1q = 1p + 1r.

Then for all operator T of rank n one hasπpq(T) ≤crπnpq(T)for 1 ≤r < 2π[n(1+ln n)]pq(T)for 2 = rπ[nr/2]pq(T)for 2 < r < ∞.A very helpful tool in the proof of this theorem is again a quotient formula for (p,q)-summingoperators which allows a reduction to the (probably worst) case q = 1.Theorem 2 Let 1 ≤q ≤p ≤∞and 1 ≤r ≤s ≤q′ with 1r = 1p + 1s. Then for all operatorT : X →Y and n ∈INπnpq(T) = sup{πnr1 (TV Dσ) | V : ℓq′ →X, Dσ : ℓ∞→ℓq′, ∥σ∥s , ∥V ∥≤1} .For instance the first case of theorem 1 is a direct consequence of theorem 2 and (2).

In theother cases a crucial observation of Jameson gives the link between limit orders and numberof vectors, see chapter 2. We are in debt to W.B.

Johnson for showing us Jameson’s paper[JAM]. There has always been a quite close connection between the theory of absolutely (p,q)-summing operators and the theory of cotype in Banach spaces.

For instances, as a consequenceof Tomczak-Jaegermann’s inequality and it’s generalization the gaussian cotype constant of ann-dimensional Banach space can be calculated with n vectors. This problem is still open inthe case of Rademacher cotype.

The presented technique allows us to reduce the number ofvectors to the order n(1+ln n)cq which indicates a positive solution for the Rademacher cotype.Unfortunately, the constant cq tends to infinity as q tends to 2.2

Theorem 3 Let 2 < q < ∞and E a n dimensional Banach space. For the Rademacher cotypeconstant one hasCq(IdE) ≤2 Cmq (IdE) ,where m satisfies the following estimate for an absolute constant c0m ≤n (c0 (1 + ln n))11−2q .Finally, we want to indicate that a linear growth is only possible if q = 2 or 1q −1p ≤12.Theorem 4 Let 1 ≤q ≤p < ∞, q ≤r ≤∞with 1q = 1p + 1r.

If q ̸= 2 and 2 < r ≤∞. Thenthere exists α > 1 such that for all sequences kn withπpq(T) ≤c πknpq (T)for all T of rank nthere exists a constant ¯c withnα ≤¯c kn .The constructed examples are very closely related to limit orders of (p,q)-summing operators.In fact, it is well known that the identity on ℓn2 yields an example for the proposition above aslong as q > 2.

In the case q < 2 we intensively use the results of Carl, Maurey and Puhl [CMP]about Benett matrices.PreliminariesIn what follows c0, c1, .. always denote universal constants. We use standard Banach spacenotation.

In particular, the classical spaces ℓq and ℓnq , 1 ≤q ≤∞, n ∈IN, are defined in theusual way. By ι : ℓnq →ℓnp we denote the canonical identity.

Let (ek)k∈IN be the sequence of unitvectors in ℓ∞. For a sequence σ = (σk)k∈IN ∈ℓ∞, τ = (τk)k∈IN ∈ℓ∞we defineDσ(τ) :=Xkσk τk ek .The standard reference on operator ideals is the monograph of Pietsch [PIE].

The ideals oflinear bounded operators, finite rank operators, integral operators are denoted by L, F, I. Herethe integral norm of T ∈I(X, Y ) is defined byι1(T) := sup{|tr(ST)| | S ∈F(Y, X), ∥S∥≤1} .Let 1 ≤q ≤p ≤∞and n ∈IN. For an operator T ∈L(X, Y ) the pq-summing norm of T withrespect to n vectors is defined byπnpq(T) := sup nX1∥Txk∥p!1/p sup∥x∗∥X∗≤1 nX1|⟨xk, x∗⟩|q!1/q≤1.An operator is said to be absolutely pq-summing, short pq-summing, (T ∈Πpq(X, Y )) ifπpq(T) := supn πnpq(T) < ∞.3

Then (Πpq, πpq) is a maximal and injective Banach ideal (in the sense of Pietsch). As usualwe abbreviate (Πq, πq) := (Πqq, πqq).For further information about absolutely pq-summingoperators we refer to the monograph of Tomczak-Jaegermann [TOJ].

In particular, we wouldlike to mention an elementary observation of Kwapien, see [TOJ].Let 1 ≤q ≤p ≤∞,1 ≤¯q ≤¯p ≤∞with q ≤¯q, p ≤¯p and 1q −1p = 1¯q −1¯p. Then one has for all Tπn¯p¯q(T) ≤πnpq(T) .For 2 ≤q < ∞, T ∈L(X, Y ) and n ∈IN the Rademacher (gaussian) Cotype q norm withrespect to n-vectors is defined byCnq (T) ( ˜Cnq (T)) := sup{ nX1∥Txk∥q!

1q ZΩnX1vkxk2dµ12≤1} ,where (vk)n1 is a sequence of independent Bernoulli (gaussian) variables on a probability space(Ω, µ). An operator is said to be of Rademacher (gaussian) cotype q if the corresponding normCq(T) := supn∈INCnq (T) ˜Cq(T) := supn∈IN˜Cnq (T)!is finite.

For further information and the relation between gaussian coype and (q,2)-summingoperators see for example [TOJ].1Positive ResultsWe start with theProof of theorem 2: ” ≤” Let x1, .., xn ∈X withsup∥α∥q′≤1nX1αkxk =sup∥x∗∥X∗ nX1|⟨xk, x∗⟩|q!1/q≤1.Therefore the operator V :=nP1ek ⊗xk : ℓq′ →X is of norm 1. By the equality case of H¨older’sinequality we obtain nX1∥Txk∥p!1/p=sup∥σ∥s≤1 nX1(|σk| ∥Txk∥)r!1/r≤sup∥σ∥s≤1πnr1(TV Dσ) .” ≥” By the maximality of the norms πnr1 there is no restriction to assume Dσ : ℓm∞→ℓmq′and V : ℓmq′ →X with ∥σ∥s , ∥V ∥≤1.

Now let U : ℓn∞→ℓm∞an operator of norm 1. By anobservation of Maurey [MAU] the extreme points of such operators are of the formU =nX1ek ⊗gk ,4

with the gk’s are of norm 1 and have disjoint support. Since we have to estimate the convexexpression nX1∥TV DσU(ek)∥r!1/rwe can assume the that U is of this form.

We define τ and J : ℓnq′ →ℓmq′ byτk := ∥Dσ(gk)∥q′ and J :=nX1ek ⊗Dσ(gk)∥Dσ(gk)∥q′ .The operator J is of norm at most 1. Since Dσ is obviously s1-summing we have∥τ∥s ≤πs1(Dσ) ∥U∥≤∥σ∥s ≤1 .Therefore we obtain by H¨older’s inequality nX1∥TV DσU(ek)∥r!1/r= nX1TV (Dσ(gk)∥Dσ(gk)∥q′ τk)r!1/r= nX1(∥TV J(ek)∥|τk|)r!1/r≤ nX1(∥TV J(ek)∥)p!1/p∥τ∥s≤πnpq(T) ∥V J∥∥τ∥s ≤πnpq(T) .✷Now we will formulate a generalization of Jameson’s lemma [JAM] which he proved in the caseq = 1, p = 2.Lemma 1.1 (Jameson) Let 1 ≤q < p ≤∞and T ∈L(X, Y ) a q-summing operator thenπpq(T) ≤21/pπnpq(T)wheren ≤ 21/p πq(T)πpq(T)!11/q−1/p.Proof: Let us assume πq(T) = 1.

For ε > 0 let x1, .., xN in X withsup∥x∗∥X∗≤1 NX1|⟨xk, x∗⟩|q!1/q≤1 and (1 −ε)πppq(T) ≤NX1∥Txk∥p .Furthermore, we assume ∥Txk∥nonincreasing. For 0 < δ we choose n ≤N minimal such that∥Txk∥≤δ holds for all k > n. Then we have n ≤δ−q because ofnδq ≤πqq(T) ≤1 .5

If δp−q ≤12(1 −ε)πppq(T) it follows that(1 −ε)πppq(T)≤NX1∥Txk∥p≤nX1∥Txk∥p + δp−qNXn+1∥Txk∥q≤nX1∥Txk∥p + δp−qπqq(T)≤nX1∥Txk∥p + 12(1 −ε)πppq(T) .This means(1 −ε)1/pπpq(T) ≤21/pπnpq(T) .Letting ε to zero we find an n ∈IN withn ≤ πppq(T)2! −qp−q= 21/p πq(T)πpq(T)!11/q−1/p.✷Remark 1.2 Exactly the same argument shows that for every operator T ∈L(X, Y ) which isof (Rademacher) Cotype 2 one hasCq(T) ≤21q Cnq (T) ,where n ∈IN satisfiesn ≤ 21/q C2(T)Cq(T)!11/q−1/2.Proof of theorem 3: Let E be a n-dimensional Banach space.

According to Jameson’s lemmawe want to compare the cotype 2 norm with the cotype q norm via the gaussian cotype. It iswell-known [PS, theorem 3.9.] that the Rademacher cotype 2 can be estimated by the gaussiancotype 2 norm in the following way.C2(IdE) ≤c0 ˜C2(IdE)q1 + ln ˜C2(IdE) .Using the inequalities ˜C2(IdE) ≤√2n12 −1q ˜Cq(IdE) and ˜Cq(IdE) ≤n1q , see [TOJ], we obtainC2(IdE)≤c0 ˜C2(IdE)q1 + ln ˜C2(IdE)≤c0 n12−1q ˜Cq(IdE)√2 + ln nCombining this estimate with Jameson’s lemma, more precisely the remark above, we see thatthere is a constant c1 > 0 such thatCq(IdE) ≤21q Cmq (IdE) ,withm≤n (c1√1 + ln n)11−2q✷6

In order to apply Jameson’s lemma an appropriate estimate of the 1-summing norm by ther1-summing norm is needed.Lemma 1.3 Let 1 ≤r ≤∞, n ∈IN and T ∈L(X, Y ) an operator of rank n. Then we haveπ1(T) ≤c0πr1(T)1r −12−1/2 n1/2for 1 ≤r < 2(n(1 + ln n))1/2r = 212 −1r−1/r′n1/r′for 2 < r < ∞.Proof: We may assume T ∈L(ℓ∞, F) with dimF = n. The inequality π2(S : F →ℓ∞) ≤√n ∥S∥, see [TOJ], implies with Tomczak-Jagermann’s inequalityπ1(T) ≤ι1(T) ≤√nπ2(T) ≤√2nπn2 (T) .For 2 < r < ∞we deduce from Maurey’s theorem, see [TOJ]πn2 (T) ≤n1/2−1/rπnr2(T) ≤c012 −1r−1/r′n1/2−1/rπr1(T) .For r = 2 we choose 2 < ¯r < ∞with 12 −1¯r =12+2 ln n. With π¯r1 ≤π21 we obtainπn2 (T) ≤2 e2 c0(1 + ln n)1/2 πn21(T) .In the case 1 < r < 2 we use the other version of Maurey’s theorem, see again [TOJ], to deduceπ2(T) ≤c01r −12−1/2πr1(T) .Combining the last three estimates with the first one gives the assertion.✷Now we can give theProof of theorem 1: First we prove the theorem in the case q = 1, hence p′ = r. FromJameson’s lemma 1.1 and lemma 1.3 we deduce for an operator T of rank nπp1(T) ≤21/pπmp1(T) ,wherem ≤(2c0)r12 −1p−1 nfor 2 < p < ∞n(1 + ln n)for r = 21p −12−r/2 nr/2for 1 ≤p < 2 .An elementary computation shows that for all α, c ≥1 one hasπ[cα]p1 (T) ≤(4c)1/pπ[α]p1 (T) .Hence we getπp1(T) ≤(16c0)r−11r −12−1/r′πnp1(T)for 1 ≤r < 2π[n(1+ln n)]21(T)for r = 212 −1r−(r−1)/2 π[nr/2]p1(T)for 2 < r < ∞.7

For an arbitrary 1 ≤q ≤p ≤∞we define ¯p = r′. Since we have 1¯p = 1p + 1q′ we can deduce fromtheorem 2 and the inequalities aboveπpq(T)=sup{π¯p1 (TV Dσ) | V : ℓq′ →X, Dσ : ℓ∞→ℓq′, ∥σ∥q′ , ∥V ∥≤1}≤cr sup{πm(r,n)¯p1(TV Dσ) | V : ℓq′ →X, Dσ : ℓ∞→ℓq′, ∥σ∥q′ , ∥V ∥≤1}=crπm(r,n)pq(T) ,where m(r, n) = n, m(r, n) = [n(1+ln n)], m(r, n) = [nr/2] for r < 2, r = 2, 2 < r, respectively.

✷Remark 1.4 The polynomial order of the vectors needed to compute the pq-summing normcan be improved for several choices of p and q, because they are close enough to the 2. Let1 ≤q ≤p, r ≤∞with 1q = 1p + 1r.

Then for all operators T of rank n one hasπpq(T) ≤(c0)r/pπ[n1+r(1/q−1/2)]pq(T)for 1 ≤q ≤2 and 2 ≤r ≤q′π[nr(1/2−1/p)]pq(T)for 2 ≤q ≤∞.Proof: First case: We choose 2 ≤s ≤∞such that 1q −1p = 12 −1s. By a result of Carl, [CAR],we have together with Tomczak-Jaegermann’s and Kwapien’s inequalityπq(T)≤n1/q−1/2 π2(T)≤√2 n1/q−1/2 πn2 (T)≤√2 n1/q−1/2 n1/2−1/s πns2(T) ≤√2 n2/q−1/2−1/p πnpq(T) .By Jameson’s lemma 1.1 and the elementry estimate in the prove above we obtainπpq(T) ≤81/p 2rp ( 1p + 12 ) π[n1+r(1/q−1/2)]pq.Second case: From Kwapien’s and Tomczak-Jaegermann’s inequality we deduceπq(T)≤π2(T) ≤√2 πn2 (T)≤√2 n1/2−1/p πnp2(T) ≤√2 n1/2−1/p πpq(T) .Again with Jameson’s lemma 1.1 this implies the assertion.✷At the end of this chapter we would like to note the followingCorollary 1.5 Let 1 ≤r ≤∞, K a compact Hausdorffspace and T ∈L(C(K), Y ) of rank n.Then we haveπp(T) ≤cp(1 + ln n)1/p′πnp (T)for 2 < p < ∞π[np′/2]p(T)for 1 < p < 2 .Proof: Using a result of Carl and Defant, see [CAD], and theorem 2 we deduceπp(T)≤c0 (1 + ln n)1/p′ πp1(T)≤cp (1 + ln n)1/p′ π[nmax(1,p′/2]p1(T) ≤cp (1 + ln n)1/p′ π[nmax(1,p′/2]p(T) .✷8

2ExamplesBy the positve results of the previous section a polynomial groth can only appear if 1q −1p > 12.Therefore we define for 1 ≤q ≤2 the critical value pq by 1pq = 1q −12. In the sequel limit orders ofpq-summing operators are of particular interest.

We intensively use the results of Carl, Maureyand Puhl, see [CMP]. The next lemma is implicitely contained there but we reproduce the easyinterpolation argument.Lemma 2.1 Let 0 < θ < 1, 1 ≤q, r ≤2 and q ≤p with 1r = 1−θq+ θ2 and 1p = 1q −θ2.

Then onehasπpq(ι : ℓnq′ →ℓnr ) ≤n1/p .Proof: Clearly one has, see [PIE],πq(ι : ℓnq′ →ℓnq ) ≤πq(ι : ℓn∞→ℓnq ) ≤n1/q .With Kwapien’s inequality, πqpq ≤π21, we deduce from the Orlicz property of ℓ2πqpq(ι : ℓnq′ →ℓn2) ≤π21(Idℓn2 )ι : ℓnq′ →ℓn2 ≤n1/q−1/2 .By interpolation, namely [ℓq(ℓq), ℓpq(ℓ2)]θ = ℓp(ℓr) see [BEL], this meansπpq(ι : ℓnq′ →ℓnr )≤n1−θqnθ( 1q −12 ) = n1/p .✷Now we can construct the counterexamplesProposition 2.2 Let 0 < θ < 1, 1 ≤q ≤2 ≤s ≤∞and q ≤t with 1s = 1−θq′ + θ2 and 1t = 1q −θ2.For all n ∈IN there exists an operator T ∈L(ℓq′, ℓn2) wich satisfiesπkpq(T)πpq(T) ≤c0√s kns/21/p−1/tfor all q ≤p ≤t and k ∈IN.Proof: Let m = [ns/2] and A : ℓmq′ →ℓn2 be a random matrix with entries ±1, a so called Benettmatrix. Obviously we haveπpq(A) ≥m1/p n1/2 ≥12 n1/2+s/2pfor all q ≤p .We will see that this estimate is sharp for some indices p. By [CMP, Lemma 5] one has∥A : ℓms′ →ℓn2∥≤c0√s max{n1/2, m1/s} ≤c0√sn .Since 1s′ = 1−θq+ θ2 we deduce from Lemma 2.1πtq(A)≤πtq(ι : ℓmq′ →ℓms′ ) ∥A : ℓms′ →ℓn2∥≤c0√s n1/2 m1/t ≤c0√s n1/2+s/2t .Therefore we obtain for arbitrary q ≤p ≤t, k ∈INπkpq(A)≤k1/p−1/t πktq(A)≤c0√s k1/p−1/t n1/2+s/2t≤c0√s kns/21/p−1/tπpq(A) .✷9

Corollary 2.3 For 1 ≤q < 2 and q ≤p < pq let 2 < s ≤∞defined by 1s = 1q′ + (1−2q′ )(1q −1p).For any sequence kn, c > 0 satisfyingπpq(T) ≤Cπknpq (T)for all T of rank nthere is a constant c1 withns/2 ≤c1ec1√1+ln n kn .Proof: We define ϑ := 2(1q −1p) < 1. For ε < 1 −ϑ we set θ := ϑ + ε and choose 2 ≤v ≤s,p ≤t ≤pq with 1v = 1−θq′ + θ2, 1t = 1q −θ2.

Now let us consider the quotient dn := ns/2k−1n . FromProposition 2.2 we deduce with an elementary computation1C≤c0√s knnv/21/p−1/t≤c0√s d−ε/2nn(s−v)ε/4≤c0√s d−ε/2nnε2s2(1/8−1/4q′) .Insetting ε = 1−ϑ2yields a constant c2 such thatln dn ≤c2 +s2(1 −ϑ)(18 −14q′ )ln n .Therefore there exists an n0 ∈IN such that for all n ≥n0 we can choose εn :=ln dns2(1/2−1/q′) ln n <1 −ϑ.

An elementary computation gives1C ≤cq exp −(ln dn)22s2(1/2 −1/q′)(ln n)!.This is only possible if there exists a constant c3 depending on s, q and C such thatns/2kn= dn≤expc3√1 + ln n.✷Remark 2.4 For q = 1 the above results can be slightly improved. For 1 ≤p < 2 ≤s < p′ ≤∞there exists an operator T ∈L(ℓ[ns/2]∞, ℓn2) such that for all k ∈INπkp(T) ≤c0√s kns/21/s−1/p′πp1(T) .In particular, the inequalityπp1(T) ≤C (1 + ln n) πknp (T)for all T of rank ncan only be satisfied, ifnp′/2 ≤¯c e¯c √1+ln n kn .This answers a conjecture of Carl and Defant.

They suggestedπp(T) ≤cp(1 + ln n)1/p′πnp1(T)for all operators T ∈L(C(K), Y ) of rank n, which turns out to be false. Furthermore, we recoverthe exponential order of vectors for π1.

More precisely, for all n, k ∈IN there is an operatorT ∈L(ℓ[n1+ln k]∞, ℓn2) withπk1(T) ≤c0s1 + ln knπ1(T) .10

Proof: Inspecting the proof of proposition 2.2 we take a Benett matrix A : ℓ[ns/2]∞→ℓn2, whosep1-summing norm satisfiesπp1(T) ≥12 n1/2+s/2p .On the other handπkp(A)≤k1/p−1/s′ πs′(A)≤k1/p−1/s′ πs′(ι : ℓ[ns/2]∞→ℓ[ns/2]s′)A : ℓ[ns/2]s′→ℓn2≤c0√s k1/p−1/s′ ns/2s′ n1/2≤2c0√s kns/21/p−1/s′πp1(T) .The logarithmic factor does not affect the calculation in the proof of corollary 2.3. For the lastassertion we note that p′ = ∞and therefore the choice s = 2(1 + ln k) implies the assertion.✷Now we will give theProof of Theorem 4: In the case 1 ≤q < 2 this follows immiadetly from corollary 2.3.

Weonly have to note that for all ε > 0 there is a constant Cε withe¯c√1+ln n ≤Cεnε .Now let 2 < q < ∞. With the help of Benett matrices it was shown in [CMP] that for 2 < q < ∞nq/2p ≤c0√q πpq(idℓn2 ) .Hence we getπkpq(idℓn2 )≤k1/p idℓn2 ≤k1/p≤c0√q knq/21/pπpq(idℓn2 ) .Therefore every sequence kn with πpq(T) ≤C πknpq (T) must satisfynq/2≤(C c0√q)p kn .✷Remark 2.5 For operators defined on n-dimensional Banach spaces the results of [JOS] and[DJ] imply that the pq-summing norm can be calculated with nq/2 (1+ln n) many vectors.

There-fore the order in the proof of the proposition above is quite correct.References[BEL]J. Bergh and J. L¨ofstr¨om: Interpolation spaces; Springer Verlag, Berlin-Heidelberg-NewYork, 1976. [CAD] B. Carl and A. Defant: An inequality between the p- and (p,1)-summing norm of finiterank operators from C(K) spaces.

Israel J. of Math. 74 (1991), 323-325.11

[CAR] B. Carl: Inequalities between absolutely (p,q)-summing norms; Studia Math. 69 (1980),143-148.

[CMP] B. Carl, B. Maurey and J. Puhl: Grenzordnungen von absolut (r,p)-summierendenOperatoren; Math. Nachr.

82 (1978), 205-218.[DJ]M. Defant and M. Junge: On absolutely summing operators with apllication to the(p,q)-summing norm with few vectors; J. of Functional Ana.

103 (1992), 62-73.[DJ2]M. Defant and M. Junge: Absolutely summing norms with n vectors; preprint[JAM]G. J. O. Jameson: The number of elements required to determine (p,1)-summing norms;to appear in Illinois J. of Math.[JOS]W.

B. Johnson and G. Schechtmann: Computing p-summing norms with few vectors;preprint. [MAU] B. Maurey: Type et cotype dans les espaces munis d’un structure localement incondi-tionelle; S´eminaire Maurey-Schwartz 73-74, Ecol- Polyt., Exp.

no. 24-25.[PIE]A.

Pietsch: Operator Ideals; Deutscher Verlag Wiss., Berlin 1978 and North Holland,Amsterdam-New York-Oxford 1980. Cambridge University Press, 1987.[PS]G.

Pisier: Factorization of linear operators and Geometry of Banach spaces; CBMSRegional Conference Series n◦60, AMS 1986.[TOJ]N. Tomczak-Jaegermann: Banach-Mazur distances and finite-dimensional operator ide-als; Longmann, 1988.1991 Mathematics Subject Classification: 47B10, 47A30,46B07.Martin DefantMarius JungeMathematisches Seminar der Universit˝at KielLudewig-Meyn-Str.

424098 KielGermanyEmail: nms06@rz.uni-kiel.d400.de12

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