Computing p–summing norms with few vectors

arXiv:math/9209215v1 [math.FA] 22 Sep 1992Computing p–summing norms with few vectorsby William B. Johnson* and Gideon Schechtman**Abstract: It is shown that the p-summing norm of any operator with n-dimensionaldomain can be well-aproximated using only “few” vectors in the definition of the p-summingnorm. Except for constants independent of n and log n factors, “few” means n if 1 < p < 2and np/2 if 2 < p < ∞.I.

IntroductionA useful result of Tomczak-Jaegermann [T-J, p. 143] states that the 2-summing normof an operator u of rank n can be well-estimated by n vectors; precisely (in the notation of[T-J, p. 140], which we follow throughout), π2(u) ≤√2π(n)2(u). No such result holds forπ1; Figiel and Pelczynski [T-J, p. 184] showed that if kn satisfies π1(u) ≤Cπ(kn)1(u) for alloperators of rank n; n = 1, 2, .

. ., then kn grows exponentially in n. The Tomczak resultreduces immediately to the case of operators whose domains are ℓn2.

Szarek [Sz] provedthat there is a 1-summing analogue to this version of Tomczak’s theorem; namely, thatπ1(u) ≤Cπ(n log n)1(u) whenever u is an operator whose domain has dimension n.In this paper we consider the case of p-summing operators. In section III we extendSzarek’s result to the range 1 < p < 2 (except that the power of log n is “3” instead of “1”).For 2 < p < ∞we show that, up to powers of log n, np2 vectors suffice to well-estimate thep-summing norm of an operator from an n-dimensional space.

The power of n is optimal,but we do not know whether a log n term is needed in either result. These results, as wellas those in section IV, shed some light on problems 24.10, 24.11, and 24.6 in [T-J].In Section IV we show that when 1 < p ̸= 2 < ∞, if kn satisfies πp(u) ≤Cπ(kn)p(u)for all operators of rank n; n = 1, 2, .

. ., then kn grows faster than any power of n. .Just as for Szarek, our main tools are sophisticated versions of embedding n-dimensional subspaces of Lp into ℓkp with k not too large.

While most of this backgroundis at least implicit in [BLM] and [T], we need more precise versions of such results thanare stated in the current literature. The necessary material is developed in Section II.Here we treat only the case of p-summing operators.There is also an extensiveliterature on related problems for (p, q)-summing operators; see [N-T] for the older historyand the recent papers [DJ1], [DJ2], [J].

In particular, Defant and Junge [DF2] show howresults for p-summing operators can be formally transformed into results for (p, q)-summingoperators.II. Preparations for the main resultBefore stating the basic entropy lemma for the main result, we set some notation.

* Supported in part by NSF #DMS90-03550 and the U.S.-Israel Binational ScienceFoundation** Supported in part by the U.S.-Israel Binational Science Foundation1

A density on a probability space (Ω, µ) is a strictly positive measurable function on Ωwhose integral is one. Given a set A, a metric δ on A, and a positive number t, E(A, δ, t)is the minimal number of open balls of radius t in the metric δ needed to cover A.We also use notation (see, for example, [T-J, p. 80]) commonly used in Banach spacetheory for measuring the expected value of the norm of Gaussian processes: If u : H →Zis a linear operator from a finite dimensional Hilbert space H into a normed space Z,ℓ(u)2 is defined to be IE∥Pmi=1 giu(ei)∥2, where e1, .

. ., em is any orthonormal basis forH and g1, .

. .

, gm are independent standard Gaussian variables. ℓis an ideal norm in thesense that if H′ is another finite dimensional Hilbert space, Z′ is another normed space,T : H′ →Z and S : Z →Z′ are linear operators, then ℓ(SuT) ≤∥S∥ℓ(u) ∥T∥.

Supposenow that ν is a probability measure on a finite set A and W is an n-dimensional subspaceof the set of scalar valued functions on A. Let Wp denote W under the Lp(ν)–norm andlet iWp,r be the formal identity mapping from Wp onto Wr (when r = 2 we abuse notationby also regarding the operator into L2(ν)).

Sudakov’s lemma [Su], stated as Proposition4.1 in [BLM], gives the entropy estimatelog E(B(Wp), ∥·∥L2(ν) , t) ≤C ℓ(iWp,2∗)t!2.The Pajor-Tomczak lemma [PT-J], stated as Proposition 4.2 in [BLM], gives the entropyestimatelog E(B(W2), ∥·∥Lp(ν) , t) ≤C ℓ(iW2,p)t!2.The ideal properties of ℓimply that if ν (respectively, µ) is a probability measure onthe finite set A (respectively, B), and Y (respectively W) is a space of scalar functions onA (respectively B), and v : Y →W is a linear operator which is an isometry from Yp ontoWp and has norm at most C as an operator from Y2 into W2, then ℓ(iWp,2∗) ≤Cℓ(iYp,2∗).The entropy lemma we use is a variation on Propositions 4.6 and 7.2 in [BLM]. Theresult we need later is different from that in [BLM] since we cannot replace the subspaceX of Lp(µ) by an (isomorphic or even isometric) copy of X in Lp(ν) but rather mustmove all of Lp(µ) isometrically onto Lp(ν).

Moreover, formally speaking, Proposition 7.2is only partly proved in [BLM] and contains some unclear statements (e.g., the claim inthe sentence immediately following (7.11) seems formally wrong and should be adjustedslightly). The accumulation of the adjustments needed to obtain Proposition 2.1 belowfrom the arguments in [BLM] required some effort on our part, so we judged it worthwhileto outline proofs of the entropy estimates we need.Proposition 2.1.

Let X be an n-dimensional subspace of Lp(N, µ) for some probabilitymeasure µ on N = {1, . .

., N}. Then there is a density α on (N, µ) satisfying the following:Put ˜X = {x/α1p : x ∈X } and let B( ˜Xr) be the closed unit ball of ˜X in Lr(N, α dµ).Then for some constant C,(i) log E(B( ˜Xp), ∥·∥∞, t) ≤C(p −1)p−22 n(log n)1−p2 (log N)p2 t−p, for 1 < p < 2.2

(ii) ∥f∥L∞≤(2n)1/p ∥f∥Lp(αdµ), for 1 < p < 2. (iii) log E(B( ˜Xp), ∥·∥∞, t) ≤C(log N)nt−2, for 2 ≤p < ∞.

(iv) ∥f∥L∞≤(2n)1/2 ∥f∥Lp(αdµ), for 2 < p < ∞.Proof. It is easy to reduce to the case of measures which are strictly positive (i.e., forwhich all points of N have positive µ measure).

The conclusion is invariant under changeof density of the original measure, so we can assume, without loss of generality, that µ isthe uniform measure on N (this simplifies slightly the notation below).Lewis [L] showed that there is a density β on N and an orthonormal (in L2(βdµ))basis f1, . .

., fn for Y = {x/α1/p1: x ∈X } so that Pni=1 f 2i = n.The density α is β+12 . Then ˜X consists of all vectors of the form v(y) with y in Y ,where v(y) =βα1/py.

The linear operator v defines an isometry from Lp(βdµ) ontoLp(αdµ) and has norm at most 21/p as an operator from L2(βdµ) into L2(αdµ).Asmentioned before the statement of Proposition 2.1, Sudakov’s lemma gives the entropyestimatelog E(B( ˜Xp), ∥·∥L2(αµ) , t) ≤Ct−2pnK(X)2.Using the fact that ( βα)1/2f1, . .

., ( βα)1/2fn is orthonormal in L2(αdµ) and the Maurey-Khintchine inequality, we get for all 1 ≤q < ∞:ℓ2(i˜X2,q) = IEnXi=1giβα1/2fi2Lq(αdµ)≤Cq(nXi=1βαf 2i )1/22Lq(αdµ)≤2Cqn.As mentioned before the statement of Proposition 2.1, the Pajor-Tomczak lemma givesthe entropy estimatelog E(B( ˜X2), ∥·∥Lq(αdµ) , t) ≤Ct−2qn. (+)Pick q = log 2N; then, since α > 1/2, ∥·∥∞≤e ∥·∥Lq(αdµ).

Since B( ˜Xp) ⊂B( ˜X2)for p ≥2, this gives (iii).The Lewis change of density forces, for f in Y , ∥f∥L∞≤n1/2 ∥f∥Lp(βdµ) (see e.g. Lemma 7.1 in [BLM]).

Since βα ≤2, we have for f in ˜X that∥f∥L∞≤(2n)1/2 ∥f∥Lp(αdµ). This gives (iv).To deal with the case 1 < p < 2, we refer to the proof of Proposition 7.2 (ii) in [BLM].By applying H¨older’s inequality and a clever duality argument, one obtains formally from(+), for 1 ≤t ≤2n, thatlog E(B( ˜Xp), ∥·∥L2(αdµ) , t) ≤C(p −1)−1(Ct )2p/(2−p)n log n.(++)3

Using, for 1 < s < t, the obvious inequalitylog E(B( ˜Xp), ∥·∥L∞, t) ≤log E(B( ˜Xp), ∥·∥L2(αdµ) , s) + log E(B( ˜X2), ∥·∥L∞, t/s),(++), and (iii) in the statement of the proposition, we obtain (i) for 1 ≤t ≤2n by minimiz-ing over s. Now the Lewis change of density forces, for f in Y , ∥f∥L∞≤n1/p ∥f∥Lp(βdµ).Since βα ≤2, we have for f in ˜X that ∥f∥L∞≤(2n)1/p ∥f∥Lp(αdµ). This gives (ii) as wellas (i) when t > 2n.The following proposition and its proof is an adjustment of results from Talagrand’spaper [T].

(The idea of “splitting the large atoms”, used also in [T], is due to the authors. )Proposition 2.2.

Let X be an n-dimensional subspace of Lp(N, τ) for some probabilitymeasure τ on N = {1, . .

., N}. Then there are N ≤M ≤32N and a probability measureν on M = {1, .

. ., M} so that:(i) There is a partition {σ1, .

. ., σn} of M with Pi∈σj ν{i} = τ{j} for j = 1, .

. ., n.(ii) E sup {PMi=1 giν{i}|yi|p : y ∈Y, ||y|| ≤1} ≤C(p −1)p−24 ( nN )12 (log n)6−p4 (log N)p4 ,for 1 < p < 2, where y1, .

. ., yn are the coordinates of the vector y and Y is theimage of X under the natural isometry Jp from Lp(N, µ) into Lp(M, ν), defined by(Jpx)i = xj if i ∈σj.

(iii) E sup {PMi=1 giν{i}|yi|p : y ∈Y, ||y|| ≤1} ≤Cpnp/4N −1/2 log n(log N)1/2,for 2 < p < ∞. Cp can be taken to be Cp22p/2.Proof.

It is easy to reduce to the case of measures which are strictly positive. Next, notethat if the proposition is true for one strictly positive probability measure on N, then it istrue for all of them.

This is because the left hand side of (ii) is invariant under a change ofdensity φ if we replace the subspace Y of Lp(ν) with its image under the natural isometryfrom Lp(ν) onto Lp(φ dν), defined by Tf = f/φ1/p. Thus we can assume that τ is themeasure αdµ given by the conclusion of Proposition 2.1.Splitting the atoms of τ of mass larger than 4/N into pieces each of size between2/N and 4/N produces M, the measure ν, and, a fortiori, the space Y along with theisometry J = Jp; (i) is thus satisfied.

Since J also defines an isometry Jr from Lr(N, τ)into Lr(M, ν) for all 0 < r ≤∞, the conclusion of Proposition 2.1 remains true for themeasure space (M, ν) (where of course ˜X is replaced by Y ).Let δ be the natural distance associated with the Gaussian process appearing in (ii),defined for y, z in Y byδ(y, z) = MXi=1[τ{i}(|yi|p −|zi|p)]2!1/2.4

Let 1 < p < 2, fix y, z in B(Yp), and set ui = |yi| ∨|zi|. Thenδ(y, z)2 ≤MXi=1ν{i}2p2u2p−2i|yi −zi|2≤∥y −z∥p∞4p2N −1MXi=1ν{i}u2p−2i|yi −zi|2−p≤4p2N −1 ∥y −z∥p∞(MXi=1ν{i}upi )2(p−1)/p(MXi=1ν{i}|yi −zi|p)(2−p)/p≤26N −1 ∥y −z∥p∞.Thus by Proposition 2.1 (ii) we get that the δ-diameter of B(Yp) is less than 24n1/2N −1/2and from Proposition 2.1 (i) that:log E(B(Yp), δ, t) ≤log E(B(Yp), ∥·∥p/2∞, 2−3N 1/2t)≤log E(B(Yp), ∥·∥∞, 2−6/pN 1/pt2/p)≤C(p −1)p−22 n(log n)1−p2 (log N)p2 N −1t−2.The last inequality in this last display requires t ≥23N −1/2; for 0 < t < 23N −1/2 usevolume considerations in the n-dimensional space B(Y∞) to getlog E(B(Yp), δ, t) ≤log E(B(Yp), ∥·∥∞, 1) + log E(B(Y∞), ∥·∥∞, 2−6/pN 1/pt2/p)≤C(p −1)p−22 n(log n)1−p2 (log N)p2 + Cn log(CN −1t−2).By Dudley’s theorem (see, e.g., [MP, p. 25]),E sup {MXi=1giν{i}|yi|p : y ∈Y,||y|| ≤1}≤24n1/2N −1/2 + C(p −1)p−24 n1/2(log n)2−p4 (log N)p4 N −1/2+ Cn1/2Z 23N−1/20log1/2(CN −1t−2) dt+ C(p −1)p−24 n1/2(log n)2−p4 (log N)p4 N −1/2Z 24n1/2N−1/223N−1/2t−1 dt≤C(p −1)p−24 n1/2(log n)6−p4 (log N)p4 N −1/2.This proves (ii).5

To prove (iii), assume now 2 < p < ∞. Fix y, z in B(Yp), and set ui = |yi|∨|zi|.

Thenδ(y, z)2 ≤MXi=1ν{i}2p2u2p−2i|yi −zi|2≤∥y −z∥2∞4p2N −1MXi=1ν{i}u2p−2i≤4p2N −1 ∥y −z∥2∞∥u∥p−2∞MXi=1ν{i}upi≤4p22p/2 n(p−2)/2N∥y −z∥2∞,where the last inequality follows from Proposition 2.1 (iv). Thus the δ diameter of B(Yp)is less than 4p2p/4np/4N −1/2 and Proposition 2.1 (iii) implies:log E(B(Yp), δ, t) ≤log E(B(Yp), ∥·∥∞, p−12−(p+4)/4n−(p−2)/4N 1/2t)≤Cp22p/2np/2N −1(log N)t−2as long as t ≥p2(p+4)/4n(p−2)/4N −1/2.

For smaller t we get by the usual volume consider-ations,log E(B(Yp), δ, t)≤log E(B(Yp), ∥·∥∞, 1) + log E(B(Y∞), ∥·∥∞, p−12−(p+4)/4n−(p−2)/4N 1/2t)≤Cp22p/2np/2N −1(log N) + Cn log(Cp2p/4n(p−2)/4N −1/2t−1).By Dudley’s theorem,E sup {MXi=1giν{i}|yi|p : y ∈Y,||y|| ≤1}≤Cp2p/4np/4N −1/2 + Cp22p/2n(p−1)/2N −1(log N)1/2+ Cn1/2Z p2(p+4)/4n(p−2)/4N−1/20log1/2(Cp2p/4n(p−2)/4N −1/2t−1) dt+ Cp2p/4np/4N −1/2(log N)1/2Z 4p2p/4np/4N−1/2p2(p+4)/4n(p−2)/4N−1/2 t−1 dt.For a fixed p the last term is dominating and one getsE sup {MXi=1giν{i}|yi|p : y ∈Y,||y|| ≤1} ≤Cpnp/4N −1/2 log n(log N)1/2where Cp can be taken to be Cp22p/2.6

Corollary 2.3. Let X be an n-dimensional subspace of Lp(N, τ) for some probabilitymeasure τ on N = {1, .

. ., N} and let Lp(M, ν), J, and Y be given from Proposition 2.2.Then there is a partition M1 ∪M2 of M into two sets of cardinality at most 78N such thatfor each y in Y and j = 1, 2:(i)1MjypLp(M,ν)≤1/2 + C(p −1)p−24 ( nN )12 (log n)6−p4 (log N)p4∥y∥pLp(M,ν) ,when 1 < p < 2; while(ii)1MjypLp(M,ν) ≤1/2 + Cpnp2N12log n(log N)12∥y∥pLp(M,ν) , for 2 < p < ∞.Moreover, (i) and (ii) hold for most such partitions of M.Proof.

First, notice that (ii) in Proposition 2.2 still holds if we substitute independentRademacher functions for the Gaussian variables gi (and replace C by, e.g., p π2 C). Thisfollows from a standard contraction principle.

Consequently, if we again enlarge C,sup {MXi=1ǫiν{i}|yi|p : y ∈Y, ||y|| ≤1} ≤C(p −1)p−24 ( nN )12 (log n)6−p4 (log N)p4holds for most choices of signs ǫi = ±1. Since also for most choices of signs the differencebetween the number of plus signs and minus signs is less than M/8, (i) follows.

(ii) followssimilarly.III. Computing p-summing normsGiven a linear operator u : X →Y of finite rank, 1 ≤q ≤∞, and positive integers n,k, defineν(n,k)q(u) = inf kXi=1ν(n)q(ui) : u =kXi=1ui,whereν(n)q(v) = inf∥A∥∥w∥∥B∥; A : X →ℓn∞; w : ℓn∞→ℓnq diagonal, B : ℓnq →Y, v = BwA.In Tomczak’s terminology [T-J, p. 181], ν(n,1)q= ν(n)q, while limk→∞ν(n,k)q= ˆν(n)qgives the cogradation which is dual to the natural gradation π(n)pof the p-summing norm[N-T, Theorem 24.2] (or something like that!

).Proposition 3.1. Let n ≤N be positive integers; u : X →Y a linear operator with Xfinite dimensional and dim(Y ) ≤n.

Then, putting q = p/(p −1),(i) For 1 < p < 2 ,ν( 78 N,2)q(u) ≤ 1 + C(p −1)p−24 nN 12(log n)6−p4 (log N)p4!ν(N,1)q(u).7

(ii) For 2 < p < ∞,ν( 78 N,2)q(u) ≤1 + Cp np2N!12log n(log N)12ν(N,1)q(u).Proof.For some probability measure τ on N, we can take A : Y ∗→Lp(N, τ),B : L1(N, τ) →X∗, so that ∥A∥∥B∥= ν(N)q(u) and u∗= Bip,1A. Apply Proposition2.2 to the subspace AY of Lp(N, τ) to get the measure space Lp(M, ν) and the naturalisometric embedding Jp : Lp(N, τ) →Lp(M, ν).By Corollary 2.3, we get a partitionM1 ∪M2 of M into two sets of cardinality at most 78N such that for each y in Y , j = 1, 2,and in the case 1 < p < 2:1MjJpAypLp(M,ν) ≤1/2 + C(p −1)p−24 ( nN )12 (log n)6−p4 (log N)p4∥Ay∥pLp(N,τ) .Denote for j = 1, 2 the injection from Lp(Mj, ν|Mj) to L1(Mj, ν|Mj) by ijp,1 and let P bethe conditional expectation projection from L1(M, ν) onto J1[L1(N, τ)] followed by J−11 .Thus u∗= BPi1p,11M1J1A + BPi2p,11M2J1A andν( 78 N,2)q(u) ≤2Xj=1ν( 78 N)q([BPijp,11MjJ1A]∗)≤2Xj=11MjJ1A ijp,1BP≤12 + C(p −1)p−24 ( nN )12 (log n)6−p4 (log N)p4 1p∥A∥∥B∥2Xj=1ν(Mj)1q≤∥A∥∥B∥1 + 2C(p −1)p−24 ( nN )12 (log n)6−p4 (log N)p4 1p≤∥A∥∥B∥1 + 2C(p −1)p−24 ( nN )12 (log n)6−p4 (log N)p4.This completes the proof when 1 < p < 2; the other case is similar.Theorem 3.2.

Suppose that dim(X) ≤n, u : X →Y is a linear operator and ǫ > 0.Then,πp(u) ≤(1 + ǫ)π(m)p(u),as long as8

(i) 1 < p < 2 andm ≥K(p −1)p−22 ǫ−2n(log n)6−p2log(p −1)p−22 ǫ−2n p2forsome absolute constant K,or(ii) 2 < p < ∞andm ≥Kpǫ−2np2 (log n)2 log(ǫ−2np2 ).Proof. Without loss of generality, we can assume that dim(Y ) ≤n.

By duality [T-J,Theorem 24.2], it is enough to prove thatˆν(m)q(v) ≤(1 + ǫ)νNq (v)for all v : Y →X and all positive integers N ≥n. Iterating Proposition 3.1, we get for allk (with ( 78)kN ≥n) and for 1 < p < 2 thatν([ 78 ]kN,2k)q(u) ≤kYj=11 + C(p −1)p−24n( 78)j−1N 12(log n)6−p4 (logh( 78)j−1 Ni)p4ν(N)q(u).The product on the right hand side of the above inequality is smaller than 1 + ǫ as long as(p −1)p−24n( 78)kN 12(log n)6−p4 (logh( 78)k Ni)p4 ≤δǫ,(where δ = δ(C) is an appropriate positive constant).

Put m = ( 78)k N; then, as long asm ≥δ′(p −1)p−22 ǫ−2n(log n)6−p2log(p −1)p−22 ǫ−2n p2,ˆν(m)q(u) ≤ν(m,2k)q(u) ≤ν(N)q(u).This completes the proof when 1 < p < 2; the case 2 < p < ∞is similar.Remark. As we have presented it, the proof of Theorem 3.2 does not recapture theresult of Szarek mentioned in the introduction.

Actually, our approach does work whenp = 1 and the technical difficulties are easier in this case because the entropy considerationsof Section II are not needed.IV. Examples and concluding remarksFor p > 2, π(k)p (ℓn2) ≤k1p , while πp(ℓn2) ≥pn/p [N-T, Theorem 10.2].

Consequently,Theorem 3.2 is precise except for the log n terms.It is natural to ask what value of k is needed for π(k)p (u) to well-estimate πp(u) for ageneral operator u of rank n. When p = 1, Figiel-Pe lczynski [T-J, p.184] checked that kmust be exponential in n. The authors and J. Bourgain checked that a result of Bourgain’s[B] yields that for 1 < p ̸= 2 < ∞, k grows faster than any power of n.9

Proposition 4.1. Let 1 < p ̸= 2 < ∞and C < ∞.

Suppose that for each s = 1, 2, . .

.,ks satisfiesπp(u) < Cπ(ks)p(u)for all operators u of rank at most s. Then for all K < ∞, kss−K →∞as N →∞.Proof: Fix 1 < p ̸= 2 < ∞, K, C, and let δ > 0 with δK < 1. Given N = 2n for some n,we identify LNp with Lp(G), where G is the group {−1, 1}n with normalized Haar measure,dg.Let E = span {wS : |S| ≥n −m} wheremn log nm ∼δ; so dim E ∼( nm)m < N δ.Here we follow Bourgain’s notation [B]; for S ⊂{1, .

. ., n}, wS = Qi∈S ri, with ri the i-thcoordinate projection (Rademacher) on G. Let jE∞,p be the formal identity from E∞toLNp .

We shall use Bourgain’s result [B] that if T is an operator on LNp which is the identityon E and ∥T∥< C, then trace T ∼N (meaning |trace (I −T)| = o(N)), to prove that ifˆN (k)p(jE∞,p) < Cνp(jE∞,p) (= C),then for large N,k > N δK ≥(dim E)K. This gives the dual form of the conclusion ofProposition 4.1.For notational convenience, set α = jE∞,p and suppose that for certain k we haveα = Pi αi with Pi νkp(αi) < C. This means that there are factorizationsEτi−→ℓk∞∆i−→ℓkpγi−→LNpof αi with ∥τi∥= ∥∆i∥= 1, ∆i diagonal, and Piγi < C. This diagram also gives thatPi ν1(αi) < Ck1−1p . Extend τi to a map ˜τi : LN∞→ℓk∞with ∥˜τi∥= 1, set ˜αi = γi∆i˜τi,and let ˜α = Pi ˜αi.

Thenν1(˜α) ≤Xiν1(˜αi) < Ck1−1p and πp(˜α) = νp(˜α) < C.Now replace ˜α by its average β over the group G, defined byβ =ZGTg ˜αTgd g(g−1 = g in G).The operator β is translation invariant (a multiplier) and satisfies the same conditionsas ˜α; namely,β|E = α,ν1(β) < Ck1−1p ,πp(β) < C.Since β is translation invariant, Haar measure on G is a suitable Pietsch measure, whichmeans that ∥βip,∞∥< C. Thus trace (βip,∞) ∼N by Bourgain’s result [B]. However,|trace (βip,∞)| ≤ν1(βip,∞) ∥ip,∞∥< Ck1−1p N1p ,which is o(N) if k ≤N δK.10

References[B] J. Bourgain, A remark on the behaviour of Lp-multipliers and the range of operatorsacting on Lp-spaces, Israel J. Math.

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[T-J] N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operatorideals, Pitman Monographs and Surveys in Pure and Applied Mathematics 38, Long-man 1989.Department of Mathematics,Texas A&M University,College Station, TX 77843, USAe-mail: wbj7835@venus.tamu.eduDepartment of Theoretical Mathematics,The Weizmann Institute of Science,Rehovot, Israele-mail: mtschech@weizmann.weizmann.ac.il11

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