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Comparing gaussian and Rademacher cotype for operators on the

arXiv:math/9302206v1 [math.FA] 4 Feb 1993Comparing gaussian and Rademacher cotype for operators on thespace of continous functionsMarius JungeAbstractWe will prove an abstract comparision principle which translates gaussian cotype in Rademachercotype conditions and vice versa. More precisely, let 2

T is of gaussian cotype q if and only if nX1 ∥T xk∥Fplog(k + 1)!q!1/q≤cnX1εkxkL2(C(K)),for all sequences with (∥T xk∥)n1 decreasing.2. T is of Rademacher cotype q if and only if nX1∥T xk∥Fplog(k + 1)q!1/q≤cnX1gkxkL2(C(K)),for all sequences with (∥T xk∥)n1 decreasing.Our methods allows a restriction to a fixed number of vectors and complements the correspondingresults of Talagrand.IntroductionOne problem in the local theory of Banach spaces consits in the description of Rademacher cotype andgaussian cotype for operators on C(K)-spaces.

A quite satisfactory answer for the Rademacher cotypewas given by Maurey. He connected cotype conditions with summing conditions (see [MAU]):Theorem 0.1 [Maurey] Let 2

Then the following are eqiuvalent:1. T is absolutely (q, 2)-summing, i.e.

for all (xk)k∈IN ⊂C(K) one has Xk∥T xk∥q!1/q≤c0 supt∈K Xk|xk(t)|2!1/2.2. T has Rademacher cotype q, i.e.

for all (xk)k∈IN ⊂C(K) one has Xk∥T xk∥q!1/q≤c0XkεkxkL2(C(K)).3. T is absolutely (q, 1)-summing, i.e.

for all (xk)k∈IN ⊂C(K) one has Xk∥T xk∥q!1/q≤c0 supt∈KXk|xk(t)|.1

Later on, Pisier gave another approach to this type of results via factorization theorems. This way waspursued by Montgomery-Smith, [MSM], and Talagrand, [TAL], to give a characterization of gaussiancotype q.Theorem 0.2 [Talagrand] Let 2

Then the following are equivalent.1. T has gaussian cotype q, i.e.

for all (xk)k∈IN ⊂C(K) one has Xk∥T xk∥q!1/q≤c1XkgkxkL2(C(K)).2. T satisfies the following summing condition, i.e.for all (xk)k∈IN ⊂C(K) such that (∥T xk∥)n1 isdecreasing one has Xk ∥T xk∥plog(k + 1)!q!1/q≤c2 supt∈KXk|xk(t)|.3.

T factors through an Orlicz space Ltq(log t)q/2, 1(µ) for some probability measure µ on K.The main new ingredient of this theorem is a factorization theorem for gaussian processes derived fromthe existence of majorizing measures, see [TA1].We will give a more abstract approach to gaussian cotype conditions which can be considered as acomplement to Talagrand’s results. Independently of him we discovered the connection between gaussiancotype and summing properties with the modified ℓq space in condition 2 of theorem 2.

In order to beprecise, let us give the following definition. For a maximal, symmetric sequence space X and T : E →Fwe defineπnX,q(T ):=supnX1∥T xk∥F ekXsupa∈BE∗ nX1|< xk, a >|q!1/q≤1,rcnX(T ):=supnX1∥T xk∥F ekXnX1εkxkL2(E)≤1,gcnX(T ):=supnX1∥T xk∥F ekXnX1gkxkL2(E)≤1.An operator is said to be (absolutely) (X, q) −summing, of Rademacher cotype X, of gaussian cotypeX if πX,q := supn∈IN πnX,q, rcX := supn∈IN rcnX, gcX := supn∈IN gcnX is finite, respectively.

In contrastto Talagrand we follow Maurey’s approach and proveTheorem 0.3 Let 2 < q < ∞, X a q-convex, maximal, symmetric sequence space and T : C(K) →F.Then the following are equivalent:1. T is (X, 2)-summing.2.

T is of Rademacher cotype X.3. T is (X, 1)-summing.2

Furthermore, there exists a constant c only depending on q and X such thatπnX,2(T ) ≤c πnX,1(T ).The main idea for the proof of the theorem above is a reduction to Maurey’s result via quotient formulas.These formulas are contained in chapter 2 and have already be seen to be helpful in the theorey ofsumming operators. Their proof goes back to a joint work of Martin Defant and the author, see [DJ].The comparision principle between gaussian and Rademacher cotype for operators on C(K)-spaces isformulated inTheorem 0.4 Let 2

If Y denotes the spaceof diagonal operators between ℓ∞,∞,1/2 and X one has for all operators T : C(K) →F and n ∈IN1crcnY (T ) ≤gcnX(T ) ≤c rcnY (T ),where c is a constant depending on q and X only.The philosophy is quite simple.The difference between gaussian and Rademacher cotype has to becorrected in the summing property with the factorplog(k + 1). This becomes clear if we apply this firstfor the space X = ℓq.

Then we see that an opertor T : C(K) →F is of gaussian cotype q if and only if Xk ∥T xk∥Fplog(k + 1)!q!1/q≤cnX1εkxkL2(C(K),for all sequences with (∥T xk∥)n1 decreasing. Applying the result for Y = ℓq we see that T is of Rademachercotype q if and only if Xk∥T xk∥Fplog(k + 1)q!1/q≤cnX1gkxkL2(C(K),for all sequences with (∥T xk∥)n1 decreasing.

Let us also note that our approach enables us to fix thenumber of vectors in consideration. For example, this restriction to n vectors can be used to prove thatfor an opertor of rank n the gaussian cotype q-norm is attained on n disjoint functions in C(K).

Anotherapplication is given in the study of weak cotype operators.PreliminariesWe use standard Banach space notations. In particular, c0, c1, .. will denote different absolute constantsand they can vary whithin the text.

The symbols X, Y, Z are reserved for sequence spaces. Standardreferences on sequence spaces and Banach lattices are the monograph of Lindenstrauss and Tzafriri,[LTI, LTII].

The symbols E, F will always denote Banach sapces with unit balls BE, BF and dualsE∗, F ∗. Basic information on operator ideals and s-numbers can be found in the monograph of Pietsch,[PIE].

The ideal of linear operators is denoted by L.The classical sequence spaces co, ℓp and ℓnp, 1 ≤p ≤∞, n ∈IN are defined in the usual way. Fromthe context it will be clear whether we mean the space co or the absolut constant c0.

A generalizationof the classical ℓp spaces is the class of Lorentz-Marcinkiewicz spaces. For a given continous functionf : IN →IR>0 with f(1) = 1 the following two indices are defined3

αf := inf{ α | ∃M < ∞∀t, s ≥1 : f(ts) ≤Mtαf(s) },βf := sup{ β | ∃c > 0 ∀t, s ≥1 : f(ts) ≥ctβf(s) }.These two indices play an important rˆole in the study of the space ℓf,q, 1 ≤q ≤∞consisting of allsequences σ ∈ℓ∞such that∥σ∥f,q := Xn(f(n) σ∗n)q n−1!1/q< ∞.For q=∞the needed modification is given by∥σ∥f,∞:= supn∈INf(n) σ∗n < ∞.Here and in the following σ∗= (σ∗n)n∈IN denotes the non-increasing rearrangement of σ.In the introduction the notions of (X, q)-summing, Rademacher cotype X and gaussian cotype X arealready defined.If X = ℓp we will shortly speak of (p, q)-summing opertors or norms, Rademachercotype p, etc. (possibly restricted to n vectors).

In this context it is convenient to use an abbreviationfor the right hand side of the definition of summing operators. For a sequence (xk)n1 in a Banach spaceE we writeωq(xk)n1 :=supa∈BE∗ nX1|< xk, a >|q!1/q.Let us note that this expression coincides with the operator norm ofu :=nX1ek ⊗xk ∈L(ℓnq′, E),where q′ is the conjugate index of q satisfying 1q +1q′ = 1.In the following (εn)n∈IN, (gn)n ∈IN will denote a sequence of independent normalized Bernoulli (Rade−macher) variables or gaussian variables respectively.

They are defined on a probability space (Ω, µ).Here Bernoulli variable meansµ(εn = +1) = µ(εn = −1) = 12 .A very deep result in the theory of gaussian processes is Talagrand’s factorization theorem, see [TA1]. (∗)There is an absolut constant c1 such that for all sequence (xk)n1 ∈C(K) withnX1gkxkL2(X)≤1.there are operators u : ℓn2 →co, R : co →C(K) with ∥u∥∥R∥≤c1 such thatRDσu(ek) = xk,where Dσ is the diagonal operator with4

σk =1plog(k + 1).Finally some s-numbers are needed. For an operator T ∈L(E, F) and n ∈IN the n-th approximationnumber is defined byan(T ) := inf{ ∥T −S∥| rank(S) < n },whereas the n-th Weyl number is given byxn(T ) := sup{ an(T u) | u ∈L(ℓ2, E) with ∥u∥≤1 }.1Maximal symmetric sequence spacesIn the following we will denote the set of all finite sequences by φ and the sequence of unit vectors in ℓ∞by (ek)k. For every sequence σ=(σk)k ⊂ℓ∞, n ∈IN we set Pn(σ) :=nP1σkek.A maximal sequence space (X, ∥·∥) is a Banach space satisfying the following conditions.1.

ℓ1 ⊂X ⊂ℓ∞and ∥ek∥= 1 for all k ∈IN.2. If σ ∈X and α ∈ℓ∞then the pointwise product ασ ∈X with ∥ασ∥≤∥σ∥X ∥α∥∞.3.

σ ∈X if and only if ( ∥Pn∥)n is bounded and in this case∥σ∥= supn∈IN∥Pn∥.For n ∈IN and σ = (σk)n1 ⊂IKn we set ∥σ∥:= ∥(σk)n1 ∥:=nP1σkek. The sequence dual of X is definedbyX+ := { τ ∈ℓ∞| ∥τ∥+ :=supσ∈BXXkσkτk < ∞}.Then (X+, ∥·∥+) is also a maximal sequence space.

We observe that ∥τ∥X∗= ∥τ∥+ holds for all τ ∈φ.Thus X++ = X with equal norms. For two maximal sequnce spaces X, Y we denote by IDL(X, Y ) thespace of continous diagonal operators from X to Y with the operator norm.

A maximal sequence spaceis symmetric if in addition σ ∈X if and only if σ∗∈X with ∥σ∗∥X = ∥σ∥X.Essentially for the following is the definition of p-convex sequence spaces. Let 1 ≤p < ∞.

A maximalsequence space is p−convex if there is a constant c>0 such that for all n ∈IN and (xk)n1 ⊂X nX1|xk|p!1/p≤c nX1∥xk∥p!1/p.The best constant c satisfying the above condition will be denoted by M p(X). Obviously, every maximalsequence space is 1-convex.

On the other hand we observeX+ = IDL(X, ℓ1) and thus X = IDL(X+, ℓ1) .More generally, one has5

Proposition 1.1 Let 1≤p<∞and X a maximal sequence space. Then the following are equivalent:1.

X is p-convex.2. The homogenous expression|σ|1/ppX is equivalent to a norm ∥·∥p with1c ∥σ∥X ≤∥|σ|p∥1/pp≤∥σ∥X.3.

There exists a maximal sequence space Y such thatX ∼= IDL(Y, ℓp).Moreover, in this case we can choose Y = IDL(X, ℓp) and have1M p(X) ∥σ∥X ≤∥Dσ∥≤∥σ∥X.Proof : The equivalence between 1. and 2. is classical and can be found for example in [LTII]. Nowwe proof 2.

⇒3. We denote by Xp the maximal sequence space defined by the norm ∥·∥p.We setY := IDL(X, ℓp).

Cleary, we have X ⊂IDL(Y, ℓp). By the observations above we have1c ∥σ∥≤∥|σ|p∥1/pp=supτ∈B(Xp)+Xk|σk|p τk1/p≤∥σ∥IDL(Y,ℓp)supτ∈B(Xp)+|τ|1/pIDL(X,ℓp)=∥σ∥IDL(Y,ℓp)supτ∈B(Xp)+supρ∈BX Xk|τk| |ρk|p!1/p=∥σ∥IDL(Y,ℓp) supρ∈BX∥|ρ|p∥1/pp≤∥σ∥IDL(Y,ℓp).For the proof of 3.

⇒1. we can asssume that X = IDL(Y, ℓp) with equal norms.

The definition of thenorm implies for (xj)n1 ⊂X nX1|xj|p!1/p=supτ∈BYXknXj=1|xj(k)|p |τk|p1/p=supτ∈BYnXj=1Xk|xj(k)τk|p1/p≤nXj=1supτ∈BY Xk|xj(k)τk|p!1/p=nXj=1∥xk∥p1/p.✷6

Remark 1.2 i) An Orlicz sequence spaceℓφ := { σ ∈ℓ∞|Xkφ(σk) < ∞}is p-convex if and only if φ(tλ) ≤cλp φ(t).ii) The criterion above is very useful to study the p-convexity of a Lorentz-Marcinkiewicz sequence spaceℓf,q. It was observed in [COB] that for p≤q and 0< βf ≤αf <1/p one has|σ|1/ppf,q∼ 1nnX1σ∗k!f p,q/p.Since the right hand side is a norm, see again [COB], the conditions above imply the p-convexity of ℓf,q.2Quotient formulas for summing propertiesWe will start with a quotient formula for (X, q)-summing operators.Proposition 2.1 Let 1≤r≤q ≤∞, Y a maximal, symmetric sequence space and X ∼= IDL(Y, ℓq).

Thenwe have for all n ∈IN and T ∈L(E, F)πnX,r(T )=sup{ πnq,r(DσRT ) | R ∈L(F, ℓ∞), Dσ ∈L(ℓ∞, ℓ∞), with ∥R∥, ∥σ∥Y ≤1 } .Proof : ” ≤” Let (xk)n1 ⊂E with. For ε>0 there exists a σ ∈BY withnX1∥T xk∥ekX≤(1 + ε) nX1|∥T xk∥σk|q!1/q.Let y∗k ∈BF ∗with < y∗k, T xk > = ∥T xk∥.

If we define R :=nP1y∗k ⊗ek ∈L(F, ℓ∞) we obtain11 + εnX1∥T xk∥ekX≤ nX1|< y∗k, T xk > σk|q!1/q≤ nX1supj< y∗j , T xk > σjq!1/q≤πnq,r(DσRT ) ωr(xk)n1.” ≥” Let σ ∈BY and R ∈L(F, ℓ∞) with ∥R∥≤1. By the maximality of (X, r)-summing operatorsthere is no restriction to assume R ∈L(F, ℓm∞) for some m ∈IN.

Now we will use a duality argument.Following the proof of theorem 1. in [DJ] there is an operator S ∈L(ℓm∞, E) withπnq,r(DσRT ) = trace(SDσRT )andS = BDτP ,where B ∈L(ℓnr′, E) with ∥B∥≤1, τ ∈Bℓnq′ and there is an increasing sequence (lk)n1 ∈{1, .., m} suchthatP =nX1elk ⊗ek ∈L(ℓm∞, ℓn∞).7

Therefore we deducetrace(SDσRT )=trace(DτPDσRT B)=nX1τk < elk , DσRT B(ek) >≤nX1|τk σlk| ∥RT B(ek)∥≤ nX1(|σlk| ∥RT B(ek)∥)q!1/q≤∥σ∥Y πnX,r(RT ) ∥B∥≤πnX,r(T ).✷We can now prove the generalized Maurey theorem.Theorem 2.2 Let 1 ≤r < q ≤∞, X a q-convex maximal, symmetric sequence space and n ∈IN. Thenfor all operators T ∈L(C(K), F) one hasπnX,r(T ) ≤c0 M q(X) 1r1r −1q−1/q′πnX,1(T ) .Proof : By proposition 1.1 we can assume that there exists a maximal, symmetric sequence space Ywith X ∼= IDL(Y, ℓq).

By the classical Maurey theorem, for the constants see [TJM], we deduce fromproposition 2.1πnX,r(T )≤M q(X) sup{ πnq,r(DσRT ) | R ∈L(F, ℓ∞), Dσ ∈L(ℓ∞, ℓ∞), with ∥R∥, ∥σ∥Y ≤1 }≤M q(X) c01r1r −1q−1/q′×sup{ πnq,1(DσRT ) | R ∈L(F, ℓ∞), Dσ ∈L(ℓ∞, ℓ∞), with ∥R∥, ∥σ∥Y ≤1 }=c0 M q(X) πnX,1(T ) .✷Remark 2.3 Now it is again well-known, see [MAU], how to derive from the above theorem the equiv-alence between Rademacher cotype conditions and summing properties as stated in the introduction astheorem 3, namelyπnX,1(T ) ≤rcnX(T ) ≤√2 πnX,2(T ) ≤c0 M q(X)12 −1q−1/q′πnX,1(T ) .At the end of this chapter we will prove another quotient formula which is more adapted for operatorson C(K)-spaces.Proposition 2.4 Let Y ,Z be maximal, symmetric sequence spaces and X = IDL(Y, Z). then we havefor all T ∈L(E, F) and n ∈INπnX,1(T )=sup{ πnZ,1(T RDσ) | R ∈L(ℓ∞, E), Dσ ∈L(ℓ∞, ℓ∞), with ∥R∥, ∥σ∥Y ≤1 } .8

Proof : ” ≤” can be proved exactly as in proposition 2.1.” ≥” Again by maximality we can assume R ∈L(ℓm∞, E) and Dσ ∈L(ℓm∞, ℓm∞) with ∥R∥, ∥σ∥Y ≤1. Wehave to show that for all S ∈L(ℓn∞, ℓm∞) with ∥S∥≤1 we havenX1∥T RDσS(ek)∥F ekZ≤πnX,1(T ) .By a lemma of Maurey, calculating essentially the extreme points of operators from ℓn∞to ℓm∞, see [MAU],and using the convexity of Z we can assume that S has the formS =nX1ek ⊗gk.Here the (gk)’s have disjoint support and satisfy 0

Now we defineJ := R nX1ek ⊗Dσgk∥Dσgk∥∞!∈L(ℓn∞, E)and τ :=Dσgk∞n1. We observe that ∥R∥≤1 and there is a subsequence (lk)n1 ⊂{1, .., m} suchthatDσgk∞=< elk, Dσgk >.

From the rearrangement invariance of Y we deduce∥τ∥Y=σlk < elk, gk >n1Y≤nX1σlk elkY≤∥σ∥Y ≤1.Hence we obtainnX1∥T RDσS(ek)∥F ekZ=nX1∥T J(ek)∥FDσgk∞ekZ≤πnX,1(T ) ∥τ∥Y ≤πnX,1(T ) .✷3Gaussian cotype conditionsAs a consequence of Talagrand’s factorization theorem for gaussian processes cotype conditions on C(K)-spaces can be reformulated with a quotient formula. This was remarked by Pisier and Montgomery-Smith,see [MSM].

We will give a prove for an arbitrary maximal, symmetric sequence space. Let us recall thatℓ∞,∞,1/2 is the space of sequences σ ∈ℓ∞with∥σ∥ℓ∞,∞,1/2 := supk∈INplog(k + 1) σ∗k < ∞.Lemma 3.1 Let X be a maximal, symmetric sequence space, T ∈L(C(K), F) and n ∈IN.

Then wehave for an absolut constant c1gcnX(T )∼c1sup{ πnX,2(T RDσ) | R ∈L(co, E), Dσ ∈L(co, co) with ∥R∥, ∥σ∥ℓ∞,∞,1/2 ≤1 } .9

Proof : ” ≥” W.l.o.g. we can assume that σk = (log(k + 1))−1/2.

Then it follows from [LIP] that forall u ∈L(ℓn2 , co) we havenX1gk RDσu(ek)L2(C(K))≤c1 ∥R∥∥u∥.With a glance on definition of gcnX we see that the first inequality is proved.” ≤” Let (xk)n1 ∈C(K) withnX1gkxkL2(C(K))≤1.By Talagrand’s factorization theorem, see (*) in the preliminaries, there are u ∈L(ℓn2, co) and R ∈L(co, C(K)) with ∥u∥≤c1, ∥R∥≤1 such thatRDσu(ek) = xk,and σk = (log(k + 1))−1/2. Hence we deduce thatnX1∥T xk∥F ekX=nX1∥T RDσu(ek)∥F ekX≤πnX,2(T RDσ) ∥u∥≤c1 πnX,2(T RDσ)nX1gkxkL2(C(K)).Taking the supremum over all sequences (xk)n1 yields the assertion.✷Now we are able to prove the comparision theorem for gaussian and Rademacher cotype.Theorem 3.2 Let 2 < q < ∞and X a q-convex maximal, symmetric sequence space.We set Y=IDL(ℓ∞,∞,1/2, X).

Then we have for all T ∈L(C(K), F) and n ∈IN1. πnY,1(T ) ≤rcnY (T ) ≤√2 πnY,2(T ) ≤c0 M q(X)12 −1q−1/q′πnY,1(T ) .2. gcnX(T )∼cqrcnY (T ).Proof : First we note that the q-convexity of X implies the q-convexity of the maximal, symmetricsequence space Y.

This can be seen exactly as in the proof of proposition1.1 . Therefore the first assertionfollows from theorem 2.2, more precisely remark 2.3, applied for Y.

With the help of the previous Lemma3.1, applying theorem 2.2 for X and with the second quotient formula 2.4 we obtaingcnX(T )∼c1sup{ πnX,2(T RDσ) | R ∈L(co, E), Dσ ∈L(co, co) with ∥R∥, ∥σ∥ℓ∞,∞,1/2 ≤1 }∼cq(X)sup{ πnX,1(T RDσ) | R ∈L(co, E), Dσ ∈L(co, co) with ∥R∥, ∥σ∥ℓ∞,∞,1/2 ≤1 }=πnY,1(T ).Using the first assertion we see that the proof of the second assertion is completed.✷10

Remark 3.3 Probably the most important applications of the above theorem are given for gaussian cotypeq and Rademacher cotype q operators when q>2.1. In the case when X = ℓq it turns out that Y is in fact the Lorentz-Marcinkiewicz space ℓq,q,−1/2.This space consists of all sequences σ ∈ℓ∞such that Xk σ∗kplog(k + 1)!q !1/q< ∞.2.

If we want to calculate the cotype conditions for (q, 1)-summing operators or Rademacher cotype qoperators we have to solve the equationℓq = IDL(ℓ∞,∞,1/2, Y ).Again this is easy with the help of Lorentz-Marcinkiewicz spaces. The space ℓq,q,−1/2 with the norm∥σ∥ℓq,q,−1/2 := Xkσ∗kplog(k + 1)q!1/qsolves the problem up to some constant.

In order to apply theorem 3.2 we have to check the r-convexity of ℓq,q,−1/2 for some r > 2. If we identify ℓq,q,−1/2 with a space ℓf,q this easily followsfrom remark 1.2.

Indeed, f is given byf(t) := t1/q plog(t + 1) ,which satisfies βf = αf =1q .In the following we will state further applications of theorem 3.2.Corollary 3.4 Let 2 < q < ∞and X a q-convex maximal, symmetric sequence space then there is aconstant c depending on q and X only such that for all n ∈IN and T ∈L(C(K), F) with rank(T ) ≤none hasgcX(T ) ≤c gcnX(T ).Moreover, the gaussian cotype constant is, up to c, attained on n disjoints functions in C(K).Proof : We set Y = IDL(ℓ∞,∞,1/2, X). By theorem 3.2 we havegcX(T ) ∼c πY,1(T ).Therefore it remains to show that the (Y, 1)-summing norm is attained on n vectors.

Using Maurey’slemma about the extreme points of operators from ℓn∞to C(K) (already used in the proof of proposition2.4), see [MAU], it is then clear from that a restriction to n disjoint blocs is possible.In theorem 3.2 it was also observed that Y is q-convex. By proposition 1.1 there is a maximal, symmetricsequence space Z with Ycong IDL(Z, ℓq).

Furthermore, it is known that for the computation of the (q, 2)-summing norm of an11

operator with rank n only n vectors are needed, see for example [DJ].Hence we can deduce fromproposition 2.1 and theorem 2.2πY,1(T )≤sup{ πq,2(DσRT ) | R ∈L(F, ℓ∞), Dσ ∈L(ℓ∞, ℓ∞), with ∥R∥, ∥σ∥Z ≤1 }≤√2 sup{ πnq,2(DσRT ) | R ∈L(F, ℓ∞), Dσ ∈L(ℓ∞, ℓ∞), with ∥R∥, ∥σ∥Z ≤1 }=√2 πnY,2(T )≤√2 cq πnY,1(T ) .✷In particular, the corollary works for X = ℓq. For the so-called ”weak” theory it is natural to replaceℓq by weak-ℓq.

More precisely, an operator T ∈L(E, F) is said to be a weak cotype q operator, if thereexists a constant c>0 such that for all u ∈L(ℓn2, E) one hassupk=1,..,nk1/q ak(T u)≤c ℓ(u).The best constant c will be denoted by ωcq(T ). It was essentially remarked by Mascioni, see [MAS], thatfor q > 2 another definition would have been possible.

An operator T ∈L(E, F) is of weak cotype q ifand only if there exists a constant c > 0 such thatsupk∈INk1/q ∥T xk∥F≤cXkgkxkL2(E)for each sequence (xk)k ⊂E such that ∥T xk∥is non-increasing (for further information see also [DJ1]).The next proposition gives a characterization of weak cotype operators on C(K)-spaces in terms of Weylnumbers.Corollary 3.5 Let 2< q<∞. An operator T ∈L(C(K), F) is of weak cotype q if and only ifsupk∈INk1/qplog(k + 1)xk(T )<∞.Proof : By remark 1.2 the space X := ℓq,∞:= ℓf,∞with f(t) = t1/q is r-convex for all 2 < r < q.We observe that Y:= IDL(ℓ∞,∞,1/2, X) coincides with ℓg,∞where g(t) = t1/q/plog(t + 1).UsingMascioni’s observation above we deduce from theorem 3.2 that T is of weak cotype q if and only if T is(Y, 2)-summing.If T is (Y, 2)-summing and u ∈L(ℓ2, C(K)) we can apply a lemma due to Lewis, see [PIE], whichguarantees for all ε>0 the existence of an orthonormal system (ok)k ⊂ℓ2 with (∥T u(ok)∥F )k decreasingandak(T u)≤(1 + ε) ∥T u(ok)∥F.Therefore we deducesupk∈INk1/qplog(k + 1)ak(T u)≤(1 + ε) supk∈INk1/qplog(k + 1)∥T u(ok)∥F≤(1 + ε) πY,2(T ) ω2(u(ok))k≤(1 + ε) πY,2(T ) ∥u∥.Taking the infimum over all ε and the supremum over all u ∈L(ℓ2, C(K)) with norm less than 1 weobtain12

supk∈INk1/qplog(k + 1)xk(T )≤πY,2(T ).Vice versa, let us assume that the sequence of Weyl numbers is in Y. Let (xk)k ∈C(K) with ω2(xk)k ≤1.There is no restriction to assume that ∥T xk∥F is decreasing.

If we define un :=nP1ek ⊗xk we can deducefrom an inequality of K¨onig, see [PIE],n1/2 ∥T xn∥≤ nX1∥T xk∥2!1/2≤π2(T un)≤c1nX1ak(T un)√k≤c1nX1(log(k + 1))1/2k1/2+1/qnX1xk(T ) ekY∥u∥≤c1plog(n + 1)n1/2−1/q1/2 −1/qXkxk(T ) ekY.Taking the supremum over all n ∈IN we have shown that T is (Y, 2)-summing.✷References[COB] F. Cobos: On the Lorentz-Marcinkiewicz Operator ideal. : Math.

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(1986), 281-300.[DJ]M. Defant and M.Junge: On absolutely summing operators with apllication to the (p,q)-summingnorm with few vectors; J. of Functional Ana.

103 (1992), 62-73.[DJ1]M. Defant and M. Junge: Random variables in weak type p spaces; Arch.

Math. 58 (1992),399-406.

[MAS] V. Mascioni: On weak cotype and weak type in Banach spaces; Note di Matematica Vol VIII-n.1(1988), 67-110. [MAU] B. Maurey: Type et cotype dans les espaces munis d’un structure localement inconditionelle;S´eminaire Maurey-Schwartz 73-74, Ecol- Polyt., Exp.

no. 24-25.

[MSM] S.J.Montgomery-Smith: The Gaussian cotype of operators from C(K); Isr. J.

Math. 68 (1989),123 - 128.[LET]M.

Ledoux and L.Talagrand: Probability in Banach spaces. Berlin Heidelberg New York: Springer1991.[LIP]W.

Linde and A. Pietsch: Mappings of gaussian cylindrical measures in Banach spaces; TheoryProbab. Appl.

19 (1974), 445-460.[LTI]J. Lindenstrauss and L. Tzafriri: Classical Banach spaces I, sequence spaces; Springer BerlinHeidelberg New York 1977.[LTII]J.

Lindenstrauss and L. Tzafriri: Classical Banach spaces II, function spaces; Springer BerlinHeidelberg New York 1979.[PIE]A. Pietsch: Eigenvalues and s-numbers of operators; Cambridge University Press, 1987.13

[TAL]M. Talagrand: Cotype of operators from C(K); Invent. math.

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Math. 159 (1987), 99-149.[TJM]N.

Tomczak-Jaegermann: Banach-Mazur distances and finite-dimensional operator ideals; Long-mann, 1988.1991 Mathematics Subject Classification: 47B38, 47A10, 46B07.Marius JungeMathematisches Seminar der Universit˝at KielLudewig-Meyn-Str. 4W-2300 Kiel 1Germany14

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