# Cohen Positive Stronglyp-Summing and p-Dominated m-Homogeneous Polynomials

Authors: Halima Hamdi; Samuel García-Hernández; Amar Belacel; Djamel Ouchenane; Sulima Ahmed Zubair
Journal: Journal of Mathematics
 (2021)

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## Abstract

We introduce the concepts of Cohen positive stronglyp-summing and positive p-dominated m-homogeneous polynomials. The version of Pietsch’s domination theorem for the first class among other results and a Bu-type theorem is proved, as well as some inclusions with other known spaces. Moreover, we present a characterization of these classes in tensor terms.


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## Body

## 1. Introduction

The identity fromℓ1 into ℓ2 is absolutely 2-summing, but its adjoint is not. Cohen introduced the concept of Cohen strongly p-summing linear operators in [1] to study the conjugate of absolutely p∗−summing linear operators. This paper had an enormous fame; in view of this, many analogies in the multilinear and the polynomial cases were investigated [2–5]. In this article, we extend this concept to positive m-homogeneous polynomials.This work is organized as follows. In Section2, we introduce the class of Cohen positive strongly p-summing m-homogeneous polynomials and prove that this class is related to different classes of p-summability Πp∗+E,F,Dpm+Xm;F,Dp+⊗^π,smX;F. By means of the adjoint operator, the associated symmetric m-linear operator and the linearization of these polynomials. Section 3 is essentially dedicated to the study of positive p-dominated m-linear operators and polynomials. This section includes our main theorem (the Bu-type theorem) which serves as a focal point in the proof of the relationship between positive p-dominated polynomials and those defined in Section 2. In Section 4, we present some applications based on Proposition 3.Finally, Section5 highlights the classes defined earlier in terms of tensor product and proves a duality between these classes and tensor spaces equipped with adequate norms through a linear mapping.Firstly, we considerX,X1,…,Xm,Y and E,E1,…,Em,F,G are Banach spaces and Banach lattices, respectively, for m∈ℕ∗.LetℒX1,…,Xm;Y is the Banach space of all continuous m-linear operators from X1×⋯×Xm to Y. If Y=K, we write ℒX1,…,Xm. In the case when X1=⋯=Xm=X, we simply write ℒXm;Y. For a Banach space X, we denote X∗ and BX are the topological dual and the closed unit ball, respectively. For 1≤p≤∞, let p∗ be its conjugate, that is, 1/p+1/p∗=1.LetE+ be the positive cone of the Banach lattice E such that x∈E,x≥0. For x∈E, let x+≔supx,0 and x−≔sup−x,0 be the positive part and the negative part of x, respectively.For anyx∈E, we have x=x+−x−andx=x++x−.Recall that anm-linear operator T:E1×⋯×Em⟶F is called positive if Tx1,…,xm∈F+, whenever x1∈E1+,…,xm∈Em+. Let ℒ+E1,…,Em;F be the collection of all positive continuous m-linear operators from E1×⋯×Em to F under the norm T=supxj∈BEj+Tx1,…,xm.We denote byℓpnX the space of all sequences xii=1n in X with the norm,(1)xii=1np=∑i=1nxip1/p,and by ℓp,weaknX the space of all sequences xii=1n in X with the norm,(2)wpxii=1n=xii=1nℓp,weaknX=supφ∈BX∗∑i=1nxi,φp1/p.Then,ℓp,weaknX is a Banach space with respect to the norm wp. For a Banach lattice E, we define(3)ℓp,weaknE≔xii=1n:xii=1n∈ℓp,weaknE,xii=1nℓp,weaknE=wpxii=1n.LetBE∗+=BE∗∩E∗+. If x1,…,xn≥0, we have that(4)xii=1nℓp,weaknE=supξ∈BE∗+∑i=1nxi,ξp1/p.The definition ofm-homogeneous polynomial can be seen, for instance, in [3, 6, 7].A mapP:X⟶Y is an m-homogeneous polynomial if there exists a unique symmetric m-linear operator P^:X×⋯×X⟶Y such that Px=P^x,…,xm,for allx∈X. Both are related by the polarization formula:(5)P^x1,…,xm=1m!2m∑ϵj=±11≤j≤mϵ1…ϵmP∑j=1mϵjxj.P is bounded on the unit ball of X if and only if P^ is bounded on the unit ball of X×⋯m×X. These norms are related by the inequalities P≤P^≤mm/m!P. PXm;Y denotes the Banach space of all continuous m−homogeneous polynomials from X to Y with the norm P=supx≤1Px=infC:Px≤Cxm,x∈X.For the scalar-valued caseY=ℝ or ℂ, we use the simplified notation PXm.ByX1⊗^π⋯⊗^πXm, we denote the completed projective tensor product of X1,…,Xm. If X=X1=⋯=Xm, we write ⊗^πmX. By ⊗smX≔X⊗s⋯m⊗sX, the m−fold symmetric tensor product of X, that is, the set of all elements u∈⊗mX of the form(6)u=∑i=1nxi⊗⋯m⊗xin∈ℕ,xi∈X,1≤i≤n.By⊗^π,smX, we denote the closure of ⊗smX in ⊗^πmX. Consider the canonical polynomial δm:X⟶⊗^πmX defined by δmx=x⊗⋯m⊗x, and we have the following factorization P=P˜°δm.Recalling some definitions we need in the sequel.Let1≤p<∞. An operator u:E⟶X is said to be positive p−summing [8] if there exists a constant C>0 such that, for all n∈ℕ,x1,…,xn∈E, the inequality(7)uxii=1np≤Cxii=1nℓp,weaknE.Forp=∞,(8)sup1≤i≤nuxi≤Cxii=1nℓ∞,weaknE.The space of positivep−summing operators from E into X given by Πp+E;X,t⋅Πp+E,X becomes a Banach space with norm πp+⋅ defined by the infimum of the constants C>0 that verify inequality (7). We have Π∞+E;X=ℒE;X.The next definition is due to Achour and Saadi in [3]. Let 1<p≤+∞. An m−homogeneous polynomial P:X⟶Y is Cohen strongly p−summing if ∃C>0 such that, for all x1,…,xn∈X and y1∗,…,yn∗∈Y∗,(9)∑i=1nPxi,yi∗≤C∑i=1nximp1/psupy∈BYyi∗yℓp∗n.The class of such polynomials is denoted byPCoh,pXm,Y; it is equipped with the norm dp⋅, i.e., the smallest constant C such that inequality (9) holds. For p=1, we have PCoh,1Xm;Y=PXm;Y.


## 2. Cohen Positive Stronglyp-Summing m-Homogeneous Polynomials

In this section, we give a natural generalization of the notion of Cohen positive stronglyp−summing to the category of homogeneous polynomial mappings. As in the multilinear case, there is an analogue connection between polynomial mapping and its linearization and associated symmetric m−linear operator.Definition 1 (see [4]).
Let1≤p≤+∞. An m−linear operator T:X1×⋯×Xm⟶F is Cohen positive strongly p−summing multilinear operator if ∃C>0, such that, for all x1j,…,xnj∈Xj, 1≤j≤m, and any y1∗,…,yn∗∈F∗,(10)Txi1,…,xim,yi∗ℓ1n≤C∑i=1n∏j=1mxijXjp1/pyi∗i=1nℓp∗,weaknF∗.
Moreover, the class of all Cohen positive stronglyp−summing m−linear operators from X1×⋯×Xm into F is given by Dpm+X1,…,Xm;F. This space is a Banach space with the norm dpm+⋅ which is the smallest constant C such that inequality (10) holds. For m=1, it is the space of positive strongly p−summing operators linear operators [2].Now, we introduce the notion of Cohen positive stronglyp−summing m−homogeneous polynomials, and we study some fundamental properties.Definition 2.
Let1<p≤+∞ and m∈ℕ. An m−homogeneous polynomial P:X⟶F is Cohen positive strongly p−summing if ∃C>0, such that, for all x1,…,xn∈X and any y1∗,…,yn∗∈F∗,(11)∑i=1nPxi,yi∗≤C∑i=1nximp1/pyi∗i=1nℓp∗,weaknF∗.
The class of such polynomials is given byPCoh,p+Xm;F. It is a Banach space with the norm dp+⋅ which is the smallest constant C such that inequality (11) holds, for p=1, we have PCoh,1+Xm;F=PXm;F.Remark 1.
Fixm∈ℕ; an m−homogeneous polynomial P:X⟶F is Cohen positive strongly p− summing if and only if ∃C>0 satisfying(12)∑i=1nPxi,yi∗≤C∑i=1nximp1/pwp∗yi∗i=1n,holds for every x1,…,xn∈X and y1∗,…,yn∗∈F∗+.Example 1.
We use the same example used in [3], with slight adjustments.
Let1<p≤+∞, m∈ℕ, and u:X⟶F be a positive strongly p−summing operator. For φ∈X∗, the functional(13)P:X⟶Fx↦φm−1xux,is Cohen positive strongly p−summing.
Forx1,x2,…,xn∈X and y1∗,y2∗,…,yn∗∈F∗, because u∈Dp+X,F, we have by [2](14)∑i=1nuxi,yi∗≤dp+uxii=1npyi∗i=1nℓp∗,weaknF∗.
Then,(15)∑i=1nPxi,yi∗=∑i=1nφm−1xiuxi,yi∗=∑i=1nuφm−1xixi,yi∗≤dp+uφm−1xixii=1npyi∗i=1nℓp∗,weaknF∗≤dp+u∑i=1nφm−1xixii=1np1/pyi∗i=1nℓp∗,weaknF∗≤dp+uφm−1∑i=1nxii=1nmp1/pyi∗i=1nℓp∗,weaknF∗.
Hence,P is Cohen positive strongly p−summing and dp+P≤dp+uφm−1.Proposition 1 (ideal property).
LetP∈PXm;F, S in L+F;G, and R in ℒY;X.
IfP is Cohen positive strongly p-summing polynomial, then SPR is Cohen positive strongly p-summing polynomial and dp+SPR≤dp+PRmS.Proof.
Letxii=1n⊂E and yi∗i=1n⊂G∗+(16)∑i=1nSPRxi,yi∗=∑i=1nPRxi,S∗yi∗≤dp+P∑i=1nRximp1/psupy∗∗∈BF∗∗+∑i=1nS∗yi∗,y∗∗p∗1/p∗≤dp+PRmS∑i=1nximp1/psupy∗∗∈BF∗∗+∑i=1nyi∗,S∗∗y∗∗S∗∗p∗1/p∗≤dp+PRmS∑i=1nximp1/psupφ∈BG∗∗+∑i=1nyi∗,φp∗1/p∗.
Thus,SPR is Cohen positive strongly p-summing polynomial and dp+SPR≤dp+PRmS.Next, we present a version of Pietsch’s domination theorem, for our class of polynomials.Theorem 1 (domination theorem).
Anm−homogeneous polynomial P:X⟶F is Cohen positive strongly p−summing 1<p≤+∞ if and only if there is a Radon probability measure μ on BF∗∗+, such that, for all x∈X and y∗∈F∗, we have(17)Px,y∗≤Cxm∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗.
In addition, in this case,(18)dp+P=infC>0,for allCverifying inequality17.For the proof of this theorem, we need to give a small review aboutR−S-abstract p,p∗-summing operators.Let the sets beX,Y, and E, the Banach space ℋ is the family of mappings from X to Y and G and the compact Hausdroff topological space K. Assume that the maps(19)R:K×E×G⟶0,∞,S:ℋ×E×G⟶0,∞,satisfy.For allf∈ℋ, there is an x0∈E, such that(20)Rφ,x0,b=Sf,x0,b=0,for anyφ∈K and b∈G.The mappingRx,b:K⟶0,∞ defined by Rx,bφ=Rφ,x,b is continuous, for any x∈E and b∈G, and Rφ,x,ηb≤ηRφ,x,b and ηSf,x,b≤Sf,x,ηb, for any φ∈K, x∈E, 0≤η≤1, b∈G, and f∈ℋ.Definition 3 (see [9]).
LetR and S be as above and 0<p<∞. A mapping f∈ℋ is said to be R−S-abstract p-summing if ∃C>0 such that(21)∑j=1mSf,xj,bjp1/p≤Csupφ∈K∑j=1mRφ,xj,bjp1/p,for all x1,…,xm∈E, b1,…,bm∈G, and m∈ℕ∗.Theorem 2 (see [9]).
ConsiderR and S be before, 0<p<∞ and f∈ℋ. Then, f is R−S-abstract p-summing ⇔, >0 and a Borel probability measure μ on K such that(22)Sf,x,b≤C∫KRφ,x,bpdμ1/p,where ∀x∈E and b∈G. Furthermore, the infimum of such constant C equals πRS,pf.For our proof, we also need to recall Theorem 4.6 in [9].Theorem 3 (see [9]).
A mapf∈ℋ is R1,…,Rt−S-abstract p1,…,pt-summing ⇔, >0, and a Borel probability measures μj on Kj, such that(23)Sf,x1,…,xr,b1,…,bt≤C∏j=1t∫KjRjφ,x1,…,xr,bjpjdμj1/pj,where ∀xl∈El, l=1,…,r, and bj∈Gj with j=1,…,t.Proof.
(proof of the domination theorem). Assume thatP is Cohen positive strongly p-summing polynomial, by selecting the parameters,(24)t=2andr=m−1,E=X,K1=BX∗+andK2=BF∗∗+,G=F,q=1q1=pandq2=p∗,ℋ=PXm,F,SP,x,y∗=Px,y∗,R1φ,x=xm,R2φ,x,y∗=φy∗.
Hence,P:X⟶F is Cohen positive strongly p-summing ⇔ and P is R1−R2−S-abstract p,p∗-summing. The previous theorem tells us that P is R1−R2−S-abstract p,p∗-summing ⇔ there is C>0, and there are probability measures μk on Kk, k=1,2, such that(25)SP,x,y∗≤C∫K1R1φ,xpdμ11/p∫K2R2φ,x,y∗p∗dμ21/p∗,i.e.,(26)Px,y∗≤C∫BX∗+xmpdμ11/p∫BF∗∗+φy∗|p∗dμ21/p∗≤Cxm∫BF∗∗+φy∗p∗dμ21/p∗.Proposition 2.
Consider1<p1<p2≤+∞.
IfP∈PCoh,p2+Xm;F, then P∈PCoh,p1+Xm,F and dp1+P≤dp2+P.Proof.
Assume thatP∈PCoh,p1+Xm,F; the Pietsch domination (Theorem 1) provides a Radon probability measure μ on BF∗∗+ such that ∀x,y∗∈X×F∗ for which(27)Px,y∗≤dp2+Pxm∫BF∗∗+y∗y∗∗p2∗dμy∗∗1/p2∗≤dp2+Pxm∫BF∗∗+y∗y∗∗p1∗dμy∗∗1/p1∗.
So,P∈PCoh,p1+Xm,F and dp1+P≤dp2+P.Definition 4 (see [10]).
Given a continuousm−homogeneous polynomial P∈PXm,Y between the Banach spaces X and Y, the adjoint of P is the following continuous linear operator:(28)P∗:Y∗⟶PXm,P∗y∗x=y∗Px.
We haveP∗=P.Proposition 3.
Let1<p≤+∞. The m−homogeneous polynomial P∈PXm;F is Cohen positive strongly p−summing if and only if P∗ is positive p−summing, and we have dp+P=πp∗+P∗.Proof.
Suppose thatP∈PCoh,p+Xm;F; we have by (17) Px,y∗≤dp+Pxm∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗, for x∈X and y∗∈F∗.
Taking the supremum, we obtain(29)supx≤1P∗y∗x≤dp+P∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗.
Then,P∗y∗≤dp+P∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗. By the Pietsch domination theorem in Proposition 3.4 in [2], P∗∈Πp∗+F∗,PXm. Conversely, let P∗∈Πp∗+F∗,PXm we have, for x∈X and y∗∈F∗,(30)Px,y∗=P∗y∗x≤P∗y∗xm.
Using the theorem of Pietsch domination for positivep∗-summing linear operators, we find(31)Px,y∗≤Cxm∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗,P∈Pcoh,p+Xm,F.Proposition 4.
If1<p≤+∞,(32)PCoh,pXm;F⊆PCoh,p+Xm;F.Proof.
Take anyP in PCoh,pXm;F; hence, its adjoint operator P∗ is in Πp∗F∗,PXm by Proposition 1 in [11], and again by Proposition 1 in [8], P∗ is in Πp∗F∗,PXm; then, P∈PCoh,p+Xm;,F.Proposition 5.
The following properties are equivalent:The polynomialP∈PXm;F is Cohen positive strongly p−summingIts associated symmetricm−linear operator P^∈ℒXm;F is Cohen positive strongly p−summingProof.
First, suppose thatP^ is Cohen positive strongly p−summing; let x1,…,xn∈X and y1∗,…,yn∗∈F∗; then,(33)∑i=1nPxi,yi∗=∑i=1nP^xi,…,xi,yi∗≤dp+P^∑i=1nximp1/pyi∗i=1nℓp∗,weaknF∗.
Therefore,P is Cohen positive strongly p−summing and dp+P≤dp+P^.
Conversely, suppose thatP∈PCoh,p+Xm;F. Let xj∈X such that xj≤11≤j≤m and y∗∈F∗; we have, by (5),(34)P^x1,…,xm,y∗=1m!2m∑ϵj=±11≤j≤mϵ1⋯ϵmP∑j=1mϵjxj,y∗≤1m!2m∑ϵj=±11≤j≤mP∑j=1mϵjxj,y∗,by the domination theorem≤1m!2m∑ϵj=±11≤j≤mdp+P∑j=1mϵjxjm∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗≤1m!2mdp+P∑ϵj=±11≤j≤m∑i=1mxjm∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗≤1m!2mdp+P2mmm∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗≤mmm!dp+P∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗.
Furthermore, for everyxj∈X1≤j≤m,xj≠0, we have(35)P^x1x1,…,xmxm,y∗≤mmm!dp+P∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗.
Thus,(36)P^x1,…,xm,y∗≤mmm!dp+P∏j=1mxj∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗.
Using [4], P^ is Cohen positive strongly p−summing; furthermore, dp+P^≤mm/m!dp+P.Corollary 1.
Let1<p<∞. If the polynomial P∈PXm;F is Cohen positive strongly p−summing, then its associated symmetric m−linear operator P^∈PXm;F is multiple p−convex. If the polynomial P∈PXm;c0 is Cohen positive strongly p−summing, then its associated symmetric m−linear operator P^ is in ℒXm;c0.Proof.
The proof of this corollary is an adaptation of the proof of Proposition 3.3 in [4].Proposition 6.
Let1<p≤∞. Let P:X⟶F be a m-homogeneous polynomial and P˜ its linearization. The following properties are equivalent(i)
The polynomialP belongs to Pcoh,p+Xm;F(ii)
The operatorP˜ belongs to Dp+⊗^π,smX;F(iii)
∃Z is a Banach space, u∈Dp+Z,F, and Q∈PXm;Z such that P=uQProof.
i⇒ii Assume that P is Cohen positive strongly p-summing; let v∈⊗^π,smX such that v≠0 and y∗∈F∗, and assume that v=∑i=1nxi⊗⋯⊗xi; then,(37)P˜v,y∗≤∑i=1nPxi,…,xi,y∗≤∑i=1ndp+Pxim∫BF∗∗+y∗y∗∗p∗1/p∗=dp+P∑i=1nxim∫BF∗∗+y∗y∗∗p∗1/p∗.
Selecting the infimum over any represents ofv, we find(38)P˜v,y∗≤dp+Pvm∫BF∗∗+y∗y∗∗p∗1/p∗,and by Theorem 4.13 in [2], P˜ is positive strongly p-summing.
ii⇒iii From the factorization, P=P˜°δm gives the result directly.
iii⇒i Assume that there exists a Banach space Z, u∈Dp+Z,F and Q∈PXm;Z, such that P=uQ. Then,(39)Qx≤Qxm.
Using Theorem 4.13 in [2],(40)ux,y∗≤dp+ux∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗.
Hence,(41)Px,y∗=uQx,y∗≤dp+uQx∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗≤dp+uQxm∫BF∗∗+y∗y∗∗p∗dμy∗∗1/p∗,and according to the domination theorem, P∈Pcoh,p+Xm;F.


## 3. Positivep-Dominated m-Homogeneous Polynomials

In this section, we give a natural generalization of the notion of positivep−dominated to the category of homogeneous polynomial and the m-linear operators.


### 3.1. Positivep−Dominated m-Linear Operators

Definition 5.
Anm-linear operator T:E1×⋯×Em⟶Y is positive p−dominated if there is a positive constant C>0 such that(42)∑i=1nTxi1,…,ximp/mm/p≤Cxi1i=1nℓp,weaknE1⋯ximi=1nℓp,weaknEm,∀xiji=1n⊂Ej, 1≤j≤m and m∈ℕ. We denote ℒdp+E1,…Em;Y the space of all positive p−dominated m−linear operators and the infimum of all constants C verifying (42) is denoted πpd+T. For all p≥m, πpd+ is a norm on ℒdp+E1,…Em;Y, whereas for p<m, it is only a quasi-norm.
Form=1, the space of all positive p−dominated operators is nothing other than the space of positive p−summing operators.We should stress out that the above definition makes sense for Banach spacesY, but the more interesting case occurs when we consider a Banach lattice F, since this assumption allows to establish an inclusion relation between the classes of positive dominated multilinear operators and Cohen positive strongly summing multilinear operators and later on for polynomials.The next theorem is an analogous Pietsch’s domination-type theorem form-linear positive p-dominated operators. The proof of this result is an adaptation of that of Theorem 4.7 in [9], so we do not present it here.Theorem 4 (domination theorem for positivep-dominated multilinear operators).
Anm-linear operator T:E1×⋯×Em⟶Y is positive p−dominated if and only if there is Radon probability measures μj∈BEj∗+, 1≤j≤m and K>0, and(43)Tx1,…,xm≤K∏j=1m∫BEj∗+xj∗,xjpdμjxj∗1/p,for all xj∈Ej+. Moreover, the smallest constants K is πpd+T.The following theorem is a version of the Bu-theorem [12]; in [13], Achour and Mezrag proved an inclusion theorem between p-dominated operators and Cohen strongly q-summing m−linear operators; in our work, we prove a positive analogue of Theorem 3.2 in [13]. This theorem plays an important role in the proof of the relationship between Cohen positive strongly summing and positive dominated polynomials.Theorem 5.
Let1<p,q<∞. Let ℓ2n1,…,ℓ2nm and F be Banach lattices. Then,(44)ℒdp+ℓ2n1,…,ℓ2nm;F⊆Dqm+ℓ2n1,…,ℓ2nm;F,for all T∈ℒdp+ℓ2n1,…,ℓ2nm,Fdqm+T≤πpd+T.For the proof of this theorem, we need to recall the Khintchine [14] and the Kahane [13] multiple inequalities.


#### 3.1.1. Multiple Khintchine’s Inequality

If0<p<∞, then(45)Apm∑i1,…,im=1n1,…,nmai1,…,im21/2≤∫01⋯∫01ri1t1⋯rimtmai1,…,impdt1⋯dtm1/p≤Bpm∑i1,…,im=1n1,…,nmai1,…,im21/2,for each choice of the scalars ai1,…,imi1,…,im=1n1,…,nm, ApandBp are the Khintchine constants, and by rmm∈ℕ, we denote the sequence of Rademacher function such that rn:0,1⟶ℝ as rnt=sin2nπt.


#### 3.1.2. Multiple Kahane’s Inequality

Let1≤p<∞. Let Radpμ,X be the vector space of all almost unconditionally summable sequences xn in the Banach space X. It is a Banach space under the norm:(46)xnp=∫01∑n=1∞rntxnpdμt1/p.Let1≤p,q<∞. Let X be a Banach space and xi1,…,in be in X. We have(47)∑i1,…,inxi1,…,in⊗j=1nrijLp⊗1≤i≤nμi,X≤Kp,qn∑i1,…,inxi1,…,in⊗j=1nrijLq⊗1≤i≤nμi,X,where Kp,q is the simple constant of Kahane’s inequality μi=μ.Proof.
LetT∈ℒdp+ℓ2n1,…,ℓ2nm;F; then, by the domination theorem for positive p-dominated m-linear operators, there are probability measures μj on Bℓ2nj+, such that, for any x1,…,xm∈ℓ2n1×⋯×ℓ2nm, we have(48)Tx1,…,xm≤πpd+T∏j=1m∫Bℓ2nj+x∗,xjpdμjx∗1/p.
For everyxiji=1n⊂ℓ2nj, for j=1,…,m and yi∗i=1n⊂F∗, we have(49)xi1,…,xim=∑k1=1n1xi,k11ek1,…,∑km=1nmxi,kmmekm.
Then,(50)∑i=1nTxi1,…,xim,yi∗=∑i=1n∑k1,…,km=1n1,…,nmxi,k11⋯xi,kmmTek1,…,ekm,yi∗.
Using Hölder’s inequality, we obtain(51)∑i=1nTxi1,…,xim,yi∗≤∑i=1n∑k1=1n1xi,k1121/2⋯∑km=1nmxi,kmm21/2∑k1,…,km=1n1,…,nmTek1,…,ekm,yi∗21/2≤∑i=1n∏j=1mxijℓ2nj∑k1,…,km=1n1,…,nmTek1,…,ekm,yi∗21/2.
By the multiple Khintchine inequality, on the last inequality, we obtain(52)∑i=1nTxi1,…,xim,yi∗≤∑i=1n∏j=1mxijℓ2njAq∗m∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1≤Aq∗m∑i=1n∏j=1mxijq1/q∑i=1n∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1q∗1/q∗.
By Fubini’s theorem(53)∑i=1nTxi1,…,xim,yi∗≤Aq∗m∑i=1n∏j=1mxijq1/q∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1,ℓq∗n≤Aq∗m∑i=1n∏j=1mxijq1/qyi∗ℓq∗,weaknF∗∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1.
For the quantity,(54)∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1,F,using the multiple Kahane’s inequality(55)∑k1,…,km=1n1,…,nmrk1t1⋯rkmtmTek1,…,ekmLq∗0,1,F≤Kp,q∗m∑k1,…,km=1n1,…,nmrk1t1⋯rkmtmTek1,…,ekmLp0,1,F≤Kp,q∗m∑k1,…,km=1n1,…,nmTrk1t1ek1,…,rkmtmekmLp0,1,F≤πpd+T∫01∏j=1m∫Bℓ2nj+∑kj=1njx∗ekjrkjtjpdμjx∗dtj1/p≤πpd+T∏j=1m∫Bℓ2nj+∫01∑kj=1njx∗ekjrkjtjpdμjx∗dtj1/p.
Applying Khintchine’s inequality,(56)≤Bpmπpd+T∏j=1m∫Bℓ2nj+∑kj=1njx∗ekj2p/2dμjx∗1/p≤Bpmπpd+T∏j=1m∫Bℓ2nj+x∗ℓ2npdμjx∗1/p≤Bpmπpd+T.
Consequently,(57)∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1,F≤Kp,q∗mBpmπpd+T.
Exploiting (53) and (57) gives(58)∑i=1nTxi1,…,xim,yi∗≤Aq∗Kp,q∗BPmπpd+T∑i=1n∏j=1mxijq1/qyi∗i=1nℓq∗,weaknF∗.
Thus,T∈Dqm+ℓ2n1,…,ℓ2nm;F and dqm+T≤Aq∗Kp,q∗BPmπpd+T.
And, this completes the proof.


### 3.2. Positivep−Dominated m−Homogeneous Polynomials

Definition 6.
Let1≤p<∞; an m−homogeneous polynomial P:E⟶Y is positive p− dominated if ∃C>0 such that, for any n∈ℕ∗ and any xii=1n⊂E+,(59)∑i=1n‖Pxi‖p/mm/p≤Csupx∗∈BE∗+∑i=1nx∗,xipm/p.
We denotePd,p+Em;Y the space of positive p−dominated polynomials from E into Y, and by δp+⋅, the norm defined by the infimum of any constants verifying the above inquality, for p≥m, but for p<m, it is only a quasi-norm.Theorem 6.
(domination theorem for positivep−dominated m-homogeneous polynomials). Let 1≤p<∞; an m−homogeneous polynomial P:E⟶Y is positive p−dominated if there are K>0 and a probability measure μ on BE∗+ such that(60)Px≤K∫BE∗+x∗,xpdμx∗m/p,∀x∈E+. Moreover, the smallest K is δp+P.In Proposition7, we study the relationship between positive p−dominated m−linear operators and positive p−dominated m−homogeneous polynomialsProposition 7.
LetP:E⟶Ybe an m−homogeneous polynomial and P^ be its corresponding symmetric m−linear operator such that Px=P^x,…,x.(1)
IfP^ is positive p−dominated operator, then P is p−dominated and(61)δp+P≤πpd+P^.(2)
IfP is positive p−dominated, then P^ is positive p−dominated and(a)
Forp≥m, πpd+P^≤mm/m!δp+P(b)
Forp<m, πpd+P^≤mm/m!2m2/p−mδp+PProof.
(1) Suppose thatP^ is positive p−dominated; let xii=1n⊂E+; then,(62)∑i=1nPxip/mm/p=∑i=1nP^xi,…,xip/mm/p≤πpd+P^xii=1nℓp,weaknEm.
Therefore,P is positive p−dominated and δp+P≤πpd+P^.
(2) For1≤j≤m and n∈ℕ, let xj∈Ej satisfy xj≤1:(63)P^x1,…,xmp/mm/p=1m!2m∑ϵj=±1ϵ1⋯ϵmP∑j=1mϵjxjp/mm/p≤1m!2m∑ϵj=±1P∑j=1mϵjxjp/mm/p.
By Theorem6,(64)≤1m!2mδp+P∑ϵj=±1∫BE∗+x∗,∑j=1mϵjxjpdμx∗m/p≤1m!2mδp+P∑ϵj=±1∑j=1mϵjxjp∫BE∗+x∗,∑j=1mϵjxj∑j=1mϵjxjpdμx∗m/p≤1m!2mδp+P∑ϵj=±1∑j=1mxjm∫BE∗+x∗,ypdμx∗m/p≤1m!2mδp+P2mmm∫BE∗+x∗,ypdμx∗m/p≤mmm!δp+P∫BE∗+x∗,ypdμx∗m/p.
Forp<m, 1≤j≤m, and n∈ℕ, let x1j,…,xnj∈Ej satisfy xiji=1nℓp,weakn=1:(65)∑i=1nP^xi1,…,ximp/mm/p≤1m!2m∑i=1n∑ϵj=±1Pϵ1xi1,…,ϵmximp/mm/p≤1m!2mδp+P∑ϵj=±1ϵ1xi1,…,ϵmximi=1nℓp,weaknpm/p≤1m!2mδp+Pmm∑ϵj=±11m/p≤mmm!2m2/p−mδp+P.In Corollary2, we give an inclusion between the two classes of polynomials defined earlier.Corollary 2.
Let1<p,q<∞, and ℓ2n Banach lattice space; then,(66)Pd,p+l2nm;F⊆PCoh,q+l2nm;F,dq+P≤δp+P.Proof.
LetP∈Pd,p+l2nm;F; from Proposition 7, the associate symmetric of P denoted P^ belongs to ℒdp+l2nm;F, and by Theorem 5, P^ is in Dqm+l2nm;F, so Proposition 5 finishes the proof.


## 3.1. Positivep−Dominated m-Linear Operators

Definition 5.
Anm-linear operator T:E1×⋯×Em⟶Y is positive p−dominated if there is a positive constant C>0 such that(42)∑i=1nTxi1,…,ximp/mm/p≤Cxi1i=1nℓp,weaknE1⋯ximi=1nℓp,weaknEm,∀xiji=1n⊂Ej, 1≤j≤m and m∈ℕ. We denote ℒdp+E1,…Em;Y the space of all positive p−dominated m−linear operators and the infimum of all constants C verifying (42) is denoted πpd+T. For all p≥m, πpd+ is a norm on ℒdp+E1,…Em;Y, whereas for p<m, it is only a quasi-norm.
Form=1, the space of all positive p−dominated operators is nothing other than the space of positive p−summing operators.We should stress out that the above definition makes sense for Banach spacesY, but the more interesting case occurs when we consider a Banach lattice F, since this assumption allows to establish an inclusion relation between the classes of positive dominated multilinear operators and Cohen positive strongly summing multilinear operators and later on for polynomials.The next theorem is an analogous Pietsch’s domination-type theorem form-linear positive p-dominated operators. The proof of this result is an adaptation of that of Theorem 4.7 in [9], so we do not present it here.Theorem 4 (domination theorem for positivep-dominated multilinear operators).
Anm-linear operator T:E1×⋯×Em⟶Y is positive p−dominated if and only if there is Radon probability measures μj∈BEj∗+, 1≤j≤m and K>0, and(43)Tx1,…,xm≤K∏j=1m∫BEj∗+xj∗,xjpdμjxj∗1/p,for all xj∈Ej+. Moreover, the smallest constants K is πpd+T.The following theorem is a version of the Bu-theorem [12]; in [13], Achour and Mezrag proved an inclusion theorem between p-dominated operators and Cohen strongly q-summing m−linear operators; in our work, we prove a positive analogue of Theorem 3.2 in [13]. This theorem plays an important role in the proof of the relationship between Cohen positive strongly summing and positive dominated polynomials.Theorem 5.
Let1<p,q<∞. Let ℓ2n1,…,ℓ2nm and F be Banach lattices. Then,(44)ℒdp+ℓ2n1,…,ℓ2nm;F⊆Dqm+ℓ2n1,…,ℓ2nm;F,for all T∈ℒdp+ℓ2n1,…,ℓ2nm,Fdqm+T≤πpd+T.For the proof of this theorem, we need to recall the Khintchine [14] and the Kahane [13] multiple inequalities.


### 3.1.1. Multiple Khintchine’s Inequality

If0<p<∞, then(45)Apm∑i1,…,im=1n1,…,nmai1,…,im21/2≤∫01⋯∫01ri1t1⋯rimtmai1,…,impdt1⋯dtm1/p≤Bpm∑i1,…,im=1n1,…,nmai1,…,im21/2,for each choice of the scalars ai1,…,imi1,…,im=1n1,…,nm, ApandBp are the Khintchine constants, and by rmm∈ℕ, we denote the sequence of Rademacher function such that rn:0,1⟶ℝ as rnt=sin2nπt.


### 3.1.2. Multiple Kahane’s Inequality

Let1≤p<∞. Let Radpμ,X be the vector space of all almost unconditionally summable sequences xn in the Banach space X. It is a Banach space under the norm:(46)xnp=∫01∑n=1∞rntxnpdμt1/p.Let1≤p,q<∞. Let X be a Banach space and xi1,…,in be in X. We have(47)∑i1,…,inxi1,…,in⊗j=1nrijLp⊗1≤i≤nμi,X≤Kp,qn∑i1,…,inxi1,…,in⊗j=1nrijLq⊗1≤i≤nμi,X,where Kp,q is the simple constant of Kahane’s inequality μi=μ.Proof.
LetT∈ℒdp+ℓ2n1,…,ℓ2nm;F; then, by the domination theorem for positive p-dominated m-linear operators, there are probability measures μj on Bℓ2nj+, such that, for any x1,…,xm∈ℓ2n1×⋯×ℓ2nm, we have(48)Tx1,…,xm≤πpd+T∏j=1m∫Bℓ2nj+x∗,xjpdμjx∗1/p.
For everyxiji=1n⊂ℓ2nj, for j=1,…,m and yi∗i=1n⊂F∗, we have(49)xi1,…,xim=∑k1=1n1xi,k11ek1,…,∑km=1nmxi,kmmekm.
Then,(50)∑i=1nTxi1,…,xim,yi∗=∑i=1n∑k1,…,km=1n1,…,nmxi,k11⋯xi,kmmTek1,…,ekm,yi∗.
Using Hölder’s inequality, we obtain(51)∑i=1nTxi1,…,xim,yi∗≤∑i=1n∑k1=1n1xi,k1121/2⋯∑km=1nmxi,kmm21/2∑k1,…,km=1n1,…,nmTek1,…,ekm,yi∗21/2≤∑i=1n∏j=1mxijℓ2nj∑k1,…,km=1n1,…,nmTek1,…,ekm,yi∗21/2.
By the multiple Khintchine inequality, on the last inequality, we obtain(52)∑i=1nTxi1,…,xim,yi∗≤∑i=1n∏j=1mxijℓ2njAq∗m∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1≤Aq∗m∑i=1n∏j=1mxijq1/q∑i=1n∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1q∗1/q∗.
By Fubini’s theorem(53)∑i=1nTxi1,…,xim,yi∗≤Aq∗m∑i=1n∏j=1mxijq1/q∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1,ℓq∗n≤Aq∗m∑i=1n∏j=1mxijq1/qyi∗ℓq∗,weaknF∗∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1.
For the quantity,(54)∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1,F,using the multiple Kahane’s inequality(55)∑k1,…,km=1n1,…,nmrk1t1⋯rkmtmTek1,…,ekmLq∗0,1,F≤Kp,q∗m∑k1,…,km=1n1,…,nmrk1t1⋯rkmtmTek1,…,ekmLp0,1,F≤Kp,q∗m∑k1,…,km=1n1,…,nmTrk1t1ek1,…,rkmtmekmLp0,1,F≤πpd+T∫01∏j=1m∫Bℓ2nj+∑kj=1njx∗ekjrkjtjpdμjx∗dtj1/p≤πpd+T∏j=1m∫Bℓ2nj+∫01∑kj=1njx∗ekjrkjtjpdμjx∗dtj1/p.
Applying Khintchine’s inequality,(56)≤Bpmπpd+T∏j=1m∫Bℓ2nj+∑kj=1njx∗ekj2p/2dμjx∗1/p≤Bpmπpd+T∏j=1m∫Bℓ2nj+x∗ℓ2npdμjx∗1/p≤Bpmπpd+T.
Consequently,(57)∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1,F≤Kp,q∗mBpmπpd+T.
Exploiting (53) and (57) gives(58)∑i=1nTxi1,…,xim,yi∗≤Aq∗Kp,q∗BPmπpd+T∑i=1n∏j=1mxijq1/qyi∗i=1nℓq∗,weaknF∗.
Thus,T∈Dqm+ℓ2n1,…,ℓ2nm;F and dqm+T≤Aq∗Kp,q∗BPmπpd+T.
And, this completes the proof.


## 3.1.1. Multiple Khintchine’s Inequality

If0<p<∞, then(45)Apm∑i1,…,im=1n1,…,nmai1,…,im21/2≤∫01⋯∫01ri1t1⋯rimtmai1,…,impdt1⋯dtm1/p≤Bpm∑i1,…,im=1n1,…,nmai1,…,im21/2,for each choice of the scalars ai1,…,imi1,…,im=1n1,…,nm, ApandBp are the Khintchine constants, and by rmm∈ℕ, we denote the sequence of Rademacher function such that rn:0,1⟶ℝ as rnt=sin2nπt.


## 3.1.2. Multiple Kahane’s Inequality

Let1≤p<∞. Let Radpμ,X be the vector space of all almost unconditionally summable sequences xn in the Banach space X. It is a Banach space under the norm:(46)xnp=∫01∑n=1∞rntxnpdμt1/p.Let1≤p,q<∞. Let X be a Banach space and xi1,…,in be in X. We have(47)∑i1,…,inxi1,…,in⊗j=1nrijLp⊗1≤i≤nμi,X≤Kp,qn∑i1,…,inxi1,…,in⊗j=1nrijLq⊗1≤i≤nμi,X,where Kp,q is the simple constant of Kahane’s inequality μi=μ.Proof.
LetT∈ℒdp+ℓ2n1,…,ℓ2nm;F; then, by the domination theorem for positive p-dominated m-linear operators, there are probability measures μj on Bℓ2nj+, such that, for any x1,…,xm∈ℓ2n1×⋯×ℓ2nm, we have(48)Tx1,…,xm≤πpd+T∏j=1m∫Bℓ2nj+x∗,xjpdμjx∗1/p.
For everyxiji=1n⊂ℓ2nj, for j=1,…,m and yi∗i=1n⊂F∗, we have(49)xi1,…,xim=∑k1=1n1xi,k11ek1,…,∑km=1nmxi,kmmekm.
Then,(50)∑i=1nTxi1,…,xim,yi∗=∑i=1n∑k1,…,km=1n1,…,nmxi,k11⋯xi,kmmTek1,…,ekm,yi∗.
Using Hölder’s inequality, we obtain(51)∑i=1nTxi1,…,xim,yi∗≤∑i=1n∑k1=1n1xi,k1121/2⋯∑km=1nmxi,kmm21/2∑k1,…,km=1n1,…,nmTek1,…,ekm,yi∗21/2≤∑i=1n∏j=1mxijℓ2nj∑k1,…,km=1n1,…,nmTek1,…,ekm,yi∗21/2.
By the multiple Khintchine inequality, on the last inequality, we obtain(52)∑i=1nTxi1,…,xim,yi∗≤∑i=1n∏j=1mxijℓ2njAq∗m∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1≤Aq∗m∑i=1n∏j=1mxijq1/q∑i=1n∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1q∗1/q∗.
By Fubini’s theorem(53)∑i=1nTxi1,…,xim,yi∗≤Aq∗m∑i=1n∏j=1mxijq1/q∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekm,yi∗Lq∗0,1,ℓq∗n≤Aq∗m∑i=1n∏j=1mxijq1/qyi∗ℓq∗,weaknF∗∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1.
For the quantity,(54)∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1,F,using the multiple Kahane’s inequality(55)∑k1,…,km=1n1,…,nmrk1t1⋯rkmtmTek1,…,ekmLq∗0,1,F≤Kp,q∗m∑k1,…,km=1n1,…,nmrk1t1⋯rkmtmTek1,…,ekmLp0,1,F≤Kp,q∗m∑k1,…,km=1n1,…,nmTrk1t1ek1,…,rkmtmekmLp0,1,F≤πpd+T∫01∏j=1m∫Bℓ2nj+∑kj=1njx∗ekjrkjtjpdμjx∗dtj1/p≤πpd+T∏j=1m∫Bℓ2nj+∫01∑kj=1njx∗ekjrkjtjpdμjx∗dtj1/p.
Applying Khintchine’s inequality,(56)≤Bpmπpd+T∏j=1m∫Bℓ2nj+∑kj=1njx∗ekj2p/2dμjx∗1/p≤Bpmπpd+T∏j=1m∫Bℓ2nj+x∗ℓ2npdμjx∗1/p≤Bpmπpd+T.
Consequently,(57)∑k1,…,km=1n1,…,nmrk1⋯rkmTek1,…,ekmLq∗0,1,F≤Kp,q∗mBpmπpd+T.
Exploiting (53) and (57) gives(58)∑i=1nTxi1,…,xim,yi∗≤Aq∗Kp,q∗BPmπpd+T∑i=1n∏j=1mxijq1/qyi∗i=1nℓq∗,weaknF∗.
Thus,T∈Dqm+ℓ2n1,…,ℓ2nm;F and dqm+T≤Aq∗Kp,q∗BPmπpd+T.
And, this completes the proof.


## 3.2. Positivep−Dominated m−Homogeneous Polynomials

Definition 6.
Let1≤p<∞; an m−homogeneous polynomial P:E⟶Y is positive p− dominated if ∃C>0 such that, for any n∈ℕ∗ and any xii=1n⊂E+,(59)∑i=1n‖Pxi‖p/mm/p≤Csupx∗∈BE∗+∑i=1nx∗,xipm/p.
We denotePd,p+Em;Y the space of positive p−dominated polynomials from E into Y, and by δp+⋅, the norm defined by the infimum of any constants verifying the above inquality, for p≥m, but for p<m, it is only a quasi-norm.Theorem 6.
(domination theorem for positivep−dominated m-homogeneous polynomials). Let 1≤p<∞; an m−homogeneous polynomial P:E⟶Y is positive p−dominated if there are K>0 and a probability measure μ on BE∗+ such that(60)Px≤K∫BE∗+x∗,xpdμx∗m/p,∀x∈E+. Moreover, the smallest K is δp+P.In Proposition7, we study the relationship between positive p−dominated m−linear operators and positive p−dominated m−homogeneous polynomialsProposition 7.
LetP:E⟶Ybe an m−homogeneous polynomial and P^ be its corresponding symmetric m−linear operator such that Px=P^x,…,x.(1)
IfP^ is positive p−dominated operator, then P is p−dominated and(61)δp+P≤πpd+P^.(2)
IfP is positive p−dominated, then P^ is positive p−dominated and(a)
Forp≥m, πpd+P^≤mm/m!δp+P(b)
Forp<m, πpd+P^≤mm/m!2m2/p−mδp+PProof.
(1) Suppose thatP^ is positive p−dominated; let xii=1n⊂E+; then,(62)∑i=1nPxip/mm/p=∑i=1nP^xi,…,xip/mm/p≤πpd+P^xii=1nℓp,weaknEm.
Therefore,P is positive p−dominated and δp+P≤πpd+P^.
(2) For1≤j≤m and n∈ℕ, let xj∈Ej satisfy xj≤1:(63)P^x1,…,xmp/mm/p=1m!2m∑ϵj=±1ϵ1⋯ϵmP∑j=1mϵjxjp/mm/p≤1m!2m∑ϵj=±1P∑j=1mϵjxjp/mm/p.
By Theorem6,(64)≤1m!2mδp+P∑ϵj=±1∫BE∗+x∗,∑j=1mϵjxjpdμx∗m/p≤1m!2mδp+P∑ϵj=±1∑j=1mϵjxjp∫BE∗+x∗,∑j=1mϵjxj∑j=1mϵjxjpdμx∗m/p≤1m!2mδp+P∑ϵj=±1∑j=1mxjm∫BE∗+x∗,ypdμx∗m/p≤1m!2mδp+P2mmm∫BE∗+x∗,ypdμx∗m/p≤mmm!δp+P∫BE∗+x∗,ypdμx∗m/p.
Forp<m, 1≤j≤m, and n∈ℕ, let x1j,…,xnj∈Ej satisfy xiji=1nℓp,weakn=1:(65)∑i=1nP^xi1,…,ximp/mm/p≤1m!2m∑i=1n∑ϵj=±1Pϵ1xi1,…,ϵmximp/mm/p≤1m!2mδp+P∑ϵj=±1ϵ1xi1,…,ϵmximi=1nℓp,weaknpm/p≤1m!2mδp+Pmm∑ϵj=±11m/p≤mmm!2m2/p−mδp+P.In Corollary2, we give an inclusion between the two classes of polynomials defined earlier.Corollary 2.
Let1<p,q<∞, and ℓ2n Banach lattice space; then,(66)Pd,p+l2nm;F⊆PCoh,q+l2nm;F,dq+P≤δp+P.Proof.
LetP∈Pd,p+l2nm;F; from Proposition 7, the associate symmetric of P denoted P^ belongs to ℒdp+l2nm;F, and by Theorem 5, P^ is in Dqm+l2nm;F, so Proposition 5 finishes the proof.


## 4. Applications

LetK be a compact Hausdroff space and K,A,μ a finite measure space.As usualCK denotes the space of continuous functions on K, and ℳK denotes the Banach space of all regular Borel measures on K, and we have ℳK≅CK∗.We shall writeLpμ,Y for the space of measurables functions on K with(67)fp=∫Kftpdμ1/p,for1≤p<+∞,and for p=+∞,(68)f∞=infM≥0:ft≤Mμalmost every where.Let1≤p<∞. An operator u:E⟶X is said to be p−concave if there exists a constant C>0 such that, for all n∈ℕ,x1,…,xn∈E, the inequality(69)uxii=1np≤C∑i=1nxip1/pholds.LetCpE;X be the space of p−concave operators from E into X. CpE,X becomes a Banach space with norm defined by the infimum of the constant C>0 that satisfy inequality (69).The following results are a conclusion of the work done by Blasco in [8] and Proposition 3.Proposition 8.
Let1<p<∞:(1)
P∈Pcoh,p+Xm,L∞μ if and only if P∗∈ℒL1μ,PXm(2)
P∈Pcoh,p+Xm,ℳK if and only if P∗∈CpCK,PXm(3)
P∈Pcoh,p+Xm,Lpμ if and only if P∗∈Lp∗μ,PXm(4)
P∈Pcoh,p+Xm,ℓp∗ if and only if P∗∈ℓp∗PXm


## 5. Tensorial Approach

This section is intended to present a tensorial perspective of Cohen positive stronglyp−summing and positive p−dominated polynomials and m-linear operators. For doing this, we apply the standard technique of associate a linear functional on a tensor product to each multilinear operator or polynomial. In the case of Cohen positive strongly p−summing operators, we pay more attention on the details for polynomials than m-linear operators. Later, we pay more attention on positive p−dominated m-linear operators than polynomials. Let m be a nonnegative integer, X be a Banach space, and F be a Banach lattice, each m−homogeneous polynomial P:X⟶F defines a functional given by(70)ϕP:⊗smX⊗F∗⟶ℝ,x⊗⋯⊗x⊗y∗⟶y∗Px.The next step in our development is to define adequate normα on the tensor space ⊗smX⊗F∗ in order that P be Cohen positive strongly p−summing (positive p−dominated) exactly when ϕP is bounded on ⊗smX⊗F∗,α.


### 5.1. Cohen Positive Stronglyp−Summing Polynomials from a Tensor Viewpoint

Before introducing the respecting tensor norm, let us state an equivalence of being a Cohen positive stronglyp−summing polynomial. This is an immediate consequence of the duality between ℓ∞n and ℓ1n. The m-homogeneous polynomial P:X⟶F is Cohen positive strongly p−summing if and only if ∃C>0 such that(71)∑i=1nλiyi∗Pxi≤λii=1nℓ∞nximi=1nℓpnyi∗i=1nℓp∗,weaknF∗,for all finite sequences λii=1n⊂ℝ, xii=1n⊂X, and yi∗i=1n⊂F∗. In this case, dp+P agrees with the sharpest constant C so that the inequality holds.Definition 7.
For eachz in ⊗smX⊗F, we define(72)λp+z=infλii=1nℓ∞nxii=1nℓpnyii=1nℓp∗,weaknF,where the infimum is taken over any representations z=∑i=1nλixi⊗⋯⊗xi⊗yi. For each z in ⊗smX⊗F, we define(73)Λp+z=infλii=1nℓ∞nxii=1nℓpnwp∗yii=1n,where the infimum is taken over all representations z=∑i=1nλixi⊗⋯⊗xi⊗yi with yi≥0, for all 1≤i≤n.The next proposition is a consequence of the fact that everyy∈F can be represented as y=y+−y−. Moreover, it helps to reduce the calculations related to λp+ and Λp+.Proposition 9.
For everyz in ⊗smX⊗F, we have(74)λp+z≤Λp+z≤2λp+z.Proof.
Letz in ⊗smX⊗F. It is clear that λp+z≤Λp+z by (4). For the opposite inequality, we take z=∑i=1nλixi⊗⋯⊗xi⊗yi. Then,(75)z=∑i=1nλixi⊗⋯⊗xi⊗yi++∑i=1n−λixi⊗⋯⊗xi⊗yi−.
Thus, we obtain a new representation forz such that(76)Λp+z≤21/pλii=1n∞xii=1nℓpnsupw∗∈BF∗+∑i=1nw∗yi+p∗+w∗yi−p∗%1/p∗≤21/pλii=1n∞xii=1nℓpnsupw∗∈BF∗+∑i=1n2w∗yip∗1/p∗≤21/p+1/p∗λii=1n∞xii=1nℓpnsupw∗∈BF∗+∑i=1nw∗yip∗1/p∗≤2λii=1n∞xii=1nℓpnyii=1nℓp∗,weakn.
Therefore,Λp+z≤2λp+z, for all z.Proposition 10.
The applicationz⟶λp+z is a norm on ⊗smX⊗F.Proof.
Standard tensorial techniques can be applied to prove thatλp+ is nonnegative, homogeneous and verify the triangular inequality. So, we only prove that λp+z=0 implies z=0. Let ξ be a bounded linear functional on ⊗^π,smX and y∗≥0 bounded on F∗. Then,(77)ξ⊗y∗z=∑i=1nλiξxi⊗⋯⊗xiy∗yi≤λii=1n∞∑i=1nξxi⊗⋯⊗xiy∗yi≤λii=1n∞∑i=1nξxi⊗⋯⊗xip1/p∑i=1ny∗yip∗1/p∗≤ξλii=1n∞∑i=1nximp1/p∑i=1ny∗yip∗1/p∗≤ξy∗λiin∞ximi=1nℓpnwp∗yii=1n.
Hence,λp+z=0 implies ξ⊗y∗z=0, for all ξ and y∗≥0. For arbitrary y∗, we have that y∗=y∗+−y∗−. Then, λp+z=0 implies(78)ξ⊗y∗z=ξ⊗y∗+z−ξ⊗y∗−z=0.
Therefore,z must be zero since the set of linear functionals of the form ξ⊗y∗ is a separating set for ⊗^π,smX⊗F∗.Next, we give the characterization of Cohen positive stronglyp-summing polynomials in tensor terms.Proposition 11 (duality).
LetP:X⟶F be an m−homogeneous polynomial. The following are equivalent:(i)
P is Cohen positive strongly p-summing(ii)
ϕP is bounded on ⊗smX⊗F∗,λp+.
Under this circumstances,dp+P=ϕP.Proof.
Letz=∑i=1nλixi⊗⋯⊗xi⊗yi∗ be any element of ⊗smX⊗F∗; then,(79)ϕPz=∑i=1nλiy∗Pxi≤dp+Pλiin∞ximi=1nℓpnyi∗i=1nℓp,weaknF∗.
After taking the infimum over all the representations ofz, we have that(80)ϕPz≤dp+Pλp+z,Holds, for all z.
Conversely, take sequencesλii=1n in ℝ, xii=1n in X, and yi∗i=1n in F∗. Hence,(81)∑i=1nλiy∗Pxi=ϕPz≤ϕPλp+∑i=1nλixi⊗⋯⊗xi⊗yi∗≤ϕPzλiin∞ximi=1nℓpnyi∗i=1nℓp,weaknF∗.As we saw, the normsλp+ and Λp+ are equivalent, and one of them is more useful than the other depending on the circumstances we are interested on. The norm λp+ provides an isometry between polynomials and linear functionals, while Λp+ reduces calculations in Proposition 12.Proposition 12 (uniform property).
LetR:X⟶Y be a bounded linear operator and S:F⟶E be a positive operator. Then, the operator defined by(82)R⊗S:⊗smX⊗F,Λp+⟶⊗smY⊗E,Λp+,x⊗⋯⊗x⊗y↦Rx⊗⋯⊗Rx⊗Sy,is bounded and R⊗S≤RmS.Proof.
Letz=∑i=1nλixi⊗⋯⊗xi⊗yi with yi≥0. Then,(83)Λp+R⊗Sz≤λiin∞Rxi⊗Rxii=1nℓpnwp∗Syi≤RmSλiin∞ximi=1nℓpnwp∗yii=1n.
After taking the infimum over all the representations ofz, we obtain Λp+R⊗Sz≤RmSΛp+z.As it is expected, previous proposition is also true ifΛp+ is remplaced by λp+, but this change involves a factor of 2 in the inequality.


### 5.2. Cohen Positive Stronglyp-Summing m-Linear Operators from a Tensor Viewpoint

Bearing in mind the previous section, we are able to develop analogous ideas to characterize Cohen positive stronglyp−summing m-linear operators. As before, throughout this section, X1,…,Xm are Banach spaces and F is a Banach lattice. Next, recall that each m-linear operator T:X1×⋯×Xm⟶F defines a linear functional by(84)ϕT:X1⊗⋯⊗Xm⊗F∗⟶ℝ,x1⊗⋯⊗xm⊗y∗↦y∗Tx1,…,xm.In this case, for eachz in X1⊗⋯⊗Xm⊗F, we define(85)λpm+z=infλiin∞xi1⋯ximi=1nℓpnyii=1nℓp∗weaknF,where the infimum is taken over all representations, z=∑i=1nλixi1⊗⋯⊗xim⊗yi.Using usual tensor techniques and slight modifications of the proof of Proposition10, let us prove that the assignment z↦λpm+z is a norm on X1⊗⋯⊗Xm⊗F. The respective duality result is exhibited in the next proposition and requires an analogous expression to (71).Proposition 13 (duality).
LetT:X1×⋯×Xm⟶F be an m-linear operator. Then, the following are equivalents:(i)
T is Cohen positive strongly p-summing(ii)
ϕT is bounded on X1⊗⋯⊗Xm⊗F∗,λpm+
Under this circumstances,dpm+T=ϕT.Proof.
For anyz=∑i=1nλixi1⊗⋯⊗xim⊗yi∗ in X1⊗⋯⊗Xm⊗F∗, we have(86)ϕTz=∑i=1nλiyi∗Txi1,…,xim≤dpm+Tλiin∞xi1⋯ximi=1nℓpnyi∗i=1nℓp∗,weaknF∗.
Hence, for anyz, we have(87)ϕTz≤dpm+Tλpm+z.
On the other direction, letλii=1n in ℝ, xiji=1n in Xj, and yi∗i=1n in F∗. Then,(88)∑i=1nλiyi∗Txi1,…,xim=ϕTz≤ϕTλpm+∑i=1nλixi1⊗⋯⊗xim⊗yi∗≤ϕTλiin∞xi1⋯ximi=1nℓpnyi∗i=1nℓp∗,weaknF∗.For eachz, in X1⊗⋯⊗Xm⊗F, we define(89)Λpm+z=infλiin∞xi1⋯ximi=1nℓpnwp∗yii=1n,where the infimum is selected over any representation z=∑i=1nλixi1⊗⋯⊗xim⊗yi such that yi≥0, for all 1≤i≤n. Then, for any z in X1⊗⋯⊗Xm⊗F, we have(90)λpm+z≤Λpm+z≤2λpm+z.HavingΛpm+ at our reach and knowing it defines a norm, we present the respective uniform property it verifies.Proposition 14 (uniform property).
LetRi:Xi⟶Yi be bounded linear operators and S:F⟶G be positive operator between Banach lattices. Then, the operator,(91)R1⊗⋯⊗Rm⊗S:X1⊗⋯⊗Xm⊗F,Λpm+⟶Y1⊗⋯⊗Ym⊗G,Λpm+,x1⊗⋯⊗xm⊗y↦R1x1⊗⋯⊗Rmxm⊗Sy,is bounded and R1⊗⋯⊗Rm⊗S≤R1⋯RmS.Proof.
Givenz=∑i=1nλixi1⊗⋯⊗xim⊗yi with yi≥0, we have(92)Λpm+R1⊗⋯⊗Rm⊗Sz≤λiin∞‖R1xi1⋯Rmximi=1n‖ℓpnwpyii=1n≤R1⋯RmSλiin∞xi1⋯ximi=1nℓpnwpyii=1n.
As a consequence,Λpm+R1⊗⋯⊗Rm⊗Sz≤R1⋯RmSΛpm+z.


### 5.3. Positivep-Dominated m-Linear Operators and Polynomials Form a Tensor Viewpoint

As before, throughout this section,E,E1,…,Em are Banach lattices and Y is a Banach space. In this case, we pay more attention on m-linear operators than polynomials.Proposition 15.
For eachz in E1⊗⋯⊗Em⊗Y, we define(93)ηpm+z=inf∏j=1mxiji=1nℓp,weaknEjyii=1nℓp/p−mn,where the infimum is selected over any representations, z=∑i=1nxi1⊗⋯⊗xim⊗yi.
For eachz in ⊗smE⊗Y, we define(94)ηp+z=infwpxii=1nmyii=1nℓp/p−mn,where the infimum is taken over all representations z=∑i=1nxi⊗⋯⊗xi⊗yi with xi≥0, for all 1≤i≤n. Then, the assignment z↦ηp+z is a norm on ⊗smE⊗Y.The proof of this proposition is an adaptation of Proposition10 The next result concerns the characterization of positive p-dominated m-linear operators and polynomial in tensor terms.Proposition 16 (duality).
LetT:E1×⋯×Em⟶Y be an m-linear operator and P:E⟶Y be an m-homogeneous polynomial. Then,(i)
T is positive p-dominated if and only if ϕT is bounded on E1⊗⋯⊗Em⊗Y∗,ηpm+(ii)
P is positive p-dominated if and only if ϕP is bounded on ⊗smE⊗Y∗,ηp+
Under this circumstances,πpd+T=ϕT and δp+P=ϕP.Proof.
We only provei since ii is a straightforward adaptation. First, suppose T is positive p-dominated. Then, for any z=∑i=1nxi1⊗⋯⊗xim⊗yi∗, in E1⊗⋯⊗Em⊗Y∗, we have(95)ϕTz≤∑i=1nyi∗Txi1,…,xim≤∑i=1nyi∗p/p−mp−m/p∑i=1nTxi1,…,ximp/mm/p≤πpd+T∏j=1mxiji=1nℓp,weaknEjyi∗i=1nℓp/p−mn.
Hence,ϕTz≤πpd+Tηpm+z which means that ϕT is bounded and ϕT≤πpd+T, for the opposite direction. Let xiji=1n in Ej1≤j≤m and ϵ>0. Then, for each 1≤i≤n, there exists yi∗ in Y∗ such that yi∗Txi1,…,xim=Txi1,…,ximp/m and yi∗≤1+ϵTxi1,…,ximp/m−1. Therefore,(96)∑i=1nTxi1,…,ximp/m=∑i=1nyi∗Txi1,…,xim=ϕT∑i=1nxi1⊗⋯⊗xim⊗yi∗≤ϕT∏j=1mxiji=1nℓp,weaknEjyii=1nℓp/p−mn≤ϕT1+ϵ∏j=1mxiji=1nℓp,weaknEj∑i=1nTxi1,…,ximp/mp−m/p.
As a consequence, we obtain(97)∑i=1nTxi1,…,ximp/mm/p≤ϕT1+ϵ∏j=1mxiji=1nℓp,weaknEj,which implies that T is positive p-dominated and πpd+T≤ϕT.The proof of next propositions is adaptations of those of Propositions9 and 11.Proposition 17.
Define, for eachz, in E1⊗⋯⊗Em⊗Y,(98)Epm+=inf∏i=1nwpxiji=1nyii=1nℓp/p−mn,where the infimum is taken over all representation z=∑i=1nxi1⊗⋯⊗xim⊗yi with xij≥0, for all 1≤i≤n and 1≤j≤m. Then,(99)ηpm+z≤Epm+z≤2ηpm+z,Holds, for all z, in E1⊗⋯⊗Em⊗Y.Proposition 18 (uniform property).
LetRi:Ei⟶Mi be a positive operators and S:Y⟶Z be a bounded linear operator. Then, the operator,(100)R1⊗⋯⊗Rm⊗S:E1⊗⋯⊗Em⊗Y,Epm+⟶M1⊗⋯⊗Mm⊗Z,Epm+,x1⊗⋯⊗xm⊗y↦R1x1⊗⋯⊗Rmxm⊗Sy,is bounded andR1⊗⋯⊗Rm⊗S≤R1⋯RmS.LetR:E⟶M be a positive operator and S:Y⟶Z be a bounded linear operator. Then, the operator defined by(101)R⊗S:⊗smE⊗Y,ηp+⟶⊗smM⊗Z,ηp+,x⊗⋯⊗x⊗y↦Rx⊗⋯⊗Rx⊗Sy,is bounded and R⊗S≤RmS.


## 5.1. Cohen Positive Stronglyp−Summing Polynomials from a Tensor Viewpoint

Before introducing the respecting tensor norm, let us state an equivalence of being a Cohen positive stronglyp−summing polynomial. This is an immediate consequence of the duality between ℓ∞n and ℓ1n. The m-homogeneous polynomial P:X⟶F is Cohen positive strongly p−summing if and only if ∃C>0 such that(71)∑i=1nλiyi∗Pxi≤λii=1nℓ∞nximi=1nℓpnyi∗i=1nℓp∗,weaknF∗,for all finite sequences λii=1n⊂ℝ, xii=1n⊂X, and yi∗i=1n⊂F∗. In this case, dp+P agrees with the sharpest constant C so that the inequality holds.Definition 7.
For eachz in ⊗smX⊗F, we define(72)λp+z=infλii=1nℓ∞nxii=1nℓpnyii=1nℓp∗,weaknF,where the infimum is taken over any representations z=∑i=1nλixi⊗⋯⊗xi⊗yi. For each z in ⊗smX⊗F, we define(73)Λp+z=infλii=1nℓ∞nxii=1nℓpnwp∗yii=1n,where the infimum is taken over all representations z=∑i=1nλixi⊗⋯⊗xi⊗yi with yi≥0, for all 1≤i≤n.The next proposition is a consequence of the fact that everyy∈F can be represented as y=y+−y−. Moreover, it helps to reduce the calculations related to λp+ and Λp+.Proposition 9.
For everyz in ⊗smX⊗F, we have(74)λp+z≤Λp+z≤2λp+z.Proof.
Letz in ⊗smX⊗F. It is clear that λp+z≤Λp+z by (4). For the opposite inequality, we take z=∑i=1nλixi⊗⋯⊗xi⊗yi. Then,(75)z=∑i=1nλixi⊗⋯⊗xi⊗yi++∑i=1n−λixi⊗⋯⊗xi⊗yi−.
Thus, we obtain a new representation forz such that(76)Λp+z≤21/pλii=1n∞xii=1nℓpnsupw∗∈BF∗+∑i=1nw∗yi+p∗+w∗yi−p∗%1/p∗≤21/pλii=1n∞xii=1nℓpnsupw∗∈BF∗+∑i=1n2w∗yip∗1/p∗≤21/p+1/p∗λii=1n∞xii=1nℓpnsupw∗∈BF∗+∑i=1nw∗yip∗1/p∗≤2λii=1n∞xii=1nℓpnyii=1nℓp∗,weakn.
Therefore,Λp+z≤2λp+z, for all z.Proposition 10.
The applicationz⟶λp+z is a norm on ⊗smX⊗F.Proof.
Standard tensorial techniques can be applied to prove thatλp+ is nonnegative, homogeneous and verify the triangular inequality. So, we only prove that λp+z=0 implies z=0. Let ξ be a bounded linear functional on ⊗^π,smX and y∗≥0 bounded on F∗. Then,(77)ξ⊗y∗z=∑i=1nλiξxi⊗⋯⊗xiy∗yi≤λii=1n∞∑i=1nξxi⊗⋯⊗xiy∗yi≤λii=1n∞∑i=1nξxi⊗⋯⊗xip1/p∑i=1ny∗yip∗1/p∗≤ξλii=1n∞∑i=1nximp1/p∑i=1ny∗yip∗1/p∗≤ξy∗λiin∞ximi=1nℓpnwp∗yii=1n.
Hence,λp+z=0 implies ξ⊗y∗z=0, for all ξ and y∗≥0. For arbitrary y∗, we have that y∗=y∗+−y∗−. Then, λp+z=0 implies(78)ξ⊗y∗z=ξ⊗y∗+z−ξ⊗y∗−z=0.
Therefore,z must be zero since the set of linear functionals of the form ξ⊗y∗ is a separating set for ⊗^π,smX⊗F∗.Next, we give the characterization of Cohen positive stronglyp-summing polynomials in tensor terms.Proposition 11 (duality).
LetP:X⟶F be an m−homogeneous polynomial. The following are equivalent:(i)
P is Cohen positive strongly p-summing(ii)
ϕP is bounded on ⊗smX⊗F∗,λp+.
Under this circumstances,dp+P=ϕP.Proof.
Letz=∑i=1nλixi⊗⋯⊗xi⊗yi∗ be any element of ⊗smX⊗F∗; then,(79)ϕPz=∑i=1nλiy∗Pxi≤dp+Pλiin∞ximi=1nℓpnyi∗i=1nℓp,weaknF∗.
After taking the infimum over all the representations ofz, we have that(80)ϕPz≤dp+Pλp+z,Holds, for all z.
Conversely, take sequencesλii=1n in ℝ, xii=1n in X, and yi∗i=1n in F∗. Hence,(81)∑i=1nλiy∗Pxi=ϕPz≤ϕPλp+∑i=1nλixi⊗⋯⊗xi⊗yi∗≤ϕPzλiin∞ximi=1nℓpnyi∗i=1nℓp,weaknF∗.As we saw, the normsλp+ and Λp+ are equivalent, and one of them is more useful than the other depending on the circumstances we are interested on. The norm λp+ provides an isometry between polynomials and linear functionals, while Λp+ reduces calculations in Proposition 12.Proposition 12 (uniform property).
LetR:X⟶Y be a bounded linear operator and S:F⟶E be a positive operator. Then, the operator defined by(82)R⊗S:⊗smX⊗F,Λp+⟶⊗smY⊗E,Λp+,x⊗⋯⊗x⊗y↦Rx⊗⋯⊗Rx⊗Sy,is bounded and R⊗S≤RmS.Proof.
Letz=∑i=1nλixi⊗⋯⊗xi⊗yi with yi≥0. Then,(83)Λp+R⊗Sz≤λiin∞Rxi⊗Rxii=1nℓpnwp∗Syi≤RmSλiin∞ximi=1nℓpnwp∗yii=1n.
After taking the infimum over all the representations ofz, we obtain Λp+R⊗Sz≤RmSΛp+z.As it is expected, previous proposition is also true ifΛp+ is remplaced by λp+, but this change involves a factor of 2 in the inequality.


## 5.2. Cohen Positive Stronglyp-Summing m-Linear Operators from a Tensor Viewpoint

Bearing in mind the previous section, we are able to develop analogous ideas to characterize Cohen positive stronglyp−summing m-linear operators. As before, throughout this section, X1,…,Xm are Banach spaces and F is a Banach lattice. Next, recall that each m-linear operator T:X1×⋯×Xm⟶F defines a linear functional by(84)ϕT:X1⊗⋯⊗Xm⊗F∗⟶ℝ,x1⊗⋯⊗xm⊗y∗↦y∗Tx1,…,xm.In this case, for eachz in X1⊗⋯⊗Xm⊗F, we define(85)λpm+z=infλiin∞xi1⋯ximi=1nℓpnyii=1nℓp∗weaknF,where the infimum is taken over all representations, z=∑i=1nλixi1⊗⋯⊗xim⊗yi.Using usual tensor techniques and slight modifications of the proof of Proposition10, let us prove that the assignment z↦λpm+z is a norm on X1⊗⋯⊗Xm⊗F. The respective duality result is exhibited in the next proposition and requires an analogous expression to (71).Proposition 13 (duality).
LetT:X1×⋯×Xm⟶F be an m-linear operator. Then, the following are equivalents:(i)
T is Cohen positive strongly p-summing(ii)
ϕT is bounded on X1⊗⋯⊗Xm⊗F∗,λpm+
Under this circumstances,dpm+T=ϕT.Proof.
For anyz=∑i=1nλixi1⊗⋯⊗xim⊗yi∗ in X1⊗⋯⊗Xm⊗F∗, we have(86)ϕTz=∑i=1nλiyi∗Txi1,…,xim≤dpm+Tλiin∞xi1⋯ximi=1nℓpnyi∗i=1nℓp∗,weaknF∗.
Hence, for anyz, we have(87)ϕTz≤dpm+Tλpm+z.
On the other direction, letλii=1n in ℝ, xiji=1n in Xj, and yi∗i=1n in F∗. Then,(88)∑i=1nλiyi∗Txi1,…,xim=ϕTz≤ϕTλpm+∑i=1nλixi1⊗⋯⊗xim⊗yi∗≤ϕTλiin∞xi1⋯ximi=1nℓpnyi∗i=1nℓp∗,weaknF∗.For eachz, in X1⊗⋯⊗Xm⊗F, we define(89)Λpm+z=infλiin∞xi1⋯ximi=1nℓpnwp∗yii=1n,where the infimum is selected over any representation z=∑i=1nλixi1⊗⋯⊗xim⊗yi such that yi≥0, for all 1≤i≤n. Then, for any z in X1⊗⋯⊗Xm⊗F, we have(90)λpm+z≤Λpm+z≤2λpm+z.HavingΛpm+ at our reach and knowing it defines a norm, we present the respective uniform property it verifies.Proposition 14 (uniform property).
LetRi:Xi⟶Yi be bounded linear operators and S:F⟶G be positive operator between Banach lattices. Then, the operator,(91)R1⊗⋯⊗Rm⊗S:X1⊗⋯⊗Xm⊗F,Λpm+⟶Y1⊗⋯⊗Ym⊗G,Λpm+,x1⊗⋯⊗xm⊗y↦R1x1⊗⋯⊗Rmxm⊗Sy,is bounded and R1⊗⋯⊗Rm⊗S≤R1⋯RmS.Proof.
Givenz=∑i=1nλixi1⊗⋯⊗xim⊗yi with yi≥0, we have(92)Λpm+R1⊗⋯⊗Rm⊗Sz≤λiin∞‖R1xi1⋯Rmximi=1n‖ℓpnwpyii=1n≤R1⋯RmSλiin∞xi1⋯ximi=1nℓpnwpyii=1n.
As a consequence,Λpm+R1⊗⋯⊗Rm⊗Sz≤R1⋯RmSΛpm+z.


## 5.3. Positivep-Dominated m-Linear Operators and Polynomials Form a Tensor Viewpoint

As before, throughout this section,E,E1,…,Em are Banach lattices and Y is a Banach space. In this case, we pay more attention on m-linear operators than polynomials.Proposition 15.
For eachz in E1⊗⋯⊗Em⊗Y, we define(93)ηpm+z=inf∏j=1mxiji=1nℓp,weaknEjyii=1nℓp/p−mn,where the infimum is selected over any representations, z=∑i=1nxi1⊗⋯⊗xim⊗yi.
For eachz in ⊗smE⊗Y, we define(94)ηp+z=infwpxii=1nmyii=1nℓp/p−mn,where the infimum is taken over all representations z=∑i=1nxi⊗⋯⊗xi⊗yi with xi≥0, for all 1≤i≤n. Then, the assignment z↦ηp+z is a norm on ⊗smE⊗Y.The proof of this proposition is an adaptation of Proposition10 The next result concerns the characterization of positive p-dominated m-linear operators and polynomial in tensor terms.Proposition 16 (duality).
LetT:E1×⋯×Em⟶Y be an m-linear operator and P:E⟶Y be an m-homogeneous polynomial. Then,(i)
T is positive p-dominated if and only if ϕT is bounded on E1⊗⋯⊗Em⊗Y∗,ηpm+(ii)
P is positive p-dominated if and only if ϕP is bounded on ⊗smE⊗Y∗,ηp+
Under this circumstances,πpd+T=ϕT and δp+P=ϕP.Proof.
We only provei since ii is a straightforward adaptation. First, suppose T is positive p-dominated. Then, for any z=∑i=1nxi1⊗⋯⊗xim⊗yi∗, in E1⊗⋯⊗Em⊗Y∗, we have(95)ϕTz≤∑i=1nyi∗Txi1,…,xim≤∑i=1nyi∗p/p−mp−m/p∑i=1nTxi1,…,ximp/mm/p≤πpd+T∏j=1mxiji=1nℓp,weaknEjyi∗i=1nℓp/p−mn.
Hence,ϕTz≤πpd+Tηpm+z which means that ϕT is bounded and ϕT≤πpd+T, for the opposite direction. Let xiji=1n in Ej1≤j≤m and ϵ>0. Then, for each 1≤i≤n, there exists yi∗ in Y∗ such that yi∗Txi1,…,xim=Txi1,…,ximp/m and yi∗≤1+ϵTxi1,…,ximp/m−1. Therefore,(96)∑i=1nTxi1,…,ximp/m=∑i=1nyi∗Txi1,…,xim=ϕT∑i=1nxi1⊗⋯⊗xim⊗yi∗≤ϕT∏j=1mxiji=1nℓp,weaknEjyii=1nℓp/p−mn≤ϕT1+ϵ∏j=1mxiji=1nℓp,weaknEj∑i=1nTxi1,…,ximp/mp−m/p.
As a consequence, we obtain(97)∑i=1nTxi1,…,ximp/mm/p≤ϕT1+ϵ∏j=1mxiji=1nℓp,weaknEj,which implies that T is positive p-dominated and πpd+T≤ϕT.The proof of next propositions is adaptations of those of Propositions9 and 11.Proposition 17.
Define, for eachz, in E1⊗⋯⊗Em⊗Y,(98)Epm+=inf∏i=1nwpxiji=1nyii=1nℓp/p−mn,where the infimum is taken over all representation z=∑i=1nxi1⊗⋯⊗xim⊗yi with xij≥0, for all 1≤i≤n and 1≤j≤m. Then,(99)ηpm+z≤Epm+z≤2ηpm+z,Holds, for all z, in E1⊗⋯⊗Em⊗Y.Proposition 18 (uniform property).
LetRi:Ei⟶Mi be a positive operators and S:Y⟶Z be a bounded linear operator. Then, the operator,(100)R1⊗⋯⊗Rm⊗S:E1⊗⋯⊗Em⊗Y,Epm+⟶M1⊗⋯⊗Mm⊗Z,Epm+,x1⊗⋯⊗xm⊗y↦R1x1⊗⋯⊗Rmxm⊗Sy,is bounded andR1⊗⋯⊗Rm⊗S≤R1⋯RmS.LetR:E⟶M be a positive operator and S:Y⟶Z be a bounded linear operator. Then, the operator defined by(101)R⊗S:⊗smE⊗Y,ηp+⟶⊗smM⊗Z,ηp+,x⊗⋯⊗x⊗y↦Rx⊗⋯⊗Rx⊗Sy,is bounded and R⊗S≤RmS.


## 6. Conclusion

In this work, by giving the concepts of Cohen positive stronglyp-summing and positive p-dominated m-homogeneous polynomials. We prove a version of Pietsch’s domination theorem for the first class among other results and a Bu-type theorem, as well as some inclusions with other known spaces. Moreover, we present a characterization of these classes in tensor terms. In the future work, we will try to extend the same study with other higher polynomial order.



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