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Hamdi, Halima; García-Hernández, Samuel; Belacel, Amar; Ouchenane, Djamel; Zubair, Sulima Ahmed

doi:https://doi.org/10.1155/2021/5495605

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Volume 2021 | Article ID 5495605 | https://doi.org/10.1155/2021/5495605

Cohen Positive Strongly -Summing and -Dominated m-Homogeneous Polynomials

Halima Hamdi,¹Samuel García-Hernández,²Amar Belacel,¹Djamel Ouchenane,¹and Sulima Ahmed Zubair^3,4

Academic Editor: Heng Liu

Received04 Oct 2021

Accepted19 Oct 2021

Published08 Nov 2021

Abstract

We introduce the concepts of Cohen positive strongly -summing and positive -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.

1. Introduction

The identity from into 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 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 -homogeneous polynomials.

This work is organized as follows. In Section 2, we introduce the class of Cohen positive strongly -summing -homogeneous polynomials and prove that this class is related to different classes of -summability . By means of the adjoint operator, the associated symmetric -linear operator and the linearization of these polynomials. Section 3 is essentially dedicated to the study of positive -dominated -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 -dominated polynomials and those defined in Section 2. In Section 4, we present some applications based on Proposition 3.

Finally, Section 5 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 consider and are Banach spaces and Banach lattices, respectively, for .

Let is the Banach space of all continuous m-linear operators from to . If , we write . In the case when , we simply write . For a Banach space , we denote and are the topological dual and the closed unit ball, respectively. For , let be its conjugate, that is, .

Let be the positive cone of the Banach lattice such that . For , let and be the positive part and the negative part of , respectively.

For any , we have .

Recall that an m-linear operator is called positive if , whenever . Let be the collection of all positive continuous -linear operators from to under the norm .

We denote by the space of all sequences in with the norm,and by the space of all sequences in with the norm,

Then, is a Banach space with respect to the norm . For a Banach lattice , we define

Let . If , we have that

The definition of -homogeneous polynomial can be seen, for instance, in [3, 6, 7].

A map is an -homogeneous polynomial if there exists a unique symmetric -linear operator such that . Both are related by the polarization formula:

is bounded on the unit ball of if and only if is bounded on the unit ball of . These norms are related by the inequalities . denotes the Banach space of all continuous homogeneous polynomials from to with the norm .

For the scalar-valued case or , we use the simplified notation .

By , we denote the completed projective tensor product of . If , we write . By , the fold symmetric tensor product of , that is, the set of all elements of the form

By , we denote the closure of in . Consider the canonical polynomial defined by , and we have the following factorization .

Recalling some definitions we need in the sequel.

Let . An operator is said to be positive summing [8] if there exists a constant such that, for all , the inequality

For ,

The space of positive summing operators from into given by becomes a Banach space with norm defined by the infimum of the constants that verify inequality (7). We have .

The next definition is due to Achour and Saadi in [3]. Let . An homogeneous polynomial is Cohen strongly summing if such that, for all and ,

The class of such polynomials is denoted by ; it is equipped with the norm , i.e., the smallest constant such that inequality (9) holds. For , we have

2. Cohen Positive Strongly -Summing -Homogeneous Polynomials

In this section, we give a natural generalization of the notion of Cohen positive strongly 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 linear operator.

Definition 1 (see [4]). Let . An linear operator is Cohen positive strongly summing multilinear operator if , such that, for all , , and any ,Moreover, the class of all Cohen positive strongly summing linear operators from into is given by . This space is a Banach space with the norm which is the smallest constant such that inequality (10) holds. For , it is the space of positive strongly summing operators linear operators [2].

Now, we introduce the notion of Cohen positive strongly summing homogeneous polynomials, and we study some fundamental properties.

Definition 2. Let and . An homogeneous polynomial is Cohen positive strongly summing if , such that, for all and any ,The class of such polynomials is given by . It is a Banach space with the norm which is the smallest constant such that inequality (11) holds, for , we have .

Remark 1. Fix ; an homogeneous polynomial is Cohen positive strongly summing if and only if satisfyingholds for every and .

Example 1. We use the same example used in [3], with slight adjustments.
Let , , and be a positive strongly summing operator. For , the functionalis Cohen positive strongly summing.
For and , because , we have by [2]Then,Hence, is Cohen positive strongly summing and .

Proposition 1 (ideal property). Let , in , and in .
If is Cohen positive strongly -summing polynomial, then is Cohen positive strongly -summing polynomial and .

Proof. Let and Thus, is Cohen positive strongly -summing polynomial and

Next, we present a version of Pietsch’s domination theorem, for our class of polynomials.

Theorem 1 (domination theorem). An homogeneous polynomial is Cohen positive strongly summing if and only if there is a Radon probability measure on , such that, for all and , we haveIn addition, in this case,

For the proof of this theorem, we need to give a small review about -abstract -summing operators.

Let the sets be , and , the Banach space is the family of mappings from to and and the compact Hausdroff topological space . Assume that the mapssatisfy. For all , there is an , such that for any and . The mapping defined by is continuous, for any and , and and , for any , , , , and .

Definition 3 (see [9]). Let and be as above and . A mapping is said to be -abstract -summing if such thatfor all , , and .

Theorem 2 (see [9]). Consider and be before, and . Then, is -abstract -summing , and a Borel probability measure on such thatwhere and . Furthermore, the infimum of such constant equals .

For our proof, we also need to recall Theorem 4.6 in [9].

Theorem 3 (see [9]). A map is -abstract -summing , , and a Borel probability measures on , such thatwhere , , and with .

Proof. (proof of the domination theorem). Assume that is Cohen positive strongly -summing polynomial, by selecting the parameters,Hence, is Cohen positive strongly -summing and is -abstract -summing. The previous theorem tells us that is -abstract -summing there is , and there are probability measures on , , such thati.e.,

Proposition 2. Consider .
If , then and .

Proof. Assume that ; the Pietsch domination (Theorem 1) provides a Radon probability measure on such that for whichSo, and .

Definition 4 (see [10]). Given a continuous homogeneous polynomial between the Banach spaces and , the adjoint of is the following continuous linear operator:We have .

Proposition 3. Let . The homogeneous polynomial is Cohen positive strongly summing if and only if is positive summing, and we have .

Proof. Suppose that ; we have by (17) , for and .
Taking the supremum, we obtainThen, . By the Pietsch domination theorem in Proposition 3.4 in [2], . Conversely, let we have, for and ,Using the theorem of Pietsch domination for positive -summing linear operators, we find

Proposition 4. If ,

Proof. Take any in ; hence, its adjoint operator is in by Proposition 1 in [11], and again by Proposition 1 in [8], is in ; then, .

Proposition 5. The following properties are equivalent: The polynomial is Cohen positive strongly summing Its associated symmetric linear operator is Cohen positive strongly summing

Proof. First, suppose that is Cohen positive strongly summing; let and ; then,Therefore, is Cohen positive strongly summing and .
Conversely, suppose that . Let such that and ; we have, by (5),Furthermore, for every , we haveThus,Using [4], is Cohen positive strongly summing; furthermore, .

Corollary 1. Let . If the polynomial is Cohen positive strongly summing, then its associated symmetric linear operator is multiple convex. If the polynomial is Cohen positive strongly summing, then its associated symmetric linear operator is in .

Proof. The proof of this corollary is an adaptation of the proof of Proposition 3.3 in [4].

Proposition 6. Let . Let be a -homogeneous polynomial and its linearization. The following properties are equivalent(i)The polynomial belongs to (ii)The operator belongs to (iii) is a Banach space, , and such that

Proof. Assume that is Cohen positive strongly -summing; let such that and , and assume that ; then,Selecting the infimum over any represents of , we findand by Theorem 4.13 in [2], is positive strongly -summing.
From the factorization, gives the result directly.
Assume that there exists a Banach space , and , such that . Then,Using Theorem 4.13 in [2],Hence,and according to the domination theorem, .

3. Positive p-Dominated m-Homogeneous Polynomials

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

3.1. Positive Dominated m-Linear Operators

Definition 5. An -linear operator is positive dominated if there is a positive constant such that, and . We denote the space of all positive dominated linear operators and the infimum of all constants verifying (42) is denoted . For all , is a norm on , whereas for , it is only a quasi-norm.
For , the space of all positive dominated operators is nothing other than the space of positive summing operators.

We should stress out that the above definition makes sense for Banach spaces , but the more interesting case occurs when we consider a Banach lattice , 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 for m-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 positive p-dominated multilinear operators). An -linear operator is positive dominated if and only if there is Radon probability measures , and , andfor all . Moreover, the smallest constants is .

The following theorem is a version of the Bu-theorem [12]; in [13], Achour and Mezrag proved an inclusion theorem between -dominated operators and Cohen strongly -summing 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. Let . Let and be Banach lattices. Then,for all .

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

If , thenfor each choice of the scalars , are the Khintchine constants, and by , we denote the sequence of Rademacher function such that as .

3.1.2. Multiple Kahane’s Inequality

Let . Let be the vector space of all almost unconditionally summable sequences in the Banach space . It is a Banach space under the norm:

Let . Let be a Banach space and be in . We havewhere is the simple constant of Kahane’s inequality .

Proof. Let ; then, by the domination theorem for positive -dominated -linear operators, there are probability measures on , such that, for any , we haveFor every , for and , we haveThen,Using Hölder’s inequality, we obtainBy the multiple Khintchine inequality, on the last inequality, we obtainBy Fubini’s theoremFor the quantity,using the multiple Kahane’s inequalityApplying Khintchine’s inequality,Consequently,Exploiting (53) and (57) givesThus, and .
And, this completes the proof.

3.2. Positive Dominated Homogeneous Polynomials

Definition 6. Let ; an homogeneous polynomial is positive dominated if such that, for any and any ,We denote the space of positive dominated polynomials from into , and by , the norm defined by the infimum of any constants verifying the above inquality, for , but for , it is only a quasi-norm.

Theorem 6. (domination theorem for positive dominated m-homogeneous polynomials). Let ; an homogeneous polynomial is positive dominated if there are and a probability measure on such that. Moreover, the smallest is .

In Proposition 7, we study the relationship between positive dominated linear operators and positive dominated homogeneous polynomials

Proposition 7. Let be an homogeneous polynomial and be its corresponding symmetric linear operator such that .(1)If is positive dominated operator, then is dominated and(2)If is positive dominated, then is positive dominated and(a)For , (b)For ,

Proof. (1) Suppose that is positive dominated; let ; then,Therefore, is positive dominated and .
(2) For and , let satisfy :By Theorem 6,For , , and , let satisfy :

In Corollary 2, we give an inclusion between the two classes of polynomials defined earlier.

Corollary 2. Let , and Banach lattice space; then,

Proof. Let ; from Proposition 7, the associate symmetric of denoted belongs to , and by Theorem 5, is in , so Proposition 5 finishes the proof.

4. Applications

Let be a compact Hausdroff space and a finite measure space.

As usual denotes the space of continuous functions on , and denotes the Banach space of all regular Borel measures on , and we have .

We shall write for the space of measurables functions on withand for ,

Let . An operator is said to be concave if there exists a constant such that, for all , the inequalityholds.

Let be the space of concave operators from into . becomes a Banach space with norm defined by the infimum of the constant that satisfy inequality (69).

The following results are a conclusion of the work done by Blasco in [8] and Proposition 3.

Proposition 8. Let :(1) if and only if (2) if and only if (3) if and only if (4) if and only if

5. Tensorial Approach

This section is intended to present a tensorial perspective of Cohen positive strongly summing and positive 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 summing operators, we pay more attention on the details for polynomials than m-linear operators. Later, we pay more attention on positive dominated m-linear operators than polynomials. Let be a nonnegative integer, be a Banach space, and be a Banach lattice, each homogeneous polynomial defines a functional given by

The next step in our development is to define adequate norm on the tensor space in order that be Cohen positive strongly summing (positive dominated) exactly when is bounded on .

5.1. Cohen Positive Strongly Summing Polynomials from a Tensor Viewpoint

Before introducing the respecting tensor norm, let us state an equivalence of being a Cohen positive strongly summing polynomial. This is an immediate consequence of the duality between and . The -homogeneous polynomial is Cohen positive strongly summing if and only if such thatfor all finite sequences , , and . In this case, agrees with the sharpest constant so that the inequality holds.

Definition 7. For each in , we definewhere the infimum is taken over any representations . For each in , we definewhere the infimum is taken over all representations with , for all .

The next proposition is a consequence of the fact that every can be represented as . Moreover, it helps to reduce the calculations related to and .

Proposition 9. For every in , we have

Proof. Let in . It is clear that by (4). For the opposite inequality, we take . Then,Thus, we obtain a new representation for such thatTherefore, , for all .

Proposition 10. The application is a norm on .

Proof. Standard tensorial techniques can be applied to prove that is nonnegative, homogeneous and verify the triangular inequality. So, we only prove that implies . Let be a bounded linear functional on and bounded on . Then,Hence, implies , for all and . For arbitrary , we have that . Then, impliesTherefore, must be zero since the set of linear functionals of the form is a separating set for .

Next, we give the characterization of Cohen positive strongly -summing polynomials in tensor terms.

Proposition 11 (duality). Let be an homogeneous polynomial. The following are equivalent:(i) is Cohen positive strongly -summing(ii) is bounded on Under this circumstances, .

Proof. Let be any element of ; then,After taking the infimum over all the representations of , we have thatHolds, for all .
Conversely, take sequences in , in , and in . Hence,

As we saw, the norms and are equivalent, and one of them is more useful than the other depending on the circumstances we are interested on. The norm provides an isometry between polynomials and linear functionals, while reduces calculations in Proposition 12.

Proposition 12 (uniform property). Let be a bounded linear operator and be a positive operator. Then, the operator defined byis bounded and .

Proof. Let with . Then,After taking the infimum over all the representations of , we obtain .

As it is expected, previous proposition is also true if is remplaced by , but this change involves a factor of 2 in the inequality.

5.2. Cohen Positive Strongly -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 strongly summing m-linear operators. As before, throughout this section, are Banach spaces and is a Banach lattice. Next, recall that each -linear operator defines a linear functional by

In this case, for each in , we definewhere the infimum is taken over all representations, .

Using usual tensor techniques and slight modifications of the proof of Proposition 10, let us prove that the assignment is a norm on . The respective duality result is exhibited in the next proposition and requires an analogous expression to (71).

Proposition 13 (duality). Let be an -linear operator. Then, the following are equivalents:(i) is Cohen positive strongly -summing(ii) is bounded on Under this circumstances, .

Proof. For any in , we haveHence, for any , we haveOn the other direction, let in , in , and in . Then,

For each , in , we definewhere the infimum is selected over any representation such that , for all . Then, for any in , we have

Having at our reach and knowing it defines a norm, we present the respective uniform property it verifies.

Proposition 14 (uniform property). Let be bounded linear operators and be positive operator between Banach lattices. Then, the operator,is bounded and .

Proof. Given with , we haveAs a consequence, .

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

As before, throughout this section, are Banach lattices and is a Banach space. In this case, we pay more attention on m-linear operators than polynomials.

Proposition 15. For each in , we definewhere the infimum is selected over any representations, .
For each in , we definewhere the infimum is taken over all representations with , for all . Then, the assignment is a norm on .

The proof of this proposition is an adaptation of Proposition 10 The next result concerns the characterization of positive -dominated m-linear operators and polynomial in tensor terms.

Proposition 16 (duality). Let be an -linear operator and be an -homogeneous polynomial. Then,(i) is positive -dominated if and only if is bounded on (ii) is positive -dominated if and only if is bounded on Under this circumstances, and .

Proof. We only prove since is a straightforward adaptation. First, suppose is positive -dominated. Then, for any , in , we haveHence, which means that is bounded and , for the opposite direction. Let in and . Then, for each , there exists in such that and . Therefore,As a consequence, we obtainwhich implies that is positive -dominated and .

The proof of next propositions is adaptations of those of Propositions 9 and 11.

Proposition 17. Define, for each , in ,where the infimum is taken over all representation with , for all and . Then,Holds, for all , in .

Proposition 18 (uniform property). Let be a positive operators and be a bounded linear operator. Then, the operator, is bounded and . Let be a positive operator and be a bounded linear operator. Then, the operator defined byis bounded and .

6. Conclusion

In this work, by giving the concepts of Cohen positive strongly p-summing and positive -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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

J. S. Cohen, “Absolutely p-summing, p-nuclear operators and their conjugates,” Mathematische Annalen, vol. 201, no. 3, pp. 177–200, 1973.
View at: Google Scholar
D. Achour and A. Belacel, “Domination and factorization theorems for positive strongly p-summing operators,” Positivity, vol. 18, no. 4, pp. 785–804, 2014.
View at: Publisher Site | Google Scholar
D. Achour and K. Saadi, “A polynomial characterization of Hilbert spaces,” Collectanea Mathematica, vol. 61, no. 3, pp. 291–301, 2010.
View at: Publisher Site | Google Scholar
A. Bougoutaia and A. Belacel, “Cohen positive strongly p-summing and p-convex multilinear operators,” Positivity, vol. 23, no. 2, pp. 379–395, 2019.
View at: Publisher Site | Google Scholar
J. R. Campos, “Cohen and multiple Cohen strongly summing multilinear operators,” Linear and Multilinear Algebra, vol. 62, no. 3, pp. 322–346, 2014.
View at: Publisher Site | Google Scholar
R. Cilia and J. M. Gutierrez, “Polynomial factorization through Lγ(μ) spaces,” Journal of the Korean Mathematical Society, vol. 46, no. 6, pp. 1293–1307, 2009.
View at: Publisher Site | Google Scholar
Y. Meléndez and A. Tonge, “Polynomials and the Pietsch domination theorem,” Mathematical Proceedings of the Royal Irish Academy, vol. 99A, no. 2, pp. 195–212, 1999.
View at: Google Scholar
O. Blasco, “Positive p-summing operators on L p -spaces,” Proceedings of the American Mathematical Society, vol. 100, no. 2, p. 275, 1987.
View at: Publisher Site | Google Scholar
D. Pellegrino, J. Santos, and J. B. Seoane-Sepúlveda, “Some techniques on nonlinear analysis and applications,” Advances in Mathematics, vol. 229, no. 2, pp. 1235–1265, 2012.
View at: Publisher Site | Google Scholar
R. M. Aron and M. Schottenloher, “Compact holomorphic mappings on banach spaces and the approximation property,” Bulletin of the American Mathematical Society, vol. 80, no. 6, pp. 1245–1250, 1974.
View at: Publisher Site | Google Scholar
D. Popa, “Cohen-Kwapień type theorems for multiple summing operators,” Quaestiones Mathematicae, vol. 36, no. 3, pp. 399–410, 2013.
View at: Publisher Site | Google Scholar
Q. Bu, “Some mapping properties of p-summing operators with Hilbertian domain,” in Contemporary Mathematics, K. Jarosz, Ed., vol. 328, pp. 145–149, American Mathematical Society, Providence, RI, USA, 2003.
View at: Publisher Site | Google Scholar
D. Achour and L. Mezrag, “On the cohen stronglyp-summing multilinear operators,” Journal of Mathematical Analysis and Applications, vol. 327, pp. 550–563, 2007.
View at: Google Scholar
A. Defant and K. Floret, “Tensor norms and operator ideals,” Number 176 in North-Holland Mathematics Studies, North-Holland, Amsterdam, Netherlands, 1993.
View at: Google Scholar

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Copyright © 2021 Halima Hamdi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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