# Positive ideals of multilinear operators

Athmane Ferradi, Abdelaziz Belaada and Khalil Saadi  Laboratoire d’Analyse
Fonctionnelle et Géométrie des Espaces, Université de M’sila, Algérie.
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###### Abstract.

We introduce and explore the concept of positive ideals for both linear and
multilinear operators between Banach lattices. This paper delineates the
fundamental principles of these new classes and provides techniques for
constructing positive multi-ideals from given positive ideals. Furthermore, we
present an example of a positive multi-ideal by introducing a new class,
referred to as positive $(p_{1},...,p_{m};r)$-dominated multilinear operators.
We establish a natural analogue of the Pietsch domination theorem and
Kwapień’s factorization theorem within this class.

††2020 Mathematics Subject Classification. Primary 47B65, 46G25, 47L20, 47B10,
46B42. Key words and phrases. Banach lattice, Positive $p$-summing operators,
Cohen positive strongly $p$-summing multilinear operators, Positive
$(p,q)$-dominated operators, Positive left ideal, Positive right ideal,
Positive ideal, Positive left multi-ideal, Positive right multi-ideal,
Positive multi-ideal, Positive $(p_{1},\ldots,p_{m};r)$-dominated.

##  1\. Introduction and preliminaries

The exploration of ideals in both linear and multilinear operators between
Banach spaces has been introduced and extensively investigated by various
researchers, see [9, 12, 13, 14, 17, 18]. In recent years, there has also been
a focus on investigating various classes of operators defined on Banach
lattice spaces, such as positive $p$-summing [5], positive strongly
$p$-summing [3], and positive $(p,q)$-dominated operators [10]. The extension
of these classes to the multilinear case has been widely studied, such as
Cohen positive strongly $p$-summing multilinear operators [6], positive Cohen
$p$-nuclear multilinear operators [7] and factorable positive strongly
$p$-summing multilinear operators [8]. However, it is important to note that
these (positive) classes are not necessarily considered as operator ideals.
The primary aim of this paper is to introduce and explore the concept of
positive ideal of linear operators and positive multi-ideal of multilinear
operators. Leveraging the groundwork laid by the definition of multilinear
ideals, our objective is to scrutinize the positive setting by investigating
diverse classes of positive linear and multilinear operators. This new
theoretical framework illustrates that several classes of linear and
multilinear operators, previously mentioned, can be classified as positive
operator ideals. We also present certain construction methods for generalizing
positive multi-ideals. Furthermore, inspired by the recent work of Chen et al.
[10], where they introduced and explored the class of positive
$(p,q)$-dominated linear operators, we extend this concept to the multilinear
case. We term this extension as positive $(p_{1},...,p_{m};r)$-dominated
multilinear mappings. The corresponding space is denoted by
$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}$. This class is a
positive multi-ideal. A proof is provided for a natural analogue of the
Pietsch domination theorem within this extended class. Finally, as a
concluding result, we present Kwapień’s factorization theorem. We demonstrate
that the space $\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}$ can be
factorized as follows

$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}=\mathcal{D}_{r^{\ast}}^{m+}\left(\Pi_{p_{1}}^{+},\ldots,\Pi_{p_{m}}^{+}\right),$

where $\Pi_{p_{j}}^{+}$ $\left(1\leq j\leq m\right)$ represents the Banach
space of positive $p$-summing operators and $\mathcal{D}_{r^{\ast}}^{m+}$ is
the Banach space of Cohen positive strongly $r^{\ast}$-summing multilinear
operators.

The paper is organized as follows.

In Section 1, we provide a concise review of fundamental concepts and
terminologies essential for our work. Additionally, we recall the definition
of the positive $p$-summing, Cohen positive strongly $p$-summing and positive
$(p,q)$-dominated linear operators.

In Section 2, we establish the groundwork for positive left ideals, denoted by
$\mathcal{B}_{L}^{+}$ , and right ideals, denoted by $\mathcal{B}_{R}^{+}$, of
linear operators. This conceptual foundation is naturally extended to the
multilinear case. We introduce the composition method to generate a positive
right ideal of multilinear operators from a given positive operator right
ideal. The multilinear operator $T$ belongs to the right multi-ideal
$\mathcal{M}_{R}^{+}$ if and only if its linearization $T_{L}$ belongs to
$\mathcal{B}_{R}^{+}$ . Additionally, we present the factorization method to
create a positive left ideal of multilinear operators from given positive
operator left ideals $\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}$. This
section is complemented by illustrative examples of positive left and right
ideals.

In Section 3, we introduce the concept of $(p_{1},...,p_{m};r)$-dominated
multilinear operators. These operators satisfy the Pietsch factorization
theorem and gives a good example of a positive multi-ideal.

Throughout this paper, we use the notations $E,F,G,E_{1},...,E_{m}$ for Banach
lattices and $X,Y,X_{1},...,X_{m}$ for Banach spaces over $\mathbb{R}$ or
$\mathbb{C}$. We will denote by $\mathcal{L}(X;Y)$ the Banach space of bounded
linear operators from $X$ into $Y$. By $B_{X}$ we denote the closed unit ball
of $X$ and by $X^{\ast}$ its topological dual. For $1\leq p\leq\infty$, let
$p^{\ast}$ be its conjugate, that is $1/p+1/p^{\ast}=1$. Let $E$ be a Banach
lattice with norm $\left\|.\right\|$ and order $\leq$. We denote by $E^{+}$
the positive cone of $E$ , i.e., $E^{+}=\\{x\in E:x\geqslant 0\\}.$ For $x\in
E$ let $x^{+}:=\sup\\{x,0\\}\geq 0\ $and $x^{-}:=\sup\\{-x,0\\}\geq 0$ be the
positive part and the negative part of $x$, respectively. For any $x\in E,$ we
have $x=x^{+}-x^{-}$ and $|x|=x^{+}+x^{-}.$ The dual $E^{\ast}$ of a Banach
lattice $E$ is a Banach lattice endowed with the natural order

$x_{1}^{\ast}\leq x_{2}^{\ast}\Leftrightarrow\langle
x,x_{1}^{\ast}\rangle\leq\langle x,x_{2}^{\ast}\rangle,\forall x\in E^{+}.$

Recall that a bounded linear operator $T:E\rightarrow F$ is called positive if
$T(x)\in F^{+}$, whenever $x\in E^{+}$. Let $\mathcal{L}^{+}(E;F)$ be the set
of all positive operators from $E$ into $F$. A linear operator $T$ is called
regular if there exist $T_{1},T_{2}\in\mathcal{L}^{+}(E;F)$ such that

$T=T_{1}-T_{2}.$

We denote by $\mathcal{L}^{r}(E;F)$ the vector space of regular operators from
$E$ into $F.$ It is easy to see that the vector space $\mathcal{L}^{r}(E;F)$
is generated by positive operators. We equipped $\mathcal{L}^{r}(E;F)$ with
the norm defined as

$\left\|T\right\|_{r}=\inf\left\\{\left\|S\right\|:S\in\mathcal{L}^{+}(E;F),\left|T\left(x\right)\right|\leq
S\left(x\right),x\in E^{+}\right\\},$

then $\mathcal{L}^{r}(E;F)$ becomes a Banach space. By [15, Section 1.3], if
$F=\mathbb{R}$, we have

$E^{\ast}=\mathcal{L}(E,\mathbb{R})=\mathcal{L}^{r}(E,\mathbb{R}).$

By a sublattice of a Banach lattice $E$ we mean a linear subspace $E_{0}$ of
$E$ so that $\sup\left\\{x,y\right\\}$ belongs to $E_{0}$ whenever $x,y\in
E_{0}$. The canonical embedding $i:E\longrightarrow E^{\ast\ast}$ such that
$\left\langle i(x),x^{\ast}\right\rangle=\left\langle x^{\ast},x\right\rangle$
of $E$ into its second dual $E^{\ast\ast}$ is an order isometry from $E$ onto
a sublattice of $E^{\ast\ast}$. If we consider $E$ as a sublattice of
$E^{\ast\ast}$ we have for $x_{1},x_{2}\in E$

$x_{1}\leq x_{2}\Longleftrightarrow\left\langle
x_{1},x^{\ast}\right\rangle\leq\left\langle
x_{2},x^{\ast}\right\rangle,\quad\forall x^{\ast}\in E^{\ast+}.$

The spaces $\mathcal{C}(K)$ where $K$ compact and $L_{p}(\mu)$, $(1\leq
p\leq\infty)$ are Banach lattices. For a Banach space $X$, we denote by
$\ell_{p}^{n}(X)$ the Banach space of all absolutely $p$-summable sequences
$(x_{i})_{i=1}^{n}\subset X$ with the norm,

$\|(x_{i})_{i=1}^{n}\|_{p}=(\sum_{i=1}^{n}\|x_{i}\|^{p})^{\frac{1}{p}},$

and by $\ell_{w,p}^{n}(X)$ the Banach space of all weakly $p$-summable
sequences $(x_{i})_{i=1}^{n}\subset X$ with the norm,

$\|(x_{i})_{i=1}^{n}\|_{w,p}=\sup_{x^{\ast}\in
B_{X^{\ast}}}(\sum_{i=1}^{n}|\langle
x^{\ast},x_{i}\rangle|^{p})^{\frac{1}{p}}.$

Consider the case where $X$ is replaced by a Banach lattice $E$, and define

$\ell_{|w|,p}^{n}(E)=\\{(x_{i})_{i=1}^{n}\subset
E:(|x_{i}|)_{i=1}^{n}\in\ell_{w,p}^{n}(E)\\}$

and

$\|(x_{i})_{i=1}^{n}\|_{|w|,p}=\|(|x_{i}|)_{i=1}^{n}\|_{w,p}.$

Let $B_{E^{\ast}}^{+}=\\{x^{\ast}\in B_{E^{\ast}}:x^{\ast}\geq
0\\}=B_{E^{\ast}}\cap E^{\ast+}$. If $(x_{i})_{i=1}^{n}\subset E^{+}$ , we
have that

$\|(x_{i})_{i=1}^{n}\|_{|w|,p}=\|(x_{i})_{i=1}^{n}\|_{w,p}=\sup_{x^{\ast}\in
B_{E^{\ast}}^{+}}(\sum_{i=1}^{n}\langle
x^{\ast},x_{i}\rangle^{p})^{\frac{1}{p}}.$

We recall certain classes of linear and multilinear operators:

\- Positive $p$-summing operator: Blasco [5] introduced the positive
generalization of $p$-summing operators as follows: An operator
$T:E\longrightarrow X$ is said to be positive $p$-summing ($1\leq p<\infty$)
if there exists a constant $C>0$ such that the inequality

$\left\|\left(T\left(x_{i}\right)\right)_{i=1}^{n}\right\|_{p}\leq
C\left\|\left(x_{i}\right)_{i=1}^{n}\right\|_{p,w},$ (1.1)

holds for all $x_{1},\ldots,x_{n}\in E^{+}.$ We denote by $\Pi_{p}^{+}(E;X)$,
the space of positive $p$-summing operators from $E$ into $X,$ which is a
Banach space with the norm $\pi_{p}^{+}(T)$ given by the infimum of the
constant $C>0$ that verify the inequality $(\ref{sec1def1}).$ We have
$\Pi_{\infty}^{+}(E;X)=\mathcal{L}(E;X)$. In [19], the author demonstrate the
Pietsch domination theorem concerning this class. The operator $T$ belongs to
$\Pi_{p}^{+}(E;X)$ if and only if there exist a Radon probability measure
$\mu$ on the set $B_{E^{\ast}}^{+}$ and a positive constant $C$ such that for
every $x\in E^{+}$

$\|T\left(x\right)\|\leq C(\int_{B_{E^{\ast}}^{+}}\langle
x,x^{\ast}\rangle^{p}d\mu(x^{\ast}))^{\frac{1}{p}}.$ (1.2)

\- Positive strongly $p$-summing operator: The positive generalization of
strongly $p$-summing operators, originally introduced by Cohen in 1973 [11],
has been further developed by Achour and Belacel [3]. An operator
$T:X\rightarrow F$ is positive strongly $p$-summing ($1<p\leq\infty$) if there
exists a constant $C>0$ , such that for all finite sets
$(x_{i})_{i=1}^{n}\subset X$ and $(y_{i}^{\ast})_{i=1}^{n}\subset F^{\ast+}$,
we have

$\sum_{i=1}^{n}|\langle T\left(x_{i}\right),y_{i}^{\ast}\rangle|\leq
C\|(x_{i})_{i=1}^{n}\|_{p}\|(y_{i}^{\ast})_{i=1}^{n}\|_{p^{\ast},w}.$

The class of all positive strongly $p$-summing operators between $X$ and $F$
is denoted by $\mathcal{D}_{p}^{+}(X;F)$. The infimum of all the constant $C$
in the inequality defines the norm $d_{p}^{+}$ on $\mathcal{D}_{p}^{+}(X;F)$.
We have $\mathcal{D}_{1}^{+}(X;F)=\mathcal{L}(X;F)$. In [3] we have: The
operator $T$ belongs to $\mathcal{D}_{p}^{+}(X;F)$ if and only if there exist
a positive constant $C>0$ and Radon probability measure $\mu$ on
$B_{F^{\ast\ast}}^{+}$ such that for all $x\in X$ and $y^{\ast}\in F^{\ast}$,
we have

$|\langle T\left(x\right),y^{\ast}\rangle|\leq
C\|x\|(\int_{B_{F^{\ast\ast}}^{+}}\langle|y^{\ast}|,\psi\rangle^{p^{\ast}}d\mu)^{\frac{1}{p^{\ast}}}.$
(1.3)

\- Positive $(p,q)$-dominated operator: The notion of $(p,q)$-dominated
operator was initiated by Pietsch [18] and generalized to positive
$(p,q)$-dominated operator by Chen et al. [10]. Let $1\leq p,q\leq\infty$ and
let $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. We say that an operator $T$ from a
Banach lattice $E$ to a Banach lattice $F$ is positive $(p,q)$-dominated if
there exists a constant $C>0$ such that

$(\sum_{i=1}^{n}|\langle
T\left(x_{i}\right),y_{i}^{\ast}\rangle|^{r})^{\frac{1}{r}}\leq
C\left\|\left(x_{i}\right)_{i=1}^{n}\right\|_{p,w}\left\|\left(y_{i}^{\ast}\right)_{i=1}^{n}\right\|_{q,w},$
(1.4)

for all finite families $(x_{i})_{i=1}^{n}$ in $E^{+}$ and
$(y_{i}^{\ast})_{i=1}^{n}$ in $F^{\ast+}$. The class of all positive
$(p,q)$-dominated operators from $E$ to $F$ is denoted by
$\mathcal{D}_{p,q}^{+}(E;F)$. In this case, we define

$d_{p,q}^{+}(T)=\inf\\{C>0\ \text{satisfying\ the\ inequality}(\ref{D01})\\}.$

######  Theorem 1.1.

Let $1\leq p,q\leq\infty$. An operator $T:E\rightarrow F$ is positive
$(p,q)$-dominated with positive constant $C$ if and only if there exist a
probability measure $\mu$ on $B_{E^{\ast}}^{+}$ and a probability measure
$\nu$ on $B_{F^{\ast\ast}}^{+}$ such that

$|\langle T\left(x\right),y^{\ast}\rangle|\leq
C(\int_{B_{E^{\ast}}^{+}}\langle
x^{\ast},x\rangle^{p}d\mu(x^{\ast}))^{\frac{1}{p}}(\int_{B_{F^{\ast\ast}}^{+}}\langle
y^{\ast\ast},y^{\ast}\rangle^{q}d\nu(y^{\ast\ast}))^{\frac{1}{q}}$

for all $x\in E^{+}$ and $y^{\ast}\in F^{\ast+}$.

\- Positive $p$-nuclear operator: If $q=p^{\ast},$ the definition of positive
$(p,p^{\ast})$-dominated operators coincides with the concept of positive
$p$-nuclear operators which introduced and studied by [7]. We denote by
$\mathcal{N}_{p}^{+}(E;F)$ the space of positive $p$-nuclear operators.

Let $X_{1},...,X_{m},Y$ be Banach spaces and $E_{1},...,E_{m},F$ be Banach
lattices. The space $\mathcal{L}(X_{1},...,X_{m};Y)$ stands the Banach space
of all bounded multilinear operators from $X_{1}\times...\times X_{n}$ into
$Y.$ We denote the complete projective tensor product of $X_{1},...,X_{m}$ by
$X_{1}\widehat{\otimes}_{\pi}...\widehat{\otimes}_{\pi}X_{m}$. For any
$T\in\mathcal{L}(X_{1},...,X_{m};Y)$, we consider its linearization
$T_{L}:X_{1}\widehat{\otimes}_{\pi}\cdots\widehat{\otimes}_{\pi}X_{m}\rightarrow
Y$, defined as $T_{L}(x^{1}\otimes\cdots\otimes x^{m})=T(x^{1},\ldots,x^{m})$.
Let $\sigma_{m}:X_{1}\times...\times X_{m}\rightarrow
X_{1}\widehat{\otimes}_{\pi}\cdots\widehat{\otimes}_{\pi}X_{m}$ be the
canonical multilinear operator, defined as
$\sigma_{m}(x_{1},...,x_{m})=x_{1}\otimes...\otimes x_{m}$. Then, we have
$T=T_{L}\circ\sigma_{m}$. The $m$-linear mapping of finite type
$T_{f}:X_{1}\times...\times X_{m}\longrightarrow Y$ is defined as follows

$T_{f}\left(x^{1},...,x^{m}\right)=\sum_{i=1}^{k}x_{i,1}^{\ast}\left(x^{1}\right)...x_{i,m}^{\ast}\left(x^{m}\right)y_{i}.$

We denote by $\mathcal{L}_{f}(X_{1},...,X_{m};Y)$ the space of all $m$-linear
mapping of finite type.

\- Cohen positive strongly $p$-summing multilinear operator: This notion have
been introduced by [6], offering a natural generalization of the multilinear
operators examined in [4]: An $m$-linear operator $T:X_{1}\times...\times
X_{m}\rightarrow F$ is said to be Cohen positive strongly $p$-summing
$(1<p\leq\infty)$, if there exists a constant $C>0,$ such that for any
$(x_{i}^{1},...,x_{i}^{m})\in X_{1}\times...\times X_{m}$ ($1\leq i\leq n$)
and $y_{1}^{\ast},...,y_{n}^{\ast}\in F^{\ast+}$,the following condition holds

$\sum_{i=1}^{n}|\langle
T(x_{i}^{1},...,x_{i}^{m}),y_{i}^{\ast}\rangle|\leq\displaystyle
C(\sum_{i=1}^{n}\prod_{j=1}^{m}\|x_{i}^{j}\|^{p})^{\frac{1}{p}}\left\|\left(y_{i}^{\ast}\right)_{i=1}^{n}\right\|_{p^{\ast},w}.$
(1.5)

We equip the space $\mathcal{D}_{p}^{m+}(X_{1},...,X_{m};F)$ of all Cohen
positive strongly $p$-summing multilinear operators with the norm
$d_{p}^{m+}(.)$, defined as the smallest constant $C$ for which the inequality
(1.5) holds. When $p=1$, we have $\mathcal{D}_{1}^{m+}(X_{1},...,X_{m};F)$
equal to $\mathcal{L}(X_{1},...,X_{m};F).$

\- Positive Cohen $p$-nuclear multilinear operator: In [7, Definition 2.1],
the authors introduced the class of positive Cohen $p$-nuclear $m$-linear
operators, providing a natural generalization of the multilinear case studied
in [2]: An $m$-linear operator $T:E_{1}\times...\times E_{m}\rightarrow F$ is
positive Cohen $p$-nuclear $(1<p<\infty)$ if there is a constant $C>0$ such
that for any $x_{1}^{j},...,x_{n}^{j}\in E_{j}^{+}$ ($1\leq j\leq m$) and
$y_{1}^{\ast},...,y_{n}^{\ast}\in F^{\ast+}$, we have

$|\sum_{i=1}^{n}\langle
T(x_{i}^{1},...,x_{i}^{m}),y_{i}^{\ast}\rangle|\leq\displaystyle
C(\underset{\underset{1\leq j\leq m}{x^{j\ast}\in
B_{E_{j}^{\ast}}^{+}}}{\sup}\sum_{i=1}^{n}\prod_{j=1}^{m}\langle
x_{i}^{j},x^{j\ast}\rangle^{p})^{\frac{1}{p}}\left\|\left(y_{i}^{\ast}\right)_{i=1}^{n}\right\|_{p^{\ast},w}.$
(1.6)

Moreover, the class $\mathcal{N}_{p}^{m+}(E_{1},...,E_{m};F)$ of all positive
Cohen $p$-nuclear $m$-linear operators from $E_{1}\times...\times E_{m}$ into
$F$ is a Banach space with the norm $\eta_{p}^{m+}(.)$, which is the smallest
constant $C$ such that (1.6) holds. If $m=1$, $\mathcal{N}_{p}^{+}(E;F)$ is
the space of positive Cohen $p$-nuclear operators.

##  2\. Positive operator ideals

Various classes of positive operators, including positive $p$-summing, Cohen
positive strongly $p$-summing, and positive $(p,q)$-dominated linear
operators, have been introduced and studied. However, these classes are not
regarded as ideal operators. Hence, in this section, we endeavor to introduce
the concept of positive operator ideals and propose abstract methods for
generating positive ideals of multilinear operators. Within this definition,
we utilize regular linear operators to establish the ideal property.

###  2.1. Positive operator ideal

A positive right ideal (or positive right ideal of linear operators), denoted
by $\mathcal{B}_{R}^{+},$ is a subclass of all continuous linear operators
from a Banach space into a Banach lattice such that for all Banach space $X$
and Banach lattice $E,$ the components

$\mathcal{B}_{R}^{+}\left(X;E\right):=\mathcal{L}\left(X;E\right)\cap\mathcal{B}_{R}^{+}$

satisfy:
$(i)$ $\mathcal{B}_{R}^{+}\left(X;E\right)$ is a linear subspace of
$\mathcal{L}\left(X;E\right)$ which contains the linear mappings of finite
type.
$(ii)$ The positive ideal property: If
$T\in\mathcal{B}_{R}^{+}\left(X;E\right),u\in\mathcal{L}\left(Y;X\right)$ and
$v\in\mathcal{L}^{r}\left(E;F\right)$, then $v\circ T\circ u$ is in
$\mathcal{B}_{R}^{+}\left(Y;F\right)$.
If
$\|\cdot\|_{\mathcal{B}_{R}^{+}}:\mathcal{B}_{R}^{+}\rightarrow\mathbb{R}^{+}$
satisfies:
a)
$\left(\mathcal{B}_{R}^{+}\left(X;E\right),\|\cdot\|_{\mathcal{B}_{R}^{+}}\right)$
is a normed (Banach) space for all Banach space $X$ and Banach lattice $E$.
b) The linear form $T:\mathbb{K}\rightarrow\mathbb{K}$ given by
$T\left(\lambda\right)=\lambda$ satisfies
$\left\|u\right\|_{\mathcal{B}_{R}^{+}}=1$,
c) $T\in\mathcal{B}_{R}^{+}\left(X;E\right),u\in\mathcal{L}\left(Y;X\right)$
and $v\in\mathcal{L}^{r}\left(E;F\right)$ then

$\left\|v\circ T\circ
u\right\|_{\mathcal{B}_{R}^{+}}\leq\|v\|\|T\|_{\mathcal{B}_{R}^{+}}\left\|u\right\|.$

The class $\left(\mathcal{B}_{R}^{+},\|\cdot\|_{\mathcal{B}_{R}^{+}}\right)$
is a normed (Banach) positive ideal.

######  Remark 2.1.

In condition $(ii)$, the space $\mathcal{L}^{r}\left(E;F\right)$ can be
replaced by the set $\mathcal{L}^{+}\left(E;F\right)$, and condition $(ii)$
remains the same. To see this, let $v\in\mathcal{L}^{r}\left(E;F\right)$, and
express it as $v=v_{1}-v_{2},$ where
$v_{1},v_{2}\subset\mathcal{L}^{+}\left(E;F\right)$. Now, considering $v\circ
T\circ u$, we can write:

$\displaystyle v\circ T\circ u$ $\displaystyle=$
$\displaystyle\left(v_{1}-v_{2}\right)\circ T\circ u$ $\displaystyle=$
$\displaystyle v_{1}\circ T\circ u-v_{2}\circ T\circ u$

Since the operators $v_{1}\circ T\circ u$ and $v_{2}\circ T\circ u$ belong to
$\mathcal{B}_{R}^{+}\left(Y;F\right)$, by the linearity of
$\mathcal{B}_{R}^{+}\left(Y;F\right)$, we conclude that $v\circ T\circ
u\in\mathcal{B}_{R}^{+}\left(Y;F\right)$. The converse is immediate.

In a manner analogous to the previous approach, we introduce the positive left
ideal, denoted by $\mathcal{B}_{L}^{+}$, by interchanging the roles of the
operators $u$ and $v$. This involves examining the composition of regular
linear operators on the left side and linear operators on the right side.
Similarly, we define the positive ideal, denoted by $\mathcal{B}^{+}$, by
considering solely the regular linear operators, with composition occurring on
both the right and left sides. It’s important to note that any positive left
or right ideal automatically qualifies as a positive ideal.

######  Proposition 2.2.

Let $\mathcal{B}_{L}^{+}$ and $\mathcal{B}_{R}^{+}$ denote positive left and
right ideals, respectively. The composition ideals
$\mathcal{B}_{R}^{+}\circ\mathcal{B}_{L}^{+}$ consists of elements $T$ that
can be factorized as $T=v\circ u,$ where $u$ belongs to
$\mathcal{B}_{L}^{+}\left(E;X\right)$ and $v$ belongs to
$\mathcal{B}_{R}^{+}\left(X;F\right)$, resulting in a positive ideal.

If $\mathcal{B}_{L}^{+}$ and $\mathcal{B}_{R}^{+}$ are normed (Banach)
positive left and right ideals, respectively. Let $E$ and $F$ be Banach
lattices and $T\in\mathcal{B}_{R}^{+}\circ\mathcal{B}_{L}^{+}(E;F),$ we define

$\left\|T\right\|_{\mathcal{B}_{R}^{+}\circ\mathcal{B}_{L}^{+}}=\inf\left\\{\left\|v\right\|_{\mathcal{B}_{R}^{+}}\left\|u\right\|_{\mathcal{B}_{L}^{+}}:T=v\circ
u\right\\}.$

The proof of the following theorem can be easily proved.

######  Theorem 2.3.

If $\mathcal{B}_{L}^{+}$ and $\mathcal{B}_{R}^{+}$ are normed (Banach)
positive left and right ideals, respectively. Then

$\left(\mathcal{B}_{R}^{+}\circ\mathcal{B}_{L}^{+},\left\|.\right\|_{\mathcal{B}_{R}^{+}\circ\mathcal{B}_{L}^{+}}\right)$

is a normed (Banach) positive ideal.

Now, we give examples of positive ideals. For every $p\geq 1$, the space of
positive $p$-summing operators $\Pi_{p}^{+}$ forms a positive left ideal,
while the space of Cohen positive strongly $p$-summing operators
$\mathcal{D}_{p}^{+}$ forms a positive right ideal. Consequently, the
composition ideal $\mathcal{D}_{p}^{+}\circ\Pi_{p}^{+},$ which is equal to
$\mathcal{N}_{p}^{+},$ the space of positive $p$-nuclear operators, forms a
positive ideal.

###  2.2. Positive right multi-ideal

We extend the previous concepts to the multilinear case by introducing a new
definition for positive ideals of multilinear operators. This new concept is
an extension of multi-ideals and utilizes techniques introduced in [9, 17]. To
begin, we introduce the notion of a positive right multi-ideal. Furthermore,
we suggest a composition method that allows us to construct a positive right
ideal of multilinear operators from a given positive right ideal.

######  Definition 2.4.

A positive right multi-ideal (or positive right ideal of multilinear
operators), denoted by $\mathcal{M}_{R}^{+},$ is a subclass of all continuous
multilinear operators from Banach spaces into a Banach lattice such that for
all $m\in\mathbb{N}^{\ast}$, Banach spaces $X_{1},\ldots,X_{m}$ and Banach
lattice $F,$ the components

$\mathcal{M}_{R}^{+}(X_{1},...,X_{m};F):=\mathcal{L}(X_{1},...,X_{m};F)\cap\mathcal{M}_{R}^{+}$

satisfy:
$(i)$ $\mathcal{M}_{R}^{+}(X_{1},...,X_{m};F)$ is a linear subspace of
$\mathcal{L}(X_{1},...,X_{m};F)$ which contains the $m$-linear mappings of
finite type.
$(ii)$ The positive ideal property: If
$T\in\mathcal{M}_{R}^{+}\left(X_{1},\ldots,X_{m};F\right),u_{j}\in\mathcal{L}\left(Y_{j};X_{j}\right)$
for $j=1,\ldots,m$ and $v\in\mathcal{L}^{r}(F;E)$, then $v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)$ is in
$\mathcal{M}_{R}^{+}\left(Y_{1},\ldots,Y_{m};E\right)$.
If
$\|\cdot\|_{\mathcal{M}_{R}^{+}}:\mathcal{M}_{R}^{+}\rightarrow\mathbb{R}^{+}$
satisfies:
a)
$\left(\mathcal{M}_{R}^{+}(X_{1},...,X_{m};F),\|\cdot\|_{\mathcal{M}_{R}^{+}}\right)$
is a normed (Banach) space for all Banach spaces $X_{1},\ldots,X_{m}$ and
Banach lattice $F$.
b) The $m$ -linear form $T^{m}:\mathbb{K}^{m}\rightarrow\mathbb{K}$ given by
$T^{m}\left(\lambda^{1},\ldots,\lambda^{m}\right)=\lambda^{1}\ldots\lambda^{m}$
satisfies $\left\|T^{m}\right\|_{\mathcal{M}_{R}^{+}}=1$ for all $m$,
c)
$T\in\mathcal{M}_{R}^{+}\left(X_{1},\ldots,X_{m};F\right),u_{j}\in\mathcal{L}\left(Y_{j};X_{j}\right)$
for $j=$ $1,\ldots,m$ and $v\in\mathcal{L}^{r}(F;E)$ then

$\left\|v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)\right\|_{\mathcal{M}_{R}^{+}}\leq\|v\|\|T\|_{\mathcal{M}_{R}^{+}}\left\|u_{1}\right\|\ldots\left\|u_{m}\right\|$

we say that $\left(\mathcal{M}_{R}^{+},\|\cdot\|_{\mathcal{M}_{R}^{+}}\right)$
is a normed (Banach) positive right multi-ideal. When $m=1$, it specifically
corresponds to the case of a positive right ideal.

######  Remark 2.5.

As mentioned in the remark 2.1, we can substitute the space
$\mathcal{L}^{r}\left(F;E\right)$ with the set
$\mathcal{L}^{+}\left(F;E\right)$. Indeed, consider $v\in\mathcal{L}^{r}(F;E)$
such that $v=v_{1}-v_{2},$ where
$v_{1},v_{2}\subset\mathcal{L}^{+}\left(E;F\right)$. Now, we can write

$\displaystyle v\circ T\circ\left(u_{1},\ldots,u_{m}\right)$ $\displaystyle=$
$\displaystyle\left(v_{1}-v_{2}\right)\circ
T\circ\left(u_{1},\ldots,u_{m}\right)$ $\displaystyle=$ $\displaystyle
v_{1}\circ T\circ\left(u_{1},\ldots,u_{m}\right)-v_{2}\circ
T\circ\left(u_{1},\ldots,u_{m}\right).$

By the linearity of $\mathcal{M}_{R}^{+}\left(Y_{1},\ldots,Y_{m};E\right)$, we
conclude that $v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)\in\mathcal{M}_{R}^{+}\left(Y_{1},\ldots,Y_{m};E\right)$.
The converse is immediate.

######  Remark 2.6.

It is evident that all multi-ideals are also positive right multi-ideals.

Here is an example of a positive right multi-ideal. In [6], the authors have
introduced the class $\mathcal{D}_{p}^{m+}$ of Cohen positive strongly
$p$-summing multilinear operators.

######  Proposition 2.7.

The class $(\mathcal{D}_{p}^{m+},d_{p}^{m+}(.))$ is a Banach positive right
multi-ideal.

###### Proof.

$(i)$ $\mathcal{D}_{p}^{m+}(X_{1},...,X_{m};F)$ is a linear subspace of
$\mathcal{L}(X_{1},...,X_{m};F)$ that contains the $m$-linear mappings of
finite type, see [6, Theorem 2.10].
$(ii)$ The positive ideal property: Proposition 2.3 [6].
The rest is obviously. In conclusion, the class
$(\mathcal{D}_{p}^{m+},d_{p}^{m+}(.))$ is indeed a Banach positive right
multi-ideal.

The composition method. Let $\mathcal{B}_{R}^{+}$ be a positive right ideal.
Suppose that $X_{j}\left(\text{for }1\leq j\leq m\right)$ are Banach spaces,
and $F$ is a Banach lattice. A multilinear operator
$T\in\mathcal{L}(X_{1},...,X_{m};F)$ belongs to
$\mathcal{B}_{R}^{+}\circ\mathcal{L}$ if there exist a Banach space $Y$, a
multilinear operator $S\in\mathcal{L}(X_{1},...,X_{m};Y)$, and an operator
$u\in\mathcal{B}_{R}^{+}(Y;F)$ such that

$\begin{array}[]{ccc}X_{1}\times...\times
X_{m}&\overset{T}{\longrightarrow}&F\\\ S\downarrow&\nearrow u&\\\
Y&&\end{array}$

i.e., $T=u\circ S$. In this case, we denote
$T\in\mathcal{B}_{R}^{+}\circ\mathcal{L}(X_{1},...,X_{m};F).$

######  Remark 2.8.

The class $\mathcal{B}_{R}^{+}\circ\mathcal{L}$ is indeed a positive right
multi-ideal.

We have the following result.

######  Proposition 2.9.

Let $\mathcal{B}_{R}^{+}$ be a positive right-ideal. Let $X_{1},...,X_{m}$ be
Banach spaces and $F$ be a Banach lattice. For
$T\in\mathcal{L}(X_{1},...,X_{m};F),$ the following statements are equivalent:
a) The operator $T$ belongs to
$\mathcal{B}_{R}^{+}\circ\mathcal{L}(X_{1},...,X_{m};F)$.
b) The linearization $T_{L}$ belongs to
$\mathcal{B}_{R}^{+}(X_{1}\widehat{\otimes}_{\pi}...\widehat{\otimes}_{\pi}X_{m};F).$

###### Proof.

A demonstration analogous to Proposition 3.2. [9].

Similar to [9, Proposition 3.7 (a)], if $\mathcal{B}_{R}^{+}$ is a normed
(Banach) right-ideal, the composition $\mathcal{B}_{R}^{+}\circ\mathcal{L}$
forms a normed (Banach) right multi-ideal. Proposition 3.1 in [6] establishes
that $\mathcal{D}_{p}^{m+}=\mathcal{D}_{p}^{+}\circ\mathcal{L}$. Consequently,
$\mathcal{D}_{p}^{m+}$ represents the Banach positive right multi-ideal
generated through the composition method from the Banach positive right-ideal
$\mathcal{D}_{p}^{+}.$

###  2.3. Positive left multi-ideal

Let’s introduce the concept of positive left multi-ideals, expanding upon the
concept of multi-ideals. Furthermore, we will present the factorization method
that allows us to construct a positive left multi-ideal using a given positive
left-ideal.

######  Definition 2.10.

A positive left multi-ideal (or positive left-ideal of multilinear operators),
denoted by $\mathcal{M}_{L}^{+},$ is a subclass of all continuous multilinear
operators from Banach lattices into a Banach space. It satisfies the property:
for all $m\in\mathbb{N}^{\ast}$, Banach lattices $E_{1},\ldots,E_{m},$ and
Banach space $X,$ the components

$\mathcal{M}_{L}^{+}(E_{1},...,E_{m};X):=\mathcal{L}(E_{1},...,E_{m};X)\cap\mathcal{M}_{L}^{+}$

satisfy:
$(i)$ $\mathcal{M}_{L}^{+}(E_{1},...,E_{m};X)$ is a linear subspace of
$\mathcal{L}(E_{1},...,E_{m};X)$ which contains the $m$-linear mappings of
finite type.
$(ii)$ The positive ideal property: If
$T\in\mathcal{M}_{L}^{+}\left(E_{1},\ldots,E_{m};X\right),u_{j}\in\mathcal{L}^{r}\left(G_{j};E_{j}\right)$
for $j=$ $1,\ldots,m$ and $v\in\mathcal{L}(X;Y)$, then $v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)$ is in
$\mathcal{M}_{L}^{+}\left(G_{1},\ldots,G_{m};Y\right)$.
If
$\|\cdot\|_{\mathcal{M}_{L}^{+}}:\mathcal{M}_{L}^{+}\rightarrow\mathbb{R}^{+}$
satisfies:
a)
$\left(\mathcal{M}_{L}^{+}(E_{1},...,E_{m};X),\|\cdot\|_{\mathcal{M}_{L}^{+}}\right)$
is a normed (Banach) space for all Banach lattices $E_{1},\ldots,E_{m}$ and
Banach space $X$.
b) The $m$ -linear form $T^{m}:\mathbb{K}^{m}\rightarrow\mathbb{K}$ given by
$T^{m}\left(\lambda^{1},\ldots,\lambda^{m}\right)=\lambda^{1}\ldots\lambda^{m}$
satisfies $\left\|T^{m}\right\|_{\mathcal{M}_{L}^{+}}=1$ for all $m$,
c)
$T\in\mathcal{M}_{L}^{+}\left(E_{1},\ldots,E_{m};X\right),u_{j}\in\mathcal{L}^{r}\left(G_{j},E_{j}\right)$
for $j=$ $1,\ldots,m$ and $v\in\mathcal{L}(X,Y)$ then

$\left\|v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)\right\|_{\mathcal{M}_{L}^{+}}\leq\|v\|\|T\|_{\mathcal{M}_{L}^{+}}\left\|u_{1}\right\|\ldots\left\|u_{m}\right\|.$

The class $\left(\mathcal{M}_{L}^{+},\|\cdot\|_{\mathcal{M}_{L}^{+}}\right)$
is referred to as a normed (Banach) positive left multi-ideal. When $m=1$, it
specifically corresponds to the case of a positive left ideal.

######  Remark 2.11.

The condition $\left(ii\right)$ is equivalent to the following assertion: For
any $T\in\mathcal{M}_{L}^{+}\left(E_{1},\ldots,E_{m};X\right)$,
$u_{j}\in\mathcal{L}^{+}\left(G_{j};E_{j}\right)$ for $j=$ $1,\ldots,m$ and
$v\in\mathcal{L}(X;Y)$, the composition $v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)$ belongs to
$\mathcal{M}_{L}^{+}\left(G_{1},\ldots,G_{m};Y\right)$. Indeed, without loss
of generality, we see this equivalence for the case $m=2.$ Consider
$u_{j}\in\mathcal{L}^{r}\left(G_{j};E_{j}\right)$ such that
$u_{j}=u_{j}^{1}-u_{j}^{2}$ where
$u_{j}^{1},u_{j}^{2}\subset\mathcal{L}^{+}\left(G_{j};E_{j}\right)$
$\left(j=1,2\right)$. Now, we can write

$\displaystyle v\circ T\circ\left(u_{1},u_{2}\right)$ $\displaystyle=$
$\displaystyle v\circ
T\circ\left(u_{1}^{1}-u_{1}^{2},u_{2}^{1}-u_{2}^{2}\right)$ $\displaystyle=$
$\displaystyle v\circ T\circ\left(u_{1}^{1},u_{2}^{1}\right)-v\circ
T\circ\left(u_{1}^{1},u_{2}^{2}\right)-v\circ
T\circ\left(u_{1}^{2},u_{2}^{1}\right)+v\circ
T\circ\left(u_{1}^{2},u_{2}^{2}\right).$

Since the operators $v\circ T\circ\left(u_{1}^{j},u_{2}^{k}\right)$ belong to
$\mathcal{M}_{L}^{+}\left(G_{1},G_{2};Y\right)$ $\left(j,k=1,2\right)$. By the
linearity of $\mathcal{M}_{L}^{+}\left(G_{1},G_{2};Y\right)$, we conclude that
$v\circ
T\circ\left(u_{1},u_{2}\right)\in\mathcal{M}_{L}^{+}\left(G_{1},G_{2};Y\right)$.
The converse is immediate.

######  Remark 2.12.

It is evident that all multi-ideals are indeed positive left multi-ideals.

The factorization method. For $m\in\mathbb{N}^{\ast},$ let
$(\mathcal{B}_{j,L}^{+},\|\cdot\|_{\mathcal{B}_{j,L}^{+}})$ be Banach positive
left ideals for $i=1,...,m$. We define the class
$\mathcal{L}(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+})$ as follows: Let
$E_{1},...,E_{m}$ be Banach lattices and $Y$ be a Banach space. An operator
$T$ belongs to
$\mathcal{L}(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+})(E_{1},...,E_{m};Y)$
if there exist Banach spaces $X_{1},...,X_{m},$ operators
$u_{j}\in\mathcal{B}_{j,L}^{+}(E_{j};X_{j})$ $\left(1\leq j\leq m\right),$ and
a multilinear operator $S\in\mathcal{L}(X_{1},...,X_{m};Y)$ such that

$\begin{array}[]{ccccc}E_{1}&\times...\times&E_{m}&\overset{T}{\longrightarrow}&Y\\\
u_{1}\downarrow&&u_{m}\downarrow&\nearrow S&\\\
X_{1}&\times...\times&X_{m}&&\end{array}$

i.e., $T=S\circ(u_{1},...,u_{m})$. In this case, we define the norm of $T$
with respect to $\mathcal{L}(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+})$
as

$\|T\|_{\mathcal{L}(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+})}=\inf\\{\|S\|\prod_{j=1}^{m}\|u_{j}\|_{\mathcal{B}_{j,L}^{+}}\\},$

where the infimum is taken over all possible factorizations of $T$ as
described above.

######  Remark 2.13.

The class $\mathcal{L}(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+})$ is
indeed a positive left multi-ideal.

###  2.4. Positive multi-ideals

In this subsection, we introduce the definition of positive multi-ideals. The
construction of these positive multi-ideals utilizes techniques inspired by
[17], employing multilinear operators exclusively defined between Banach
lattices.

######  Definition 2.14.

A positive multi-ideal (or positive ideal of multilinear operators) is a
subclass $\mathcal{M}^{+}$ of all continuous multilinear operators between
Banach lattices. It is characterized by the property that for all
$m\in\mathbb{N}^{\ast}$ and Banach lattices $E_{1},\ldots,E_{m}$ and $F$, the
components

$\mathcal{M}^{+}(E_{1},...,E_{m};F):=\mathcal{L}(E_{1},...,E_{m};F)\cap\mathcal{M}^{+}$

satisfy:
$(i)$ $\mathcal{M}^{+}(E_{1},...,E_{m};F)$ is a linear subspace of
$\mathcal{L}(E_{1},...,E_{m};F)$ which contains the $m$-linear mappings of
finite type.
$(ii)$ The positive ideal property: If
$T\in\mathcal{M}^{+}\left(E_{1},\ldots,E_{m};F\right),u_{j}\in\mathcal{L}^{r}\left(G_{j};E_{j}\right)$
for $j=$ $1,\ldots,m$ and $v\in\mathcal{L}^{r}(F;G)$, then $v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)$ is in
$\mathcal{M}^{+}\left(G_{1},\ldots,G_{m};G\right)$.
If $\|\cdot\|_{\mathcal{M}^{+}}:\mathcal{M}^{+}\rightarrow\mathbb{R}^{+}$
satisfies:
a)
$\left(\mathcal{M}^{+}(E_{1},...,E_{m};F),\|\cdot\|_{\mathcal{M}^{+}}\right)$
is a normed (Banach) space for all Banach lattices $E_{1},\ldots,E_{m},$ $F$.
b) The $m$ -linear form $T^{m}:\mathbb{K}^{m}\rightarrow\mathbb{K}$ given by
$T^{m}\left(\lambda^{1},\ldots,\lambda^{m}\right)=\lambda^{1}\ldots\lambda^{m}$
satisfies $\left\|T^{m}\right\|_{\mathcal{M}^{+}}=1$ for all $m$,
c)
$T\in\mathcal{M}^{+}\left(E_{1},\ldots,E_{m};F\right),u_{j}\in\mathcal{L}^{r}\left(G_{j};E_{j}\right)$
for $j=$ $1,\ldots,m$ and $v\in\mathcal{L}^{r}(F;G)$ then

$\left\|v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)\right\|_{\mathcal{M}^{+}}\leq\|v\|\|T\|_{\mathcal{M}^{+}}\left\|u_{1}\right\|\ldots\left\|u_{m}\right\|.$

The class $\left(\mathcal{M}^{+},\|\cdot\|_{\mathcal{M}}\right)$ is referred
to as a normed (Banach) positive multi-ideal. In particular, when $m=1$, we
specifically refer to it as a positive operator ideal.

######  Remark 2.15.

As mentioned in the remarks 2.5 and 2.11, we can substitute the space
$\mathcal{L}^{r}\left(G_{j};E_{j}\right)$ with the set
$\mathcal{L}^{+}\left(G_{j};E_{j}\right)$ $\left(1\leq j\leq m\right)$ and
$\mathcal{L}^{r}(F;G)$ with the set $\mathcal{L}^{+}(F;G).$ Then, the
condition $\left(ii\right)$ remains valid.

######  Remark 2.16.

1) It is evident that all multi-ideals are indeed positive multi-ideals.

2) Every positive right or left multi-ideal is positive multi-ideal.

Let $\mathcal{B}_{R}^{+}$ be a positive right ideal, and $\mathcal{M}_{L}^{+}$
be a positive left multi-ideal. The composition
$\mathcal{B}_{R}^{+}\circ\mathcal{M}_{L}^{+}$ is defined as the class of
multilinear operators $T$ that can be factorized through a Banach space as
$T=v\circ S$. In other words, for any Banach lattices $E_{1},...,E_{m}$ and
$F$, and for
$T\in\mathcal{B}_{R}^{+}\circ\mathcal{M}_{L}^{+}(E_{1},...,E_{m};F)$, there
exist a Banach space $X$, an element $v$ belonging to
$\mathcal{B}_{R}^{+}\left(X,E\right)$, and a multilinear operator $S$
belonging to $\mathcal{M}_{L}^{+}(E_{1},...,E_{m};X)$ such that

$\begin{array}[]{ccc}E_{1}\times...\times
E_{m}&\overset{T}{\longrightarrow}&F\\\ S\downarrow&\nearrow v&\\\
X&&\end{array}$

In other words, $T$ can be expressed as $T=v\circ S$. If $\mathcal{B}_{R}^{+}$
is normed positive right ideal and $\mathcal{M}_{L}^{+}$ is normed positive
left multi-ideal and
$T\in\mathcal{B}_{R}^{+}\circ\mathcal{M}_{L}^{+}(E_{1},...,E_{m};F)$ we define

$\left\|T\right\|_{\mathcal{B}_{R}^{+}\circ\mathcal{M}_{L}^{+}}=\inf\left\\{\left\|v\right\|_{\mathcal{B}_{R}^{+}}\left\|S\right\|_{\mathcal{M}_{L}^{+}}:T=v\circ
S\right\\}.$

The proof of the following theorem can be easily proved.

######  Theorem 2.17.

If $\mathcal{B}_{R}^{+}$ is normed (Banach) positive right ideal and
$\mathcal{M}_{L}^{+}$ is normed (Banach) positive left multi-ideal, then

$\left(\mathcal{B}_{R}^{+}\circ\mathcal{M}_{L}^{+},\left\|.\right\|_{\mathcal{B}_{R}^{+}\circ\mathcal{M}_{L}^{+}}\right)$

is a normed (Banach) positive multi-ideal.

If $\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}$ represent positive left
ideals, and $\mathcal{M}_{R}^{+}$ represents a positive right multi-ideal, the
class
$\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right)$
is defined as the class of multilinear operators $T$ that can be expressed
through Banach spaces as $T=S\left(v_{1},...,v_{m}\right)$. In other words,
for any Banach lattices $E_{1},...,E_{m}$ and $F$, we say that $T$ belongs to
$\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right)(E_{1},...,E_{m};F)$
if there exist Banach spaces $X_{1},...,X_{m}$, elements $u_{j}$ belonging to
$\mathcal{B}_{j,L}^{+}\left(E_{j},X_{j}\right)\left(1\leq j\leq m\right)$, and
a multilinear operator $S$ belonging to
$\mathcal{M}_{R}^{+}(X_{1},...,X_{m};F)$ such that

$\begin{array}[]{ccccc}E_{1}&\times...\times&E_{m}&\overset{T}{\longrightarrow}&F\\\
u_{1}\downarrow&&u_{m}\downarrow&\nearrow S&\\\
X_{1}&\times...\times&X_{m}&&\end{array}$

In other words, $T$ can be represented as $T=S(u_{1},...,u_{m})$. If
$\mathcal{B}_{j,L}^{+}$ $\left(1\leq j\leq m\right)$ are normed positive left
ideals and $\mathcal{M}_{R}^{+}$ is normed positive right multi-ideal and

$T\in\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right)(E_{1},...,E_{m};F)$

we define

$\left\|T\right\|_{\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right)}=\inf\left\\{\mathop{\displaystyle\prod}\limits_{i=1}^{m}\left\|u_{i}\right\|_{\mathcal{B}_{i,L}^{+}}\left\|S\right\|_{\mathcal{M}_{R}^{+}}:T=S(u_{1},...,u_{m})\right\\}.$

The proof of the following theorem can be easily proved.

######  Theorem 2.18.

If $\mathcal{B}_{j,L}^{+}$ $\left(1\leq j\leq m\right)$ are normed (Banach)
positive left ideals and $\mathcal{M}_{R}^{+}$ is normed (Banach) positive
right multi-ideal, then

$\left(\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right),\left\|.\right\|_{\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right)}\right)$

is a normed (Banach) positive multi-ideal.

In [7, Theorem 3.2], the authors established the following factorization

$\mathcal{N}_{p}^{m+}=\mathcal{D}_{p}^{m+}(\Pi_{p}^{+},...,\Pi_{p}^{+}).$

In other words, we can state that the class $\mathcal{N}_{p}^{m+}$ represents
the Banach positive multi-ideal of type
$\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right)$
where $\mathcal{M}_{R}^{+}=\mathcal{D}_{p}^{m+}$ and
$\mathcal{B}_{j,L}^{+}=\Pi_{p}^{+}$ for $1\leq j\leq m.$ As
$\mathcal{D}_{p}^{m+}=\mathcal{D}_{p}^{+}\circ\mathcal{L}$, thus

$\mathcal{N}_{p}^{m+}=\mathcal{D}_{p}^{+}\circ\mathcal{L}(\Pi_{p}^{+},...,\Pi_{p}^{+}).$

This means that the class $\mathcal{N}_{p}^{m+}$ consists of the multilinear
operators that can be obtained by composing of positive strongly $p$-summing
operators and multilinear operators obtained via a factorization method
utilizing positive left ideal $\Pi_{p}^{+}$.

##  3\. Positive (p1,…,pm;r)-dominated multilinear operators

The concept of $(p_{1},\ldots,p_{m};r)$-dominated multilinear operators was
introduced by Achour [1]. This notion is a natural generalization of the
concept of $(p,q)$-dominated linear operators, which was initially studied by
Pietsch in [18]. In this section, we explore the positive multilinear version
of this concept, providing a good example of a positive multi-ideal.

######  Definition 3.1.

Consider $1\leq r,p,p_{1},\ldots,p_{m}\leq\infty$ such that
$\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}+\frac{1}{r}$. Let
$E_{1},...,E_{m}$ and $F$ be Banach lattices. A mapping
$T\in\mathcal{L}\left(E_{1},\ldots,E_{m};F\right)$ is said to be positive
$(p_{1},\ldots,p_{m};r)$-dominated if there exists a constant $C>0$ such that
for any $(x_{i}^{1},...,x_{i}^{m})\in E_{1}^{+}\times...\times E_{m}^{+}$
$\left(1\leq i\leq n\right)$ and $y_{1}^{\ast},\ldots,y_{n}^{\ast}\in
F^{\ast+}$, the following inequality holds:

$\left\|\left(\left\langle
T\left(x_{i}^{1},\ldots,x_{i}^{m}\right),y_{i}^{\ast}\right\rangle\right)_{i=1}^{n}\right\|_{p}\leq
C\prod_{j=1}^{m}\left\|\left(x_{i}^{j}\right)_{i=1}^{n}\right\|_{p_{j},w}\left\|\left(y_{i}^{\ast}\right)_{i=1}^{n}\right\|_{r,w}.$
(3.1)

The space consisting of all such mappings is denoted by
$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}\left(E_{1},\ldots,E_{m};F\right)$.
In this case, we define

$d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(T)=\inf\\{C>0:C\text{ {satisfies
inequality} }(\ref{def1sec3})\\}.$

It is easy to verify that every $(p_{1},\ldots,p_{m};r)$-dominated multilinear
operator is positive $(p_{1},\ldots,p_{m};r)$-dominated. Then , by [1,
Proposition 2.4 (i)] we have

$\mathcal{L}_{f}(X_{1},...,X_{m};F)\subset\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}\left(E_{1},\ldots,E_{m};F\right).$

We readily deduce that the class
$\left(\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+},d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}\right)$
constitutes a Banach positive multi-ideal. The proof is straightforward and
will be omitted.

In the next result, we give the following equivalent definition.

######  Theorem 3.2.

Let $1\leq r,p,p_{1},\ldots,p_{m}\leq\infty$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}+\frac{1}{r}$ and
$T\in\mathcal{L}\left(E_{1},\ldots,E_{m};F\right)$. The following properties
are equivalent:
$(a)$ The operator $T$ is positive $(p_{1},\ldots,p_{m};r)$-dominated.
$(b)$ There is a constant $C>0$ such that for any
$(x_{i}^{1},...,x_{i}^{m})\in E_{1}\times...\times E_{m}$ $\left(1\leq i\leq
n\right)$ and $y_{1}^{\ast},\ldots,y_{n}^{\ast}\in F^{\ast}$, we have

$\left\|\left(\left\langle
T\left(x_{i}^{1},\ldots,x_{i}^{m}\right),y_{i}^{\ast}\right\rangle\right)_{i=1}^{n}\right\|_{p}\leq
C\prod_{j=1}^{m}\left\|\left(x_{i}^{j}\right)_{i=1}^{n}\right\|_{p_{j},\left|w\right|}\left\|\left(y_{i}^{\ast}\right)_{i=1}^{n}\right\|_{r,\left|w\right|}.$
(3.2)

In this case, we define

$d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(T)=\inf\\{C>0:C\text{ {satisfies
inequality} }(\ref{def2sec3})\\}.$

###### Proof.

$(b)\Rightarrow(a):$ Immediately applying Definition 3.1 for
$(x_{i}^{1},...,x_{i}^{m})\in E_{1}^{+}\times...\times E_{m}^{+}$, $1\leq
i\leq n$ and $y_{1}^{\ast},...,y_{n}^{\ast}\in F^{\ast+}$.
$(a)\Rightarrow(b):$ Suppose that $T$ is positive
$(p_{1},\ldots,p_{m};r)$-dominated. For convenience, we prove only the
inequality for the case when $m=2$. Let $\left(x_{i}^{1},x_{i}^{2}\right)\in
E_{1}\times E_{2},(1\leq i\leq n)$ $y_{1}^{\ast},\ldots,y_{n}^{\ast}\in
F^{\ast}$, then one has

$\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T(x_{i}^{1},x_{i}^{2}),y_{i}^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}=(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1+}-x_{i}^{1-},x_{i}^{2+}-x_{i}^{2-}\right),y_{i}^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1+},x_{i}^{2+}\right),y_{i}^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1+},x_{i}^{2-}\right),y_{i}^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}+$
$\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2+}\right),y_{i}^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2-}\right),y_{i}^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}$

which is less than or equal to

$\displaystyle\leq$ $\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1+},x_{i}^{2+}\right),y_{i}^{\ast+}\right\rangle\right|^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1+},x_{i}^{2+}\right),y_{i}^{\ast-}\right\rangle\right|^{p})^{\frac{1}{p}}+$
$\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1+},x_{i}^{2-}\right),y_{i}^{\ast+}\right\rangle\right|^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1+},x_{i}^{2-}\right),y_{i}^{\ast-}\right\rangle\right|^{p})^{\frac{1}{p}}+$
$\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2+}\right),y_{i}^{\ast+}\right\rangle\right|^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2+}\right),y_{i}^{\ast-}\right\rangle\right|^{p})^{\frac{1}{p}}+$
$\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2+}\right),y_{i}^{\ast+}\right\rangle\right|^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2+}\right),y_{i}^{\ast-}\right\rangle\right|^{p})^{\frac{1}{p}}+$
$\displaystyle(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2-}\right),y_{i}^{\ast+}\right\rangle\right|^{p})^{\frac{1}{p}}+(\sum_{i=1}^{n}\left|\left\langle
T\left(x_{i}^{1-},x_{i}^{2-}\right),y_{i}^{\ast-}\right\rangle\right|^{p})^{\frac{1}{p}}$

finally we have

$(\sum_{i=1}^{n}\left|\left\langle
T(x_{i}^{1},x_{i}^{2}),y_{i}^{\ast}\right\rangle\right|^{p})^{\frac{1}{p}}\leq
8d_{\left(p_{1},p_{2};r\right)}^{+}(T)\|(x_{i}^{1})_{i=1}^{n}\|_{p_{1},|w|}\|(x_{i}^{2})_{i=1}^{n}\|_{p_{2},|w|}\|(y_{i}^{\ast})_{i=1}^{n}\|_{r,|w|}.$

######  Proposition 3.3.

The class
$(\mathcal{D}_{\left(p_{1},...,p_{m};r\right)}^{+},d_{\left(p_{1},...,p_{m};r\right)}^{+})$
is a Banach positive multi-ideal.

###### Proof.

We will verify the positive ideal property; the rest is straightforward. Let
$E_{1},\ldots,E_{m}$ and $F$ are Banach lattices. Let
$T\in\mathcal{D}_{\left(p_{1},...,p_{m};r\right)}^{+}\left(E_{1},\ldots,E_{m};F\right),u_{j}\in\mathcal{L}^{+}\left(G_{j};E_{j}\right)$
$\left(1\leq j\leq m\right)$ and $v\in\mathcal{L}^{+}(F;G)$ where
$G_{1},\ldots,G_{m}$ and $G$ are Banach lattices. Let
$(x_{i}^{1},...,x_{i}^{m})\in G_{1}^{+}\times...\times G_{m}^{+}$ $\left(1\leq
i\leq n\right)$ and $y_{1}^{\ast},\ldots,y_{n}^{\ast}\in G^{\ast+}.$ Since
$T\in\mathcal{D}_{\left(p_{1},...,p_{m};r\right)}^{+}\left(E_{1},\ldots,E_{m};F\right),$
$u_{j}\left(x_{i}^{j}\right)\geq 0$ and $v^{\ast}\left(y_{i}^{\ast}\right)\geq
0$ $\left(1\leq j\leq m,1\leq i\leq n\right)$ we have

$\displaystyle\left\|\left(\left\langle v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)\left(x_{i}^{1},\ldots,x_{i}^{m}\right),y_{i}^{\ast}\right\rangle\right)_{i=1}^{n}\right\|_{p}$
$\displaystyle=$ $\displaystyle\left\|\left(\left\langle
T\left(u_{1}\left(x_{i}^{1}\right),\ldots,u_{m}\left(x_{i}^{m}\right)\right),v^{\ast}\left(y_{i}^{\ast}\right)\right\rangle\right)_{i=1}^{n}\right\|_{p}$
$\displaystyle\leq$ $\displaystyle
d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(T)\prod_{j=1}^{m}\left\|\left(u_{j}\left(x_{i}^{j}\right)\right)_{i=1}^{n}\right\|_{p_{j},w}\left\|\left(v^{\ast}\left(y_{i}^{\ast}\right)\right)_{i=1}^{n}\right\|_{r,w}$
$\displaystyle\leq$ $\displaystyle
d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(T)\left\|u_{1}\right\|\ldots\left\|u_{m}\right\|\|v\|\prod_{j=1}^{m}\left\|\left(x_{i}^{j}\right)_{i=1}^{n}\right\|_{p_{i},w}\left\|\left(y_{i}^{\ast}\right)_{i=1}^{n}\right\|_{r,w}$

thus $v\circ T\circ\left(u_{1},\ldots,u_{m}\right)$ is in
$\mathcal{D}_{\left(p_{1},...,p_{m};r\right)}^{+}\left(G_{1},\ldots,G_{m};G\right)$
and we have

$d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}\left(v\circ
T\circ\left(u_{1},\ldots,u_{m}\right)\right)\leq
d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(T)\left\|u_{1}\right\|\ldots\left\|u_{m}\right\|\|v\|.$

Now, we characterize the positive $(p_{1},\ldots,p_{m};r)$-dominated
multilinear operators by giving the Pietsch domination theorem. To establish
this, we employ the full general Pietsch domination theorem presented by
Pellegrino et al. in [16, Theorem 4.6].

######  Theorem 3.4 (Pietsch domination theorem).

Let $1\leq r,p,p_{1},\ldots,p_{m}\leq\infty$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}+\frac{1}{r}$. Let
$E_{1},...,E_{m}$ and $F$ be Banach lattices. The following statements are
equivalent:

1) The operator $T\in\mathcal{L}\left(E_{1},\ldots,E_{m};F\right)$ is positive
$(p_{1},\ldots,p_{m};r)$-dominated.

2) There is a constant $C>0$ and Borel probability measures $\mu_{j}$ on
$B_{E_{j}^{\ast}}^{+}$ ($1\leq j\leq m$) and $\mu_{m+1}$ on
$B_{F^{\ast\ast}}^{+}$ such that

$\begin{array}[]{ll}&|\langle T(x^{1},...,x^{m}),y^{\ast}\rangle|\\\
&\leq\displaystyle
C\prod_{j=1}^{m}\Big{(}\int_{B_{E_{j}^{\ast}}^{+}}\langle|x^{j}|,\varphi_{j}\rangle^{p_{j}}d\mu_{j}\Big{)}^{\frac{1}{p_{j}}}\Big{(}\int_{B_{F^{\ast\ast}}^{+}}\langle|y^{\ast}|,\varphi_{m+1}\rangle^{r}d\mu_{m+1}\Big{)}^{\frac{1}{r}}\end{array}$
(3.3)

for all $(x^{1},...,x^{m},y^{\ast})\in E_{1}\times...\times E_{m}\times
F^{\ast}.$

3) There is a constant $C>0$ and Borel probability measures $\mu_{j}$ on
$B_{E_{j}^{\ast}}^{+}$ ($1\leq j\leq m$) and $\mu_{m+1}$ on
$B_{F^{\ast\ast}}^{+}$ such that

$\begin{array}[]{ll}&|\langle T(x^{1},...,x^{m}),y^{\ast}\rangle|\\\
&\leq\displaystyle C\prod_{j=1}^{m}\Big{(}\int_{B_{E_{j}^{\ast}}^{+}}\langle
x^{j},\varphi_{j}\rangle^{p_{j}}d\mu_{j}\Big{)}^{\frac{1}{p_{j}}}\Big{(}\int_{B_{F^{\ast\ast}}^{+}}\langle
y^{\ast},\varphi_{m+1}\rangle^{r}d\mu_{m+1}\Big{)}^{\frac{1}{r}}\end{array}$
(3.4)

for all $(x^{1},...,x^{m},y^{\ast})\in E_{1}^{+}\times...\times
E_{m}^{+}\times F^{\ast+}$.

###### Proof.

$1)\Leftrightarrow 2):$ We will choose the parameters as specified in [16,
Theorem 4.6]

$\left\\{\begin{array}[]{l}t=m+1\\\ E_{i}=\mathbb{R},\text{ }i=1,\ldots,k\\\
G_{j}=E_{j},\text{ }j=1,\ldots,m\\\ G_{m+1}=F^{\ast}\\\
K_{j}=B_{E_{j}^{\ast}}^{+},\text{ }j=1,\ldots,m\\\
K_{m+1}=B_{F^{\ast\ast}}^{+}\\\
\mathcal{H}=\mathcal{L}\left(E_{1},...,E_{m};F\right)\\\
S\left(T,\lambda,x^{1},\ldots,x^{m},y^{\ast}\right)=|\langle
T(x^{1},...,x^{m}),y^{\ast}\rangle|\\\
R_{j}(\varphi_{j},\lambda,x^{j})=\langle|x^{j}|,\varphi_{j}\rangle,j=1,\ldots,m\\\
R_{m+1}(\varphi_{m+1},\lambda,y^{\ast})=\langle|y^{\ast}|,\varphi_{m+1}\rangle.\end{array}\right.$

We can easily conclude that $T:E_{1}\times...\times E_{m}\rightarrow F$ is
positive $(p_{1},\ldots,p_{m};r)$-dominated if, and only if, $T$ is
$R_{1},...,R_{m+1}$-$S$-abstract $(p_{1},...,p_{m};r)$-summing. As outlined in
[16, Theorem 4.6], this implies that $T$ is $R_{1},...,R_{m+1}$-$S$-abstract
$(p_{1},...,p_{m};r)$-summing if, and only if, there exists a positive
constant $C$ and probability measures $\mu_{j}$ on $K_{j}$, $j=1,...,m+1,$
such that

$\begin{array}[]{lll}\displaystyle
S\left(T,\lambda,x^{1},\ldots,x^{m},y^{\ast}\right)&\leq&\displaystyle
C\prod_{j=1}^{m}\left(\int_{K_{j}}R_{j}(\varphi_{j},\lambda,x^{j})^{p_{j}}d\mu_{j}\right)^{\frac{1}{p_{j}}}\\\
&&\displaystyle\times\left(\int_{K_{m+1}}R_{m+1}(\varphi_{m+1},\lambda,y^{\ast})^{r}\right)^{\frac{1}{r}}.\end{array}$

Consequently

$\begin{array}[]{ll}&|\langle T(x^{1},...,x^{m}),y^{\ast}\rangle|\\\
&\leq\displaystyle
C\prod_{j=1}^{m}\Big{(}\int_{B_{E_{j}^{\ast}}^{+}}\langle|x^{j}|,\varphi_{j}\rangle^{p_{j}}d\mu_{j}\Big{)}^{\frac{1}{p_{j}}}\Big{(}\int_{B_{F^{\ast\ast}}^{+}}\langle|y^{\ast}|,\varphi_{m+1}\rangle^{r}d\mu_{m+1}\Big{)}^{\frac{1}{r}}\end{array}$
(3.5)

and with this, we conclude the proof.

$2)\Leftrightarrow 3):$ Straightforward by using the idea of the proof of
Theorem 3.2.

As an immediate consequence of Theorem 3.4, we can show that if $p_{j}\leq
q_{j}$ and $r\leq s$ then

$\mathcal{D}_{\left(p_{1},...,p_{m};r\right)}^{+}\left(E_{1},\ldots,E_{m};F\right)\subset\mathcal{D}_{\left(q_{1},...,q_{m};s\right)}^{+}\left(E_{1},\ldots,E_{m};F\right).$

The following result demonstrates that the class of positive
$\left(p_{1},\ldots,p_{m};r\right)$-dominated multilinear operators can be
construed as

$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}=\mathcal{D}_{r^{\ast}}^{m+}\left(\Pi_{p_{1}}^{+},\ldots,\Pi_{p_{m}}^{+}\right).$

This represents a positive variant of the Kwapień factorization.

######  Theorem 3.5.

Let $1\leq r,p,p_{1},\ldots,p_{m}\leq\infty$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}+\frac{1}{r}$. Then,
$T\in\mathcal{L}(E_{1},...,E_{m};F)$ is positive
$\left(p_{1},\ldots,p_{m};r\right)$-dominated if and only if there exist
Banach spaces $X_{1},\ldots,X_{m}$, a Cohen positive strongly
$r^{\ast}$-summing multilinear operator $S:X_{1}\times...\times
X_{m}\rightarrow F$ and linear operators
$R_{j}\in\Pi_{p_{j}}^{+}\left(E_{j};X_{j}\right)$ so that
$T=S\circ\left(R_{1},\ldots,R_{m}\right)$, i.e.,

$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(E_{1},...,E_{m};F)=\mathcal{D}_{r^{\ast}}^{m+}\left(\Pi_{p_{1}}^{+},\ldots,\Pi_{p_{m}}^{+}\right)(E_{1},...,E_{m};F).$

Moreover

$d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(T)=\inf\left\\{d_{r^{\ast}}^{m+}(S)\prod_{j=1}^{m}\pi_{p_{j}}^{+}\left(R_{j}\right):T=S\circ\left(R_{1},\ldots,R_{m}\right)\right\\}.$

###### Proof.

First we prove the converse. Suppose that
$T=S\circ\left(R_{1},\ldots,R_{m}\right)$ where $R_{j}$ is positive
$p_{j}$-summing and $S$ is Cohen positive strongly $r^{\ast}$-summing
multilinear operator. By [6, Theorem 2.5], there exists $\mu$ on
$B_{F^{\ast\ast}}^{+}$ such that, for all $x^{j}\in E_{j}$ $(1\leq j\leq m)$
and $y^{\ast}\in B_{F^{\ast}},\ $we have

$\left|\left\langle T(x^{1},...,x^{m}),y^{\ast}\right\rangle\right|$ | $=$ | $\left|\left\langle S\left(R_{1}\left(x^{1}\right),\ldots,R_{m}\left(x^{m}\right)\right),y^{\ast}\right\rangle\right|$
---|---|---
| $\leq$ | $d_{r^{\ast}}^{m}(S)\prod\limits_{j=1}^{m}\left\|R_{j}\left(x^{j}\right)\right\|\displaystyle\left(\int_{B_{F^{\ast\ast}}^{+}}\left|\left\langle y^{\ast},y^{\ast\ast}\right\rangle\right|^{r}d\mu\right)^{\frac{1}{r}}.$

Since $R_{j}$ is positive $p_{j}$-summing then, by (1.2) there is a
probability measure $\mu_{j}$ on
$K_{j}=\left(B_{E_{j}^{\ast}}^{+},w^{\ast}\right)$ such that for all $x^{j}\in
E_{j}^{+}$

$\left\|R_{j}\left(x^{j}\right)\right\|\leq\pi_{p_{j}}^{+}\left(R_{j}\right)\left(\int_{K_{j}}\langle
x^{j},\varphi_{j}\rangle^{p_{j}}d\mu_{j}\right)^{\frac{1}{p_{j}}}.$

Consequently

$\displaystyle\left|\left\langle
T(x^{1},\cdots,x^{m}),y^{*}\right\rangle\right|$ $\displaystyle\leq
d_{r^{*}}^{m+}(S)\prod\limits_{j=1}^{m}\pi_{p_{j}}^{+}\left(R_{j}\right)\left(\int_{K_{j}}\langle
x^{j},\varphi_{j}\rangle^{p_{j}}d\mu_{j}\right)^{\frac{1}{p_{j}}}\left(\int_{B_{F^{**}}^{+}}\left|\left\langle
y^{*},y^{**}\right\rangle\right|^{r}d\mu\right)^{\frac{1}{r}}.$

Therefore, $T$ is positive $(p_{1},...,p_{m};r)$-dominated by Theorem 3.4 and

$d_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(T)\leq
d_{r^{\ast}}^{m+}(S)\prod\limits_{j=1}^{m}\pi_{p_{j}}^{+}\left(R_{j}\right).$

To prove the first implication. Let
$T\in\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}(E_{1},...,E_{m};F).$
By Theorem 3.4, there are probability measures $\mu_{j}$ on
$K_{j}=\left(B_{E_{j}^{\ast}}^{+},w^{\ast}\right)$ and $\mu$ on
$B_{F^{\ast\ast}}^{+}$ such that for all $x^{j}\in E_{j}^{+}$ and $y^{\ast}\in
F^{\ast+}$ we have

$\displaystyle\left|\left\langle
T\left(x^{1},\ldots,x^{m}\right),y^{*}\right\rangle\right|\leq$ $\displaystyle
d_{p_{1},...,p_{m};r}^{+}(T)\prod_{j=1}^{m}\left(\int_{K_{j}}\langle
x^{j},\varphi_{j}\rangle^{p_{j}}d\mu_{j}\right)^{\frac{1}{p_{j}}}\left(\int_{B_{F^{**}}^{+}}\langle
y^{*},y^{**}\rangle^{r}d\mu\right)^{\frac{1}{r}}.$

We now consider the operator $R_{j}^{0}:E_{j}\rightarrow
L_{p_{j}}\left(K_{j},\mu_{j}\right)$ which is given by
$R_{j}^{0}\left(x^{j}\right)(\varphi_{j})=\varphi_{j}(x^{j})$. Notice that for
all $x^{j}\in E_{j}^{+},1\leq j\leq m$ we have

$\left\|R_{j}^{0}\left(x^{j}\right)\right\|=\left(\int_{K_{j}}\langle
x^{j},\varphi_{j}\rangle^{p_{j}}d\mu_{j}\right)^{\frac{1}{p_{j}}}\leq\left\|x^{j}\right\|.$

Let $X_{j}=\overline{R_{j}^{0}(E_{j})}^{L_{p_{j}}(K_{j},\mu_{j})}$ be the
closure in $L_{p_{j}}(K_{j},\mu_{j})$ of the range of $R_{j}^{0}$, and let
$R_{j}:E_{j}\rightarrow X_{j}$ be the induced operator. Note that $R_{j}$ is
positive $p_{j}$-summing with $\pi_{p_{j}}^{+}\left(R_{j}\right)\leq 1$. Let
$S_{0}$ be the multilinear operator defined on
$R_{1}^{0}\left(E_{1}\right)\times\ldots\times R_{m}^{0}\left(E_{m}\right)$ by

$S_{0}\left(R_{1}^{0}\left(x^{1}\right),\ldots,R_{m}^{0}\left(x^{m}\right)\right)=T\left(x^{1},...,x^{m}\right)$

this definition makes sense because

$\displaystyle\left|\left\langle
S_{0}\left(R_{1}^{0}\left(x^{1}\right),\ldots,R_{m}^{0}\left(x^{m}\right)\right),y^{*}\right\rangle\right|$
$\displaystyle\leq
d_{p_{1},...,p_{m};r}^{+}(T)\prod_{j=1}^{m}\left\|R_{j}^{0}\left(x^{j}\right)\right\|\left(\int_{B_{F^{**}}^{+}}\left\langle
y^{*},y^{**}\right\rangle^{r}d\mu\right)^{\frac{1}{r}}.$

It follows that $S_{0}$ is continuous on
$R_{1}^{0}\left(E_{1}\right)\times\ldots\times R_{m}^{0}\left(E_{m}\right)$
and has a unique bounded multilinear extension $S$ to $X_{1}\times\cdots\times
X_{m}$. This implies that $S$ is Cohen positive strongly $r^{\ast}$-summing
multilinear operator. Finally, note also that
$T=S\circ\left(R_{1},\ldots,R_{m}\right)$ where
$R_{j}\in\Pi_{p_{j}}^{+}\left(E_{j};X_{j}\right),(1\leq j\leq m)$ and
$S\in\mathcal{D}_{r^{\ast}}^{m+}(X_{1},...,X_{m};F)$ and this ends the proof.

Every positive $\left(p_{1},\ldots,p_{m};r\right)$-dominated multilinear
operator can be factored through a Cohen positive strongly $r^{\ast}$-summing
multilinear operator and positive $p_{j}$-summing linear operators $(1\leq
j\leq m)$. Consequently, the class
$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}$ forms a positive muti-
ideal of type
$\mathcal{M}_{R}^{+}\left(\mathcal{B}_{1,L}^{+},...,\mathcal{B}_{m,L}^{+}\right)$
where

$\mathcal{M}_{R}^{+}=\mathcal{D}_{r^{\ast}}^{m+}\text{ and
}\mathcal{B}_{j,L}^{+}=\Pi_{p_{j}}^{+}\left(1\leq j\leq m\right).$

###  3.1. Positive (p1,…,pm)-dominated

An interesting special case of positive $(p_{1},...,p_{m};r)$-dominated
operators arises when $r=\infty,$ i.e., $1/p=1/p_{1}+...+1/p_{m}$. These
operators are referred to as positive $(p_{1},...,p_{m})$-dominated. We will
precisely define and explore these operators as they map from Banach lattices
into a Banach space.

######  Definition 3.6.

Let $1\leq p,p_{1},\ldots,p_{m}\leq\infty$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}$. Let $E_{1},...,E_{m}$ be
Banach lattices and $Y$ be a Banach space. An $m$-linear operator
$T:E_{1}\times...\times E_{m}\rightarrow Y$ is positive
$(p_{1},...,p_{m})$-dominated, if there is a constant $C>0$ such that for any
$(x_{i}^{1},...,x_{i}^{m})\in E_{1}^{+}\times...\times E_{m}^{+}$ ($1\leq
i\leq n$), we have

$(\sum_{i=1}^{n}\|T(x_{i}^{1},...,x_{i}^{m})\|^{p})^{\frac{1}{p}}\leq
C\prod_{j=1}^{m}\|(x_{i}^{j})_{i=1}^{n}\|_{p_{j},w}.$ (3.6)

We denote the space of all such mappings by
$\Pi_{p_{1},\ldots,p_{m}}^{+}\left(E_{1},\ldots,E_{m};Y\right)$. In this case,
we define the norm

$\pi_{p_{1},\ldots,p_{m}}^{+}(T)=\inf\\{C>0:\quad C\ \text{satisfying\ the\
inequality}\ (\ref{ppdm})\\}.$

It is straightforward to demonstrate the equivalence of the formula (3.6) with

$(\sum_{i=1}^{n}\|T(x_{i}^{1},...,x_{i}^{m})\|^{p})^{\frac{1}{p}}\leq\prod_{j=1}^{m}\|(x_{i}^{j})_{i=1}^{n}\|_{p_{j},|\omega|},$

for any $(x_{i}^{1},...,x_{i}^{m})\in E_{1}\times...\times E_{m},$
$\left(1\leq i\leq n\right)$.

######  Proposition 3.7.

The class $(\Pi_{p_{1},\ldots,p_{m}}^{+},\pi_{p_{1},\ldots,p_{m}}^{+})$ is a
Banach positive left multi-ideal.

Similar to the earlier section, we can establish Pietsch’s theorem concerning
this class.

######  Theorem 3.8 (Pietsch domination theorem).

Let $1\leq p,p_{1},\ldots,p_{m}<\infty$ with
$\frac{1}{p}=\frac{1}{p_{1}}+\ldots+\frac{1}{p_{m}}$. Let $E_{1},...,E_{m}$ be
Banach lattices and $Y$ be a Banach space. The following properties are
equivalent:

1) The operator $T\in\mathcal{L}\left(E_{1},\ldots,E_{m};Y\right)$ is positive
$(p_{1},\ldots,p_{m})$-dominated.

2) There is a constant $C>0$ and Borel probability measures $\mu_{j}$ on
$B_{E_{j}^{\ast}}^{+}$ ($1\leq j\leq m$), such that

$\|T(x^{1},...,x^{m})\|\leq\displaystyle
C\prod_{j=1}^{m}\Big{(}\int_{B_{E_{j}^{\ast}}^{+}}\langle|x^{j}|,\varphi_{j}\rangle^{p_{j}}d\mu_{j}\Big{)}^{\frac{1}{p_{j}}},$
(3.7)

for all $(x^{1},...,x^{m})\in E_{1}\times...\times E_{m}.$

3) There is a constant $C>0$ and Borel probability measures $\mu_{j}$ on
$B_{E_{j}^{\ast}}^{+}$ ($1\leq j\leq m$), such that

$\|T(x^{1},...,x^{m})\|\leq\displaystyle
C\prod_{j=1}^{m}\Big{(}\int_{B_{E_{j}^{\ast}}^{+}}\langle
x^{j},\varphi_{j}\rangle^{p_{j}}d\mu_{j}\Big{)}^{\frac{1}{p_{j}}},$

for all $(x^{1},...,x^{m})\in E_{1}^{+}\times...\times E_{m}^{+}.$

4) (Kwapień’s factorization) There exist Banach spaces $X_{1},\ldots,X_{m}$, a
multilinear operator $S:X_{1}\times...\times X_{m}\rightarrow Y$ and linear
operators $R_{j}\in\Pi_{p_{j}}^{+}\left(E_{j};X_{j}\right)$, so that
$T=S\circ\left(R_{1},\ldots,R_{m}\right)$, i.e.
$\Pi_{p_{1},\ldots,p_{m}}^{+}=\mathcal{L}\left(\Pi_{p_{1}}^{+},\ldots,\Pi_{p_{m}}^{+}\right).$
Moreover

$\pi_{p_{1},\ldots,p_{m}}^{+}(T)=\inf\left\\{\|S\|\prod_{j=1}^{m}\pi_{p_{j}}^{+}\left(R_{j}\right):T=S\circ\left(R_{1},\ldots,R_{m}\right)\right\\}.$

In other words, we say that the class
$\Pi_{p_{1},\ldots,p_{m}}^{+}=\mathcal{L}(\Pi_{p_{1}}^{+},...,\Pi_{p_{m}}^{+})$
is the Banach positive left multi-ideal generated by the factorization method
from the Banach positive operator left ideals
$\Pi_{p_{1}}^{+},...,\Pi_{p_{m}}^{+}.$

######  Remark 3.9.

The composition class
$\mathcal{D}_{r^{\ast}}^{+}\circ\Pi_{p_{1},\ldots,p_{m}}^{+}$ is equal to
$\mathcal{D}_{r^{\ast}}^{+}\circ\mathcal{L}(\Pi_{p_{1}}^{+},...,\Pi_{p_{m}}^{+})$
which is, in turn, equal to
$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}$. Consequently, an
alternative expression for the class
$\mathcal{D}_{\left(p_{1},\ldots,p_{m};r\right)}^{+}$ is given by
$\mathcal{B}_{R}^{+}\circ\mathcal{M}_{L}^{+}$ where
$\mathcal{B}_{R}^{+}=\mathcal{D}_{r^{\ast}}^{+}$ and
$\mathcal{M}_{L}^{+}=\Pi_{p_{1},\ldots,p_{m}}^{+}$.

Declarations

Conflict of interest. The authors declare that they have no conflicts of
interest.

## References

  * [1] D. Achour, Multilinear extensions of absolutely $(p;q;r)$-summing operators. Rend. Circ. Mat. Palermo 60, 337-350 (2011).
  * [2] D. Achour, A. Alouani, On multilinear generalizations of the concept of nuclear operators. Colloq. Math 120(1), 85-102 (2010).
  * [3] D. Achour and A. Belacel, Domination and factorization theorems for positive strongly $p$-summing operators. Positivity 18, 785-804 (2014).
  * [4] D. Achour and L. Mezrag, On the Cohen strongly $p$-summing multilinear operators. J. Math. Anal. Appl 327, 550-563 (2007).
  * [5] O. Blasco, Positive $p$-summing operators on $L_{p}$-spaces. Proceedings of the American Mathematical Society 100.2, 275-280 (1987).
  * [6] A. Bougoutaia and A. Belacel, Cohen positive strongly $p$-summing and $p$-convex multilinear operators. Positivity 23.2, 379-395 (2019).
  * [7] A. Bougoutaia, A. Belacel and H. Hamdi, Domination and Kwapién factorization theorems for positive Cohen nuclear linear operators. Moroccan Journal of Pure and Applied Analysis 7.1, 100-115 (2021).
  * [8] A. Bougoutaia, A. Belacel and R. Pilar, Summability of multilinear operators and their linearizations on Banach lattices. J. Math. Anal. Appl. 527, 127459, (2023).
  * [9] G. Botelho, D. Pellegrino and R. Pilar, On composition ideals of multilinear mappings and homogeneous polynomials. Publications of the Research Institute for Mathematical Sciences 43.4, 1139-1155, (2007).
  * [10] D. Chen, A. Belacel, J. A. C. Domínguez, Positive $p$-summing operators and disjoint $p$-summing operators. Positivity 25, 1045-1077, (2021).
  * [11] J.S. Cohen, Absolutely $p$-summing, $p$-nuclear operators and their conjugates. Math. Ann 201, 177-200 (1973).
  * [12] A. Defant and K. Floret, Tensor Norms and Operator Ideals. North-Holland Publishing, North-Holland, 1993.
  * [13] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators. Cambridge University Press, Cambridge, 1995.
  * [14] S. Geiss, Ideale multilinearer Abbildungen, Diplomarbeit, 1984.
  * [15] P. Meyer Nieberg, Banach lattices. Springer, Berlin, 1991\.
  * [16] D. Pellegrino, J. Santos and J.B.S. Seplveda, Some techniques on nonlinear analysis and applications. Adv. Math 229, 1235-1265 (2012)
  * [17] A. Pietsch, Ideals of multilinear functionals. In: Proceedings of the second international conference on operator algebras, ideals, and their applications in theoretical physics (Leipzig, 1983). vol. 67, 185-199 (1983).
  * [18] A. Pietsch, Operator Ideals. North-Holland Publications, North-Holland, 1980.
  * [19] O.I. Zhukova, On modifications of the classes of $p$-nuclear, $p$-summing, and $p$-integral operators. Sib Math J 39, 894-907 (1998).