Generalized multiple summing multilinear operators on Banach spaces

TL;DR

This paper introduces a unified abstract framework for multiple summing multilinear operators on Banach spaces, encompassing known classes and defining new subclasses with shared properties.

Contribution

It provides a unifying approach to various classes of multiple summing operators and introduces new subclasses within this framework.

Findings

Unified treatment of multiple summing operator classes

Extension of known classes to a broader abstract setting

Introduction of new subclasses with inherited properties

Abstract

In this paper we provide an abstract aproach to the study of classes of multiple summing multilinear operators between Banach spaces. The main purpose is unify the study of several known classes and results, for example multiple $(p, q_1,\dots, q_n)$-summing operators, multiple mixing $(s, q, p)$-summing operators and multiple strong $(s, q, p)$-mixing summing operators. We also define new class of multiple summing multilinear operator that are particular cases of our construction and, therefore, satisfy the results proved in the paper.

Equations

$$c_{00}(E)\subset {\gamma }_s(E)\stackrel{1}{\hookrightarrow }{\ell }_{\infty }(E)\text{and}\Vert e_j{\Vert }_{{\gamma }_s(\mathbb{K})}=1,\forall j=1,\dots ,n.$$

$$(x_j{)}_{j=1}^{\infty }\in {\gamma }_s(E)⟺\underset{k}{\sup }{\Vert {\left(x_j\right)}_{j=1}^k\Vert }_{{\gamma }_s(E)}<\infty ,$$

$${\Vert {\left(x_j\right)}_{j=1}^{\infty }\Vert }_{{\gamma }_s(E)}:=\underset{k}{\sup }{\Vert {\left(x_j\right)}_{j=1}^k\Vert }_{{\gamma }_s(E)}.$$

$${\left(u\left(x_j\right)\right)}_{j=1}^{\infty }\in {\gamma }_s(F)$$

$${\Vert {\left({\lambda }_j^{(1)}\cdots {\lambda }_j^{(n)}\right)}_{j=1}^{\infty }\Vert }_{{\gamma }_s(\mathbb{K})}\le \prod _{i=1}^{\infty }{\Vert {\left({\lambda }_j^{(i)}\right)}_{j=1}^{\infty }\Vert }_{{\gamma }_{s_i}(\mathbb{K})}$$

$$\Vert \cdot {\Vert }_{{\mathcal{M}}_n}:{\mathcal{M}}_n⟶\mathbb{R}$$

$${\mathcal{M}}_n(E_1,\dots ,E_n;F):=\mathcal{L}(E_1,\dots ,E_n;F)\cap {\mathcal{M}}_n$$

$$\Vert t\circ T\circ (u_1,\dots ,u_n){\Vert }_{{\mathcal{M}}_n}\le \Vert t\Vert {\Vert T\Vert }_{{\mathcal{M}}_n}\Vert u_1\Vert \cdots \Vert u_n\Vert .$$

$$f(j_1,\dots ,j_n)=x_{j_1,\dots ,j_n}\mathrm{for}\mathrm{all}{j}_1,\dots ,j_n\in \mathbb{N},$$

$$(x_{1,1},x_{2,1},x_{3,1},\dots ,x_{1,2},x_{2,2},x_{3,2},\dots ,x_{1,3},x_{2,3},x_{3,3},\dots ).$$

$$x_{i,j}=\left\{\begin{array}[]{rc}1,&\text{if $i$ is odd and any $j\in\mathbb{N}$}\\ 0,&\text{if $i$ is pair and any $j\in\mathbb{N}$}.\end{array}\right.$$

$$(1,0,1,0,\dots )\text{ and }(0,1,0,1,\dots ).$$

$$c_{00}\left(E;{\mathbb{N}}^n\right):=\left\{(x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\text{; }{x}_{j_1,\dots ,j_n}\ne 0\text{ for only finitely many }{j}_1,\dots ,j_n\right\}$$

$${\ell }_{\infty }\left(E;{\mathbb{N}}^n\right):=\left\{(x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\text{; }\underset{j_1,\dots ,j_n\in \mathbb{N}}{\sup }\Vert x_{j_1,\dots ,j_n}\Vert <\infty \right\}.$$

$$c_{00}\left(E;{\mathbb{N}}^n\right)\subset {\gamma }_s\left(E;{\mathbb{N}}^n\right)\stackrel{1}{\hookrightarrow }{\ell }_{\infty }\left(E;{\mathbb{N}}^n\right)\text{and}\Vert e_{k_1,\dots ,k_n}{\Vert }_{{\gamma }_s\left(\mathbb{K};{\mathbb{N}}^n\right)}=1,$$

$$x_{j_{1},\dots,j_{n}}=\left\{\begin{array}[]{rc}1,&\text{if}\quad j_{1}=k_{1},\dots,j_{n}=k_{n}\\ 0,&\text{otherwise}.\end{array}\right.$$

$$(x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\in {\gamma }_s\left(E;{\mathbb{N}}^n\right)⟺\underset{m_1,\dots ,m_n\in \mathbb{N}}{\sup }{\Vert {\left(x_{j_1,\dots ,j_n}\right)}_{j_1,\dots ,j_n=1}^{m_1,\dots ,m_n}\Vert }_{{\gamma }_s\left(E;{\mathbb{N}}^n\right)}<\infty$$

$${\Vert (x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\Vert }_{{\gamma }_s\left(E;{\mathbb{N}}^n\right)}=\underset{m_1,\dots ,m_n\in \mathbb{N}}{\sup }{\Vert {\left(x_{j_1,\dots ,j_n}\right)}_{j_1,\dots ,j_n=1}^{m_1,\dots ,m_n}\Vert }_{{\gamma }_s\left(E;{\mathbb{N}}^n\right)}.$$

$$\Vert (x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }{\Vert }_{\infty }:=\underset{j_1,\dots ,j_n\in \mathbb{N}}{\sup }\Vert x_{j_1,\dots ,j_n}\Vert .$$

$${\ell }_p^w\left(E;{\mathbb{N}}^n\right):=\left\{(x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\text{; }\sum _{j_1,\dots ,j_n}|\phi \left(x_{j_1,\dots ,j_n}\right){|}^p<\infty \text{, }\forall \phi \in E^{\prime }\right\},$$

$${\ell }_p\left(E;{\mathbb{N}}^n\right):=\left\{(x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\text{; }\sum _{j_1,\dots ,j_n}\Vert x_{j_1,\dots ,j_n}{\Vert }^p<\infty \right\},$$

$${\ell }_p⟨E;{\mathbb{N}}^n⟩$$

$$:=$$

$$\Vert (x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }{\Vert }_{Coh,p}:=\underset{({\phi }_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\in B_{{\ell }_p^w\left(E^{\prime };{\mathbb{N}}^n\right)}}{\sup }\sum _{j_1,\dots ,j_n}\left|{\phi }_{j_1,\dots ,j_n}\left(x_{j_1,\dots ,j_n}\right)\right|.$$

$${\ell }_p^{mid}\left(E;{\mathbb{N}}^n\right):=\left\{(x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\text{; }\sum _n\sum _{j_1,\dots ,j_n}|{\phi }_n(x_{j_1,\dots ,j_n}){|}^p<\infty ,\forall ({\phi }_n{)}_{n=1}^{\infty }\in {\ell }_p^w(E^{\prime })\right\},$$

$$\frac{1}{s(p{)}^{\prime }}+\frac{1}{s}=\frac{1}{p}.$$

$${\Vert {\left(x_{j_1,\dots ,j_n}\right)}_{j_1,\dots ,j_n=1}^{\infty }\Vert }_{m(s;q)}=\inf {\Vert {\left({\tau }_{j_1,\dots ,j_n}\right)}_{j_1,\dots ,j_n=1}^{\infty }\Vert }_{s(q{)}^{\prime }}{\Vert {\left(x_{j_1,\dots ,j_n}^0\right)}_{j_1,\dots ,j_n=1}^{\infty }\Vert }_{w,s},$$

$$(x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\in {\gamma }_s\left(E;{\mathbb{N}}^n\right)⟺(x_{j_{\sigma (1)},\dots ,j_{\sigma (n)}}{)}_{j_1,\dots ,j_n=1}^{\infty }\in {\gamma }_s\left(E;{\mathbb{N}}^n\right)$$

$${\Vert (x_{j_1,\dots ,j_n}{)}_{j_1,\dots ,j_n=1}^{\infty }\Vert }_{{\gamma }_s\left(E;{\mathbb{N}}^n\right)}={\Vert (x_{j_{\sigma (1)},\dots ,j_{\sigma (n)}}{)}_{j_1,\dots ,j_n=1}^{\infty }\Vert }_{{\gamma }_s\left(E;{\mathbb{N}}^n\right)},$$

$${\left(T\left(x_{j_1}^{(1)},\dots ,x_{j_n}^{(n)}\right)\right)}_{j_1,\dots ,j_n=1}^{\infty }\in {\gamma }_s\left(F;{\mathbb{N}}^n\right)$$

$$\hat{T}:{\gamma }_{s_1}(E_1)\times \cdots \times {\gamma }_{s_n}(E_n)\to {\gamma }_s\left(F;{\mathbb{N}}^n\right)$$

Full text

1 Introduction and background

In the 1970’s, Pietsch [24] introduced the abstract theory of operators ideals and in 1983 he presented in [25] the concept of ideals of multilinear operators, whose adaptation to the case of homogeneous polynomials is immediate.

The notion of multiple summing multilinear operators was introduced, independently, in [10, 19]. This notion, which is based on the successful theory of absolutely summing operators, has been extensively studied recently. Several aspects of the theory of summing multilinear operators show that the class of multiple summing multilinear operators is one of the most suitable and useful approaches to the nonlinear theory of absolutely summing operators. Details can be found, e.g., in [1, 5, 9, 12, 16, 22, 23, 19].

In [26] it was presented an abstract approach to absolutely summing operators, that generalizes some concepts of absolutely summing operators already studied in the literature. But the task of generalization is not easy. For example, this work has small gaps that were filled by Botelho and Campos in [6]. Such generalizations deal with abstract classes of vector-valued sequences, abstract finitely determined sequence classes and abstract linearly stable sequence classes. Up to the corrections pointed out in [6], it was proved in [26] that the abstract classes of summing absolutely multilinear operators are Banach ideals of multilinear operators.

The main goal of this paper is to construct an abstract approach to the classes of multiple summing multilinear operators and to show that the resulting classes are Banach multi-ideals. Moreover, coherence and compatibility of these multi-ideals will be investigated. In the last section of the paper we will recover some well studied classes as particular instances of our abstract construction and new classes will be introduced as well.

Our abstract construction is strongly based on the concept of sequence classes introduced in [6]. However, to deal with multiple summing operators we have to extend this notion to what we call $n$-sequences classes in Section 2. For the moment, let us recall the original definition from [6]:

Definition 1.1**.**

A class of vector-valued sequences $\gamma_{s}$ is a rule that assigns to each Banach space $E$ a Banach space $\gamma_{s}(E)$ of $E$-valued sequences, that is $\gamma_{s}(E)$ is a vector subspace of $E^{\mathbb{N}}$ with the coordinatewise operations, such that:

[TABLE]

where $e_{j}$ is the vector with $1$ in the $j$-th coordinate and zero in the other coordinates, and the symbol $E\overset{1}{\hookrightarrow}F$ means that $E$ is a linear subspace of $F$ and $\|x\|_{F}\leq\|x\|_{E}$ .

Still according to [6], the sequence class $\gamma_{s}$ is said to be:

$\bullet$ finitely determined if for every $(x_{j})_{j=1}^{\infty}\in E^{\mathbb{N}}$, it holds

[TABLE]

and, in this case,

[TABLE]

$\bullet$ linearly stable if for every $u\in\mathcal{L}(E;F)$ it holds

[TABLE]

whenever $\left(x_{j}\right)_{j=1}^{\infty}\in\gamma_{s}(E)$ and $\|\hat{u}:\gamma_{s}(E)\rightarrow\gamma_{s}(F)\|=\|u\|$, where $\hat{u}$ is the linear operator induced, in the obvious way, by $u$.

Given sequence classes $\gamma_{s_{1}},\dots,\gamma_{s_{n}},\gamma_{s}$, we write $\gamma_{s_{1}}(\mathbb{K})\cdots\gamma_{s_{n}}(\mathbb{K})\overset{1}{\hookrightarrow}\gamma_{s}(\mathbb{K})$ if $\left(\lambda_{j}^{(1)}\cdots\lambda_{j}^{(n)}\right)_{j=1}^{\infty}\in\gamma_{s}(\mathbb{K})$ and

[TABLE]

whenever $\left(\lambda_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(\mathbb{K}),i=1,\dots,n$.

The classical notion of ideal of multilinear operators (multi-ideal) is presented in the next definition.

Definition 1.2**.**

Let $n\in\mathbb{N}$. A Banach ideal of $n$-linear operators is a pair $\left(\mathcal{M}_{n},\|\cdot\|_{\mathcal{M}_{n}}\right)$ where $\mathcal{M}_{n}$ is as subclass of the class of all $n$-linear operators between Banach spaces and

[TABLE]

it is a function such that, for all Banach spaces $E_{1},\dots,E_{n},F$, the component

[TABLE]

is a subspace of $\mathcal{L}(E_{1},\dots,E_{n};F)$ on which $\|\cdot\|_{\mathcal{M}_{n}}$ is a complete norm and:

The subspace of the $n$-linear operators of finite type is contained in $\mathcal{M}_{n}(E_{1},\dots,E_{n};F)$. 2. 2.

The operator $I_{n}:\mathbb{K}^{n}\rightarrow\mathbb{K}$, given by $I_{n}(\lambda_{1},\dots,\lambda_{n})=\lambda_{1}\cdots\lambda_{n}$, belongs to $\mathcal{M}_{n}(\mathbb{K}^{n};\mathbb{K})$ and $\|I_{n}\|_{\mathcal{M}_{n}}=1$. 3. 3.

$($Multi-ideal property$)$* If $T\in\mathcal{M}_{n}(E_{1},\dots,E_{n};F),u_{i}\in\mathcal{L}(G_{i};E_{i}),i=1,\dots,n$, and $t\in\mathcal{L}(F;H)$ then $t\circ T\circ(u_{1},\dots,u_{n})\in\mathcal{M}_{n}(G_{1},\dots,G_{n};H)$ and*

[TABLE]

The notion of ideals of homogeneous polynomials can be defined in a similar way (see, e.g., [21, 4]).

2 Multiple $\gamma_{s,s_{1},\dots,s_{n}}$-summing operators

The theory of multiple summing multilinear operators, which has been intensively studied, see, e.g. [1, 5, 22, 23], serves as a prototype of the general theory we introduce in this section. We begin presenting the notions of $n$-sequences and classes of vector-valued $n$-sequences.

Definition 2.1**.**

Given $n\in\mathbb{N}$, an $n$-sequence in a Banach space $E$ is a map $f:\mathbb{N}^{n}\rightarrow E$. Writing

[TABLE]

the $n$-sequence $f$ can be denoted by $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}$.

It is worth observing that, for $n\geq 2$, an $n$-sequence is not a sequence. For example, given a $2$-sequence $(x_{i,j})_{i,j=1}^{\infty}$, it is useless to try to display it like a sequence in the following fashion:

[TABLE]

Of course this is not a sequence. Note also that an $n$-sequence can be transformed into a sequence in several ways. For example, consider the $2$-sequence $(x_{i,j})_{i,j\in\mathbb{N}}$ given by

[TABLE]

It can be transformed into a sequence in many ways, for example

[TABLE]

Now we start the construction of our abstract framework. Throughout this paper, we will consider:

[TABLE]

and

[TABLE]

Definition 2.2**.**

Let $n\in\mathbb{N}$. A class of vector-valued $n$-sequences $\gamma_{s}\left(\cdot,\mathbb{N}^{n}\right)$, or simply an $n$-sequence class $\gamma_{s}\left(\cdot,\mathbb{N}^{n}\right)$, is a rule that assigns to each Banach space $E$ a Banach space $\gamma_{s}(E;\mathbb{N}^{n})$ of $E$-valued $n$-sequences, that is $\gamma_{s}(E;\mathbb{N}^{n})$ is a complete linear subspace of the space of all $E$-valued $n$-sequences with the coordinatewise operations, such that:

[TABLE]

for all $k_{1},\dots,k_{n}\in\mathbb{N}$, where $e_{k_{1},\dots,k_{n}}=\left(x_{j_{1},\dots,j_{n}}\right)_{j_{1},\dots,j_{n}=1}^{\infty}$ is the $n$-sequence defined by:

[TABLE]

An $n$-sequence class $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ is finitely determined if for every $E$-valued $n$-sequence $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}$,

[TABLE]

and, in this case,

[TABLE]

Note that the concept of $n$-sequence class generalizes the concept of sequence class introduced in the literature by Botelho and Campos in [6].

Next we give some examples of finitely determined $n$-sequence classes.

Examples 2.3**.**

$(a)$* The correspondence $E\mapsto\ell_{\infty}\left(E;\mathbb{N}^{n}\right)$, endowed with the norm*

[TABLE]

$(b)$* The correspondence $E\mapsto\ell_{p}^{w}\left(E;\mathbb{N}^{n}\right)$, $1\leq p<\infty$, where*

[TABLE]

endowed with the norm $\|(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}\|_{w,p}:=\displaystyle\sup_{\varphi\in B_{E^{\prime}}}\left(\displaystyle\sum_{j_{1},\dots,j_{n}}|\varphi\left(x_{j_{1},\dots,j_{n}}\right)|^{p}\right)^{\frac{1}{p}}$.

$(c)$* The correspondence $E\mapsto\ell_{p}\left(E;\mathbb{N}^{n}\right)$, $1\leq p<\infty$, where*

[TABLE]

endowed with the norm $\|(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}\|_{p}:=\left(\displaystyle\sum_{j_{1},\dots,j_{n}}\|x_{j_{1},\dots,j_{n}}\|^{p}\right)^{\frac{1}{p}}$.

$(d)$* The correspondence $E\mapsto\ell_{p}\left<E;\mathbb{N}^{n}\right>$, $1\leq p<\infty$, where*

[TABLE]

endowed with the norm

[TABLE]

$(e)$* The correspondence $E\mapsto\ell_{p}^{mid}\left(E;\mathbb{N}^{n}\right)$, $1\leq p<\infty$, where*

[TABLE]

endowed with the norm $\|(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}\|_{mid,p}:=\displaystyle\sup_{(\varphi_{n})_{n=1}^{\infty}\in B_{\ell_{p}^{w}(E^{\prime})}}\left(\displaystyle\sum_{n}\displaystyle\sum_{j_{1},\dots,j_{n}}|\varphi_{n}(x_{j_{1},\dots,j_{n}})|^{p}\right)^{\frac{1}{p}}$.

$(f)$* The correspondence $E\mapsto\ell_{m(s,p)}\left(E;\mathbb{N}^{n}\right)$, $1\leq p<\infty$, where $\ell_{m(s,p)}\left(E;\mathbb{N}^{n}\right)$ is the set of all $E$-valued $n$-sequences $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}$ such that $x_{j_{1},\dots,j_{n}}=\tau_{j_{1},\dots,j_{n}}x_{j_{1},\dots,j_{n}}^{0}$ for some $(\tau_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}\in\ell_{s(q)^{\prime}}\left(\mathbb{K};\mathbb{N}^{n}\right),(x_{j_{1},\dots,j_{n}}^{0})_{j_{1},\dots,j_{n}=1}^{\infty}\in\ell_{p}^{w}\left(E;\mathbb{N}^{n}\right)$ and*

[TABLE]

Consider $\ell_{m(s,q)}\left(E;\mathbb{N}^{n}\right)$ endowed with the norm

[TABLE]

where the infimum ranges over all representations $x_{j_{1},\dots,j_{n}}=\tau_{j_{1},\dots,j_{n}}x_{j_{1},\dots,j_{n}}^{0}$, $j_{1},\dots,j_{n}\in\mathbb{N}$.

Note that considering $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}$ in different orders of the indexes, one may end up with different $n$-sequences. In order to avoid this dependence on the order the indexes are taken, we shall henceforth consider only $n$-sequence classes $\gamma_{s}\left(\cdot,\mathbb{N}^{n}\right)$ enjoying the following symmetry condition: for any $E$-valued $n$-sequence $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}$,

[TABLE]

and

[TABLE]

for every permutation $\sigma$ of the set $\{1,\ldots,n\}$.

Henceforth $\gamma_{s_{1}},\dots,\gamma_{s_{n}}$ are linearly stable finitely determined sequence classes and $\gamma_{s}\left(\cdot,\mathbb{N}^{n}\right)$ is a finitely determined $n$-sequence class enjoying the symmetry condition above.

Now we are ready to introduce our main definition:

Definition 2.4**.**

Let $n\in\mathbb{N}$. A continuous $n$-linear operator $T\in\mathcal{L}(E_{1},\dots,E_{n};F)$ is said to be multiple $\gamma_{s,s_{1},\dots,s_{n}}$-summing if

[TABLE]

whenever $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(E_{i}),i=1,\dots,n$.

Note that the symmetry condition of the $n$-sequence class $\gamma_{s}\left(\cdot,\mathbb{N}^{n}\right)$ guarantees that this concept is well defined.

The set of all multiple $\gamma_{s,s_{1},\dots,s_{n}}$-summing $n$-linear operators from $E_{1}\times\cdots\times E_{n}$ to $F$, which is clearly a linear space, is denoted by $\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$. To give this space a suitable complete norm, we need the following result.

Lemma 2.5**.**

Let $E_{1},\dots,E_{n}$, $F$ be Banach spaces and $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$. Then, the induced map

[TABLE]

given by

[TABLE]

is well-defined continuous $n$-linear operator.

Proof.

It is clear that $\hat{T}$ is well-defined and it is easy to check its $n$-linearity. To prove that $\hat{T}$ is a continuous operator, we will use the Closed Graph Theorem. We shall use the sum norm in the cartesian product. Let

[TABLE]

be a sequence in $\gamma_{s_{1}}(E_{1})\times\cdots\times\gamma_{s_{n}}(E_{n})$ converging to

[TABLE]

and such that

[TABLE]

So, given $\epsilon>0$ there exists $k_{0}\in\mathbb{N}$ such that for any $k\geq k_{0}$ and $i=1,\dots,n$,

[TABLE]

Thus, $x_{j_{i}}^{(i),k}\rightarrow x_{j_{i}}^{(i)}$ for every $i=1,\dots,n$. It follows from the continuity of $T$ that

[TABLE]

for all $j_{1},\dots,j_{n}\in\mathbb{N}$. On the other hand, by (1) we can take $k_{1}\in\mathbb{N}$ such that for any $k\geq k_{1}$ and $i=1,\dots,n$,

[TABLE]

So,

[TABLE]

for all $j_{1},\dots,j_{n}\in\mathbb{N}$. Consequently,

[TABLE]

proving that $\hat{T}$ has closed graph, hence it is continuous. ∎

The converse of Lemma 2.5 is obviously true. The next proposition will be useful to introduce the norm that will make the linear space $\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$ a Banach space.

Theorem 2.6**.**

Let $E_{1},\dots,E_{n},F$ be Banach spaces and $T\in\mathcal{L}(E_{1},\dots,E_{n};F)$. The following statements are equivalent:

$($a$)$* $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$.*

$($b$)$* There is $C>0$, such that*

[TABLE]

whenever $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(E_{i}),i=1,\dots,n$.

$($c$)$* There is $C>0$ such that*

[TABLE]

for any $m_{1},\dots,m_{n}\in\mathbb{N}$ and $x_{j}^{(i)}\in E_{i}$, $i=1,\dots,n$.

Proof.

$(\text{b})\Rightarrow(\text{c})$ is straightforward and $(\text{c})\Rightarrow(\text{a})$ follows easily from the fact that the underlying sequence classes and $n$-sequence class are finitely determined. Thus, we have only to prove $(\text{a})\Rightarrow(\text{b})$. To do so, suppose that $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$ and define

[TABLE]

as in Lemma 2.5. Thus, $\hat{T}$ is a well-defined $n$-linear continuous operator, from which it follows that

[TABLE]

∎

Now standard arguments give the following result:

Proposition 2.7**.**

*Let $E_{1},\dots,E_{n}$ and $F$ be Banach spaces.

(a) The infimum of the constant $C>0$ satisfying (2) in Proposition 2.6 defines a norm on $\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$, which is denoted by $\|\cdot\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}$.

(b) $\|T\|\leq\|T\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}$ for every $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$.

(c) $\left(\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F),\|\cdot\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}\right)$ is a Banach space.*

3 $\left(\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m},\|\cdot\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}\right)$ is a Banach ideal

After proving that $\left(\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m},\|\cdot\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}\right)$ is a Banach ideal of multilinear operators, we will establish the relationship between the class of multiple $\gamma_{s,s_{1},\dots,s_{n}}$-summing operators and the class of $\gamma_{s,s_{1},\dots,s_{n}}$-summing operators at the origin, introduced in [26].

The proof that $\left(\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m},\|\cdot\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}\right)$ is a Banach ideal shall be splitted into several steps. The first step was taken in Proposition 2.7. Before proceeding to the next steps, let us define the conditions the sequences classes shall be supposed to satisfy.

Definition 3.1**.**

*We say that an $n$-sequence class $\gamma_{s}\left(\cdot,\mathbb{N}^{n}\right)$ is linearly stable if

$\left(u(x_{j_{1},\dots,j_{n}})\right)_{j_{1},\dots,j_{n}=1}^{\infty}\in\gamma_{s}\left(F;\mathbb{N}^{n}\right)$ whenever $\left(x_{j_{1},\dots,j_{n}}\right)_{j_{1},\dots,j_{n}=1}^{\infty}\in\gamma_{s}\left(E;\mathbb{N}^{n}\right)$ and $u\in\mathcal{L}(E;F)$, and, in this case,*

[TABLE]

All $n$-sequence classes presented in Example 2.3 are linearly stable. From now on, all $n$-sequence classes are supposed to be finitely determined and linearly stable.

Given sequence classes $\gamma_{s_{1}},\dots,\gamma_{s_{n}}$ and an $n$-sequence class $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$, we write

[TABLE]

if $\left(\lambda_{j_{1}}^{(1)}\cdots\lambda_{j_{n}}^{(n)}\right)_{j_{1},\dots,j_{n}=1}^{\infty}\in\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$ and

[TABLE]

whenever $\left(\lambda_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(\mathbb{K}),i=1,\dots,n$.

Proposition 3.2**.**

Suppose that $\gamma_{s_{1}}(\mathbb{K})\cdots\gamma_{s_{n}}\left(\mathbb{K}\right)\overset{mult,1}{\hookrightarrow}\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$. Then, for all Banach spaces $E_{1},\dots,E_{n}$ and $F$, the finite type $n$-linear continuous operators are contained in $\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$.

Proof.

Consider the $n$-linear operator

[TABLE]

where $b\in F$, $\varphi^{(r)}\in E_{r}^{\prime}$, $r=1,\dots,n$. Consider also the linear operator

[TABLE]

It is clear that $\|f\|=\|b\|$. Given $\left(x_{j}^{(r)}\right)_{j=1}^{\infty}\in\gamma_{s_{r}}(E_{r}),r=1,\dots,n$, it follows from linear stability of $\gamma_{s_{r}}$ that $\left(\varphi^{(r)}\left(x_{j}^{(r)}\right)\right)_{j=1}^{\infty}\in\gamma_{s_{r}}(\mathbb{K})$, $r=1,\dots,n$. Now, from $\gamma_{s_{1}}(\mathbb{K})\cdots\gamma_{s_{n}}(\mathbb{K})\overset{mult,1}{\hookrightarrow}\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$ it follows that

[TABLE]

Therefore, from linear stability of $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ we have that

[TABLE]

what proves that $B\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1}\dots,E_{n};F)$. Since $\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1}\dots,E_{n};F)$ is a linear space, we conclude that it contains that the finite type operators. ∎

The next result proves the multi-ideal property.

Proposition 3.3**.**

Let $E_{1},\dots,E_{n},G_{1},\dots,G_{n},F$ and $H$ be Banach spaces, $t\in\mathcal{L}(F;H),T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$ and $u_{i}\in\mathcal{L}(G_{i};E_{i}),i=1,\dots,n$. Then

[TABLE]

and

[TABLE]

Proof.

Let $\left(x_{j_{i}}^{(i)}\right)_{j_{i}=1}^{\infty}\in\gamma_{s_{i}}(G_{i})$, $i=1,\dots,n$. Since each $\gamma_{s_{i}}$ is linearly stable, we have

[TABLE]

for $i=1,\ldots n$. As $\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1}\dots,E_{n};F)$, we get

[TABLE]

It follows from the linear stability of $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ that

[TABLE]

whenever $\left(x_{j_{i}}^{(i)}\right)_{j_{i}=1}^{\infty}\in\gamma_{s_{i}}(G_{i})$, $i=1,\dots,n$. Therefore

[TABLE]

Thus, from linear stability of $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ and $\gamma_{s_{i}}$, follows that

[TABLE]

which completes the proof. ∎

Now we take the last step:

Proposition 3.4**.**

Consider the map

[TABLE]

and suppose that $\gamma_{s_{1}}(\mathbb{K})\cdots\gamma_{s_{n}}(\mathbb{K})\overset{mult,1}{\hookrightarrow}\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$. Then $\|I_{n}\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}=1$.

Proof.

Given $\left(\lambda_{j_{i}}^{(i)}\right)_{j_{i}=1}^{\infty}\in\gamma_{s_{i}}(\mathbb{K})$, $i=1,\dots,n$, it follows from $\gamma_{s_{1}}(\mathbb{K})\cdots\gamma_{s_{n}}(\mathbb{K})\overset{mult,1}{\hookrightarrow}\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$ that $\left(\lambda_{j_{1}}^{(1)}\cdots\lambda_{j_{n}}^{(n)}\right)_{j_{1},\dots,j_{n}=1}^{\infty}\in\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$ and

[TABLE]

So, $I_{n}\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(\mathbb{K}^{n};\mathbb{K})$ and $\|I_{n}\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}\leq 1$. The other inequality follows Proposition 2.7(b). ∎

Just combine Propositions 2.7, 3.2, 3.3 and 3.4 to obtain the following result:

Theorem 3.5**.**

Let $\gamma_{s_{1}},\dots,\gamma_{s_{n}}$ be linearly stable, finitely determined sequence classes and $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ be a linearly stable, finitely determined $n$-sequence classes. Suppose that $\gamma_{s_{1}}\left(\mathbb{K}\right)\cdot\cdot\cdot\gamma_{s_{n}}\left(\mathbb{K}\right)\overset{mult,1}{\hookrightarrow}\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$. Then $\left(\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m},\|\cdot\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}\right)$ is a Banach multi-ideal.

It is worth mentioning that we used the condition $\gamma_{s_{1}}\left(\mathbb{K}\right)\cdot\cdot\cdot\gamma_{s_{n}}\left(\mathbb{K}\right)\overset{mult,1}{\hookrightarrow}\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$ twice, in the proofs of Propositions 3.3 and 3.4. We also note that this condition is not restrictive, since the main classes known in the literature enjoy this property.

In order to compare the class we are studying with the class of absolutely $\gamma_{s,s_{1},\dots,s_{n}}$-summing multilinear operators at the origin $a=(0,\dots,0)\in E_{1}\times\cdots\times E_{n}$, which was introduced and denoted by $\prod_{\gamma_{s,s_{1},\dots,s_{n}}}(E_{1},\dots,E_{n};F)$ in [26], we need the following definition.

Definition 3.6**.**

For every $n\in\mathbb{N}$, let an $n$-sequence class $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ be given. We say that the $n$-sequence class $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ is sequentially compatible with $\gamma_{s}(\cdot):=\gamma_{s}\left(\cdot;\mathbb{N}^{1}\right)$ if, for every $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}\in\gamma_{s}\left(E;\mathbb{N}^{n}\right)$, we have $(x_{j})_{j=1}^{\infty}:=(x_{j,\dots,j})_{j=1}^{\infty}\in\gamma_{s}(E)$ and

[TABLE]

Example 3.7**.**

All classes mentioned in Example 2.3 satisfy Definition 3.6.

Proposition 3.8**.**

Let $E_{1},\dots,E_{n},F$ be Banach spaces and suppose that $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ sequentially compatible with $\gamma_{s}(\cdot)$. Then $\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)\subset\prod_{\gamma_{s,s_{1},\dots,s_{n}}}(E_{1},\dots,E_{n};F)$ and

[TABLE]

for any $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$.

Proof.

Given $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}(E_{1},\dots,E_{n};F)$ and $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(E_{i}),i=1,\dots,n$, it follows directly from Definition 3.6 that $\left(T\left(x_{j}^{(1)},\dots,x_{j}^{(n)}\right)\right)_{j=1}^{\infty}\in\gamma_{s}(F)$, which gives $T\in\prod_{\gamma_{s,s_{1},\dots,s_{n}}}(E_{1},\dots,E_{n};F)$, and

[TABLE]

which proves that $\pi_{\gamma_{s,s_{1},\dots,s_{n}}}(T)\leq\|T\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m}}$. ∎

We finish this section giving another consequence of definition above.

Proposition 3.9**.**

Let $\gamma_{s_{1}},\dots,\gamma_{s_{n}}$ be finitely determined sequence classes, $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ be a finitely determined $n$-sequence class and suppose that $\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)$ sequentially compatible with $\gamma_{s}(\cdot)$.  If $\gamma_{s_{1}}(\mathbb{K})\cdots\gamma_{s_{n}}(\mathbb{K})\overset{mult,1}{\hookrightarrow}\gamma_{s}\left(\mathbb{K};\mathbb{N}^{n}\right)$, then $\gamma_{s_{1}}(\mathbb{K})\cdots\gamma_{s_{n}}(\mathbb{K})\overset{1}{\hookrightarrow}\gamma_{s}(\mathbb{K})$.

Proof.

Given $\left(\lambda_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(\mathbb{K})$, $i=1,\dots,n$, by Definition 3.6 we have $\left(\lambda_{j}^{(1)}\cdots\lambda_{j}^{(n)}\right)_{j=1}^{\infty}\in\gamma_{s}(\mathbb{K})$ and

[TABLE]

∎

4 Coherence and compatibility

The concept of coherence and compatibility was introduced in the literature initially by D. Pellegrino and G. Botelho in [4, 8] (with a different terminology) and also was studied by D. Carando, V. Dimant and S. Muro in [13, 14, 15]. The terms ”coherence and compatibility” were coined in [13]. In [21] it was introduced a new approach to ”coherence and compatibility”, which considers the sequence formed by the pairs of ideals of multilinear applications and homogeneous polynomials. For this reason, some information about homogeneous polynomials is needed.

The class of all continuous homogeneous polynomials between Banach spaces is denoted by $\cal P$. Given an $n$-homogeneous polynomial $P\colon E\longrightarrow F$, by $\check{P}$ we denote the unique symmetric continuous $n$-linear operator associated to $P$. For any unexplained notation about polynomials we refer to [2, 21]. The next result is folklore.

Proposition 4.1**.**

Let $\left(\mathcal{M},\|\cdot\|_{\mathcal{M}}\right)$ be a Banach ideal of multilinear operators. Then, the class

[TABLE]

is a Banach ideal of homogeneous polynomials.

The Banach ideal $\left(\mathcal{P}_{\mathcal{M}},\|\cdot\|_{\mathcal{P}_{\mathcal{M}}}\right)$ of homogeneous polynomials is called the Banach ideal of polynomials generated by $\left(\mathcal{M},\|\cdot\|_{\mathcal{M}}\right)$. This class has been studied extensively in several works, of which we highlight [2, 11, 17].

In this section, the class of the multiple $\gamma_{s,s_{1},\ldots,s_{1}}$-summing $n$-linear operators shall be denoted by $\mathcal{L}_{\gamma_{s,s_{1}}}^{m,n}$.

Definition 4.2**.**

Given $E$ and $F$ Banach spaces, the class of the multiple $\gamma_{s,s_{1}}$-summing $n$-homogeneos polynomials is defined by

[TABLE]

Since $\left(\mathcal{L}_{\gamma_{s,s_{1}}}^{m,n},\left\|\cdot\right\|_{\mathcal{L}_{\gamma_{s,s_{1}}}^{m,n}}\right)$ is Banach ideal of multilinear operators, from Proposition 4.1 we have the following result.

Corollary 4.3**.**

Let $E$ and $F$ be Banach spaces. Then $\left(\mathcal{P}_{\gamma_{s,s_{1}}}^{m,n},\left\|P\right\|_{\mathcal{P}_{\gamma_{s,s_{1}}}^{m,n}}\right)$ is a Banach ideal of $n$-homogeneous polynomials endowed with the norm

[TABLE]

The definition below shall prove to be the correct condition the sequence classes should satisfy for coherence to hold.

Definition 4.4**.**

*A sequence $\left(\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right)\right)_{n=1}^{M}$, where $M\in\mathbb{N}\cup\{\infty\}$ and each $\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right)$ is an $n$-sequence class, is said to be:

(a) multiple regular with the sequence class $\gamma_{s_{i}}$ if the following condition holds: if $\left(\lambda_{j}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(\mathbb{K})$, $i=1,\dots,n$, and $\left(a_{j_{1},\dots,j_{i-1},j_{i+1},\dots,j_{n}}\right)_{j_{1},\dots,j_{i-1},j_{i+1},\dots,j_{n}=1}^{\infty}\in\gamma_{n-1}\left(F;\mathbb{N}^{n-1}\right)$, regardless of the Banach space $F$, then $\left(\lambda_{j_{i}}a_{j_{1},\dots,j_{i-1},j_{i+1},\dots,j_{n}}\right)_{j_{1},\dots,j_{n}=1}^{\infty}\in\gamma_{n}\left(F;\mathbb{N}^{n}\right)$ and*

[TABLE]

(b) down regular if, for any Banach space $E$ and every $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{n}=1}^{\infty}\in\gamma_{n}\left(E;\mathbb{N}^{n}\right)$ with $n\geq 2$, for any fixed $j_{i}$, $i=1,\dots,n$, it holds that $(x_{j_{1},\dots,j_{n}})_{j_{1},\dots,j_{i-1},j_{i+1},\dots,j_{n}=1}^{\infty}\in\gamma_{n-1}\left(E;\mathbb{N}^{n-1}\right)$ and

[TABLE]

Example 4.5**.**

All classes presented in Example 2.3 are multiple regular and down regular.

Besides of guaranteeing coherence, as we shall prove soon, the definitions above avoid artificial sequences of $n$-sequences, as the following example illustrates.

Example 4.6**.**

Let $\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right)$ be defined by: $\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right):=\ell_{p}^{w}\left(\cdot;\mathbb{N}^{n}\right)$ if $n$ is even and $\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right):=\ell_{p}\left(\cdot;\mathbb{N}^{n}\right)$ if $n$ is odd. It is plain that the sequence $\left(\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right)\right)_{n=1}^{M}$ is neither multiple regular nor down regular.

A classic result that will be important to next Theorem is as following.

Lemma 4.7**.**

Let $P\in\mathcal{P}(^{n}E;F)$ and $a\in E$. Then $(P_{a})^{\vee}=\check{P}_{a}$.

The main result of this sections reads as follows.

Theorem 4.8**.**

Let $\gamma_{s_{i}}$ be a finitely determined and linearly stable sequence class and, for every $n\in\mathbb{N}$, let $\gamma_{s}\left(\cdot,\mathbb{N}^{n}\right)$ be a finitely determined and linearly stable $n$-sequence class. Suppose that the sequence of $n$-sequence classes $\left(\gamma_{s}\left(\cdot;\mathbb{N}^{n}\right)\right)_{n=1}^{\infty}$ is multiple regular with $\gamma_{s_{i}}$ and down regular. Then the sequence of pairs

[TABLE]

is coherent and compatible with $\mathcal{L}_{\gamma_{s,s_{i}}}$ in the sense of [21].

Proof.

To prove (CH1), we will do only the case $i=1$. The general case is analogous. Let $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n+1}}}^{m,n+1}(E_{1},\dots,E_{n+1};F)$ and $a_{1}\in E_{1}$.

Consider the sequence $\left(x_{j}^{(1)}\right)_{j=1}^{\infty}$, such that, $x_{1}^{(1)}=a_{1}$ and $x_{j}^{(1)}=0$ for $j\neq 1$. It is easy to see that, $\left(x_{j}^{(1)}\right)_{j=1}^{\infty}\in\gamma_{s_{1}}(E_{1})$. Let $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(E_{i})$, $i=2,\dots,n+1$. Thus

[TABLE]

Assuming $j_{1}=1$, since $\left(\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right)\right)_{n=1}^{\infty}$ is down regular and the sequence class $\gamma_{s_{1}}$ is linearly stable, we have

[TABLE]

and

[TABLE]

Therefore, $T_{a_{1}}\in\mathcal{L}_{\gamma_{s,s_{2},\dots,s_{n+1}}}^{m,n}(E_{2},\dots,E_{n+1};F)$ and $\|T_{a_{1}}\|_{\mathcal{L}_{\gamma_{s,s_{2},\dots,s_{n+1}}}^{m,n}}\leq\|T\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n+1}}}^{m,n+1}}\|a_{1}\|.$

Now we will check (CH3). Let $T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m,n}(E_{1},\dots,E_{n};F)$ and $\gamma\in E_{n+1}^{\prime}$. Let $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(E_{i})$, $i=1,\dots,n+1$. Since $\left(\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right)\right)_{n=1}^{\infty}$ is multiple regular with $\gamma_{s_{i}}(\cdot)$ and the sequence classes are linearly stable, then $\left(\gamma T\left(x_{j_{1}}^{(1)},\dots,x_{j_{n+1}}^{(n+1)}\right)\right)_{j_{1},\dots,j_{n+1}=1}^{\infty}=\left(\gamma\left(x_{j_{n+1}}^{(n+1)}\right)T\left(x_{j_{1}}^{(1)},\dots,x_{j_{n}}^{(n)}\right)\right)_{j_{1},\dots,j_{n+1}=1}^{\infty}\in\gamma_{s}\left(F;\mathbb{N}^{n+1}\right)$ and

[TABLE]

Therefore, $\gamma T\in\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n+1}}}^{m,n+1}(E_{1},\dots,E_{n+1};F)$ and $\|\gamma T\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n+1}}}^{m,n+1}}\leq\|\gamma\|\|T\|_{\mathcal{L}_{\gamma_{s,s_{1},\dots,s_{n}}}^{m,n}}.$

Now we will prove (CH2). Let $P\in\mathcal{P}_{\gamma_{s,s_{i}}}^{m,n+1}(^{n+1}E;F)$ and $a\in E$. To see that $P_{a}\in\mathcal{P}_{\gamma_{s,s_{i}}}^{m,n}(^{n}E;F)$ is enough to show that $(P_{a})^{\vee}\in\mathcal{L}_{\gamma_{s,s_{i}}}^{m,n}(E^{n};F)$. Since $P\in\mathcal{P}_{\gamma_{s,s_{i}}}^{n+1}(^{n+1}E;F)$, then

[TABLE]

thus, by (CH1),

[TABLE]

By the Lemma 4.7, we have that

[TABLE]

Like this,

[TABLE]

Now we will prove (CH4). Let $P\in\mathcal{P}_{\gamma_{s,s_{i}}}^{m,n}(^{n}E;F)$ and $\varphi\in E^{\prime}$. As done in (CH2), to see that $\varphi P\in\mathcal{P}_{\gamma_{s,s_{i}}}^{m,n+1}(^{n+1}E;F)$ is enough to show that $(\varphi P)^{\vee}\in\mathcal{L}_{\gamma_{s,s_{i}}}^{m,n+1}(E^{n+1};F)$. Note that

[TABLE]

Since $P\in\mathcal{P}_{\gamma_{s,s_{i}}}^{m,n}(^{n}E;F)$, then $\check{P}\in\mathcal{L}_{\gamma_{s,s_{i}}}^{m,n}(E^{n};F)$. Thus, for any $\left(x_{j}^{(k)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(E)$, $k=1,\dots,n+1$

[TABLE]

So, $\left(\gamma_{n}\left(\cdot;\mathbb{N}^{n}\right)\right)_{n=1}^{\infty}$ is multiple regular with $\gamma_{s_{i}}(\cdot)$, then for any $\left(x_{j}^{(k)}\right)_{j=1}^{\infty}\in\gamma_{s_{i}}(E)$, $k=1,\dots,n+1$

[TABLE]

Therefore

[TABLE]

Like this, $(\varphi P)^{\vee}\in\mathcal{L}_{\gamma_{s,s_{i}}}^{m,n+1}(E^{n+1};F)$. Note also that, by (CH3)

[TABLE]

The condition (CH5) follows from Definition 4.2.

∎

For the definition of global holomorphy type, see, e.g., [4].

Corollary 4.9**.**

Under the assumptions of Theorem 4.8, $\left(\mathcal{P}_{\gamma_{s,s_{i}}}^{m,n},\|\cdot\|_{\mathcal{P}_{\gamma_{s,s_{i}}}^{m,n}}\right)_{n=1}^{\infty}$ is a global holomorphy type.

5 Applications

In this section we show that some well studied classes of multilinear operators can be recovered as particular cases of our abstract approach, and we also introduce new classes of multilinear operators that arise from our abstract point of view. Examples 3.7 and 4.5 assure that the main results of this paper apply to all classes listed in this section.

5.1 Multiple $(p,q_{1},\dots,q_{n})$-summing operators

The class of multiple $(p,q_{1},\dots,q_{n})$-summing operators, denoted by $\mathcal{L}_{ms(p,q_{1},\dots,q_{n})}(E_{1},\dots,E_{n};F)$, has been largely studied, see, e.g., [5, 9, 16]. This class is recovered by abstract framework by choosing

[TABLE]

for $1\leq q\leq p\leq\infty$ and $k=1,\dots,n$.

5.2 Multiple Cohen strongly $p$-summing operators

The class of multiple Cohen strongly $p$-summing operators, denoted by $\mathcal{L}_{mCoh,p}(E_{1},\dots,E_{m};F)$, was studied in [12]. In our abstract approach, it is recovered by choosing

[TABLE]

where $1<p<\infty$ and $k=1,\dots,n$.

5.3 Multiple mixing $(s,q,p)$-summing operators

The concepts of mixed $m(s,q)$-summing sequences and multiple mixing $(s,q,p)$-summing operators were studied in [1, 18, 20]. Just reminding the main definition, for $1\leq p_{1},\dots,p_{n}\leq q\leq s<\infty$, a continuous multilinear operator $T:E_{1}\times\cdots\times E_{n}\rightarrow F$ is multiple $(s,q,p_{1},\dots,p_{n})$-mixing summing if

[TABLE]

whenever $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\ell_{p_{i}}^{w}(E_{i})$, $i=1,\dots,n$. Note that, considering

[TABLE]

for $i=1,\dots,n$, this class is a particular case of our general construction.

The next three subsections introduce new classes of multilinear operators which are particular cases of our abstract framework, making clear that our results can also be applied to classes that had not been considered in the literature yet.

5.4 Multiple strong $(s,q,p)$-mixing summing operators

Let $1\leq q\leq s\leq\infty$, $E_{1},\dots,E_{n},F$ be Banach spaces and $p\leq q$. A continuous multilinear mapping $T:E_{1}\times\cdots\times E_{n}\rightarrow F$ is said to be multiple strong $(s,q,p)$-mixing summing if

[TABLE]

wherever $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\ell_{m(s,q)}(E_{i}),i=1,\dots,n$.

Choosing

[TABLE]

we conclude that the class of all multiple $(s,q,p)$-mixing summing multilinear operators is a Banach multi-ideal for which the results of this paper apply.

5.5 Multiple strong mid $p$-summing operators

Let $1\leq p<\infty$, $n\in\mathbb{N}$ and $E_{1},\dots,E_{n},F$ be Banach spaces. A continuous multilinear operator $T:E_{1}\times\cdots\times E_{n}\longrightarrow F$ is said to be multiple strong mid $p$-summing if

[TABLE]

whenever $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\ell_{p}^{mid}(E_{i})$, $i=1,\dots,n$.

Since the sequence classes $\ell_{p}$ and $\ell_{p}^{mid}$ are finitely determined and linearly stable (see [6, 7]), the class of multiple strong mid $p$-summing multilinear operators is one more particular case of the classes studied in this paper.

5.6 Multiple mid weakly $p$-summing operator

Let $1<p<\infty$, $n\in\mathbb{N}$ and $E_{1},\dots,E_{n},F$ be Banach spaces. A continuous multilinear application $T:E_{1}\times\cdots\times E_{n}\longrightarrow F$ is said to be multiple weakly mid $p$-summing if

[TABLE]

whenever $\left(x_{j}^{(i)}\right)_{j=1}^{\infty}\in\ell_{p}^{w}(E_{i})$, $i=1,\dots,n$.

For the same reasons, the class of all multiple mid weakly $p$-summing multilinear operators is a particular instance of the classes studied in this paper as well.