Ideals of polynomials between Banach spaces revisited

TL;DR

This paper introduces a unified framework for studying ideals of polynomials and multilinear operators between Banach spaces, extending previous research and establishing a generalized Bohnenblust--Hille inequality.

Contribution

It presents a more general approach to ideals of polynomials and multilinear operators, including new results like a generalized Bohnenblust--Hille inequality.

Findings

Established a generalized Bohnenblust--Hille inequality

Proposed a unified approach to polynomial ideals

Suggested new lines of investigation in the framework

Abstract

Ideals of polynomials and multilinear operators between Banach spaces have been exhaustively investigated in the last decades. In this paper, we introduce a unified (and more general) approach and propose some lines of investigation in this new framework. Among other results, we prove a Bohnenblust--Hille inequality in this more general setting.

Equations

$$A:E\times \cdots \times E\to F$$

$$P(x)=A(x,\dots ,x)$$

$$\Vert P\Vert :=\underset{\Vert x\Vert \le 1}{\sup }\Vert P(x)\Vert <\infty .$$

$$\mathcal{I}\left(E;F\right):=\mathcal{L}\left(E;F\right)\cap \mathcal{I}$$

$$t\circ u\circ v\in \mathcal{I}\left(G;H\right).$$

$${\Vert t\circ u\circ v\Vert }_{\mathcal{I}}\le \Vert t\Vert {\Vert u\Vert }_{\mathcal{I}}\Vert v\Vert \text{.}$$

$$T\left(x_1,\dots ,x_m\right)=\stackrel{k}{\sum _{i=1}}{\phi }_i^{\left(1\right)}\left(x_1\right)\cdots {\phi }_i^{\left(m\right)}\left(x_m\right)b_i\text{.}$$

$${\mathcal{M}}_m\left(E_1,\dots ,E_m;F\right):={\mathcal{L}}_m\left(E_1,\dots ,E_m;F\right)\cap \mathcal{M}$$

$$t\circ T\circ \left(u_1,\dots ,u_m\right)\in {\mathcal{M}}_m\left(G_1,\dots ,G_m;H\right).$$

$${\Vert t\circ T\circ \left(u_1,\dots ,u_m\right)\Vert }_{\mathcal{M}}\le \Vert t\Vert {\Vert T\Vert }_{\mathcal{M}}\Vert u_1\Vert \cdots \Vert u_m\Vert \text{.}$$

$${\mathcal{M}}_m:=\bigcup _{E_1,\dots ,E_m,F}{\mathcal{M}}_m\left(E_1,\dots ,E_m;F\right)$$

$$P\left(x\right)=\stackrel{k}{\sum _{i=1}}{\phi }_i{\left(x\right)}^m{b}_i\text{.}$$

$${\mathcal{Q}}_m\left({}^{m}E;F\right):={\mathcal{P}}_m\left({}^{m}E;F\right)\cap \mathcal{Q}$$

$$t\circ P\circ u\in {\mathcal{Q}}_m\left({}^{m}G;H\right).$$

$${\Vert t\circ P\circ u\Vert }_{\mathcal{Q}}\le \Vert t\Vert {\Vert P\Vert }_{\mathcal{Q}}{\Vert u\Vert }^m\text{.}$$

$${\mathcal{Q}}_m:=\bigcup _{E,F}{\mathcal{Q}}_m\left({}^{m}E;F\right)$$

$$T:E_1\times \cdots \times E_m\to F$$

$$P$$

$$P(x_1,\dots ,x_m)$$

$$\Vert T(x_1,\dots ,x_m)\Vert \le \Vert T\Vert {\Vert (x_1,\dots ,x_m)\Vert }^m,$$

$$\Vert T(x_1,\dots ,x_m)\Vert \le \Vert T\Vert \Vert x_1\Vert \cdots \Vert x_m\Vert .$$

$$\Vert P\Vert :=\sup \left\{\Vert P\left(x_1,\dots ,x_m\right)\Vert ;x_i\in E_i,\underset{i}{\max }{\Vert x_i\Vert }_{E_i}\le 1\right\}.$$

$$A\left(x_1,\dots ,x_m\right)=\frac{1}{m!2^m}\sum _{{\epsilon }_j=\pm 1}{\epsilon }_1\cdots {\epsilon }_{m}A{\left(x_0+{\epsilon }_1{x}_1+\cdots +{\epsilon }_m{x}_m\right)}^m\text{.}$$

$$\Vert \hat{A}\Vert \le \Vert A\Vert \le \frac{m^m}{m!}\Vert \hat{A}\Vert \text{,}$$

$$\Vert P\left(x_1,\dots ,x_m\right)\Vert$$

$$=n_m^{n_m}\Vert A\left(x_1,\dots ,x_{m-1}\right){\left(\frac{x_m}{n_m}\right)}^{n_m}\Vert$$

$$=n_m^{n_m}\Vert \frac{1}{n_m!2^{n_m}}\sum _{{\epsilon }_j=\pm 1}{\epsilon }_1\cdots {\epsilon }_{n_m}A\left(x_1,\dots ,x_{m-1}\right){\left(a_m+\left({\epsilon }_1+\cdots +{\epsilon }_{n_m}\right)\frac{x_m}{n_m}\right)}^{n_m}\Vert$$

$$\le \frac{n_m^{n_m}}{n_m!2^{n_m}}\sum _{{\epsilon }_j=\pm 1}\Vert A\left(x_1,\dots ,x_{m-1}\right){\left(a_m+\left({\epsilon }_1+\cdots +{\epsilon }_{n_m}\right)\frac{x_m}{n_m}\right)}^{n_m}\Vert$$

$$=\frac{n_m^{n_m}}{n_m!2^{n_m}}\sum _{{\epsilon }_j=\pm 1}\Vert P\left(\underset{m-1}{\underbrace{\cdot ,\dots ,\cdot }},a_m+\left({\epsilon }_1+\cdots +{\epsilon }_{n_m}\right)\frac{x_m}{n_m}\right)\left(x_1,\dots ,x_{m-1}\right)\Vert$$

$$\le \frac{n_m^{n_m}}{n_m!2^{n_m}}{2}^{n_m}C\frac{n_1^{n_1}}{n_1!}\cdots \frac{n_{m-1}^{n_{m-1}}}{n_{m-1}!}$$

$$=C\frac{n_1^{n_1}}{n_1!}\cdots \frac{n_{m-1}^{n_{m-1}}}{n_{m-1}!}\frac{n_m^{n_m}}{n_m!}\text{.}$$

$$\Vert P\left(x_1,\dots ,x_m\right)\Vert \le C{\Vert x_1\Vert }^{n_1}\cdots {\Vert x_m\Vert }^{n_m}\text{,}$$

$$\left(x_1,\dots ,x_m\right)\in E_1\times \cdots \times E_m,\Vert \left(x_1,\dots ,x_m\right)\Vert <\delta \Rightarrow \Vert P\left(x_1,\dots ,x_m\right)\Vert <1\text{.}$$

$$\Vert \left(\frac{\delta x_1}{2\Vert x_1\Vert },\dots ,\frac{\delta x_m}{2\Vert x_m\Vert }\right)\Vert =\frac{\delta }{2}<\delta$$

Full text

2010 Mathematics Subject Classification: Primary 46G25, 47L22, 47H60

T. Velanga was supported by FAPERO and CAPES

1. 1 Introduction
2. 2 Ideals of polynomials and multilinear operators: the classic definitions
3. 3 Basic results
4. 4 The unified approach: multipolynomial ideals
5. 5 Multipolynomial hyper-ideals
6. 6 A Bohnenblust–Hille inequality for multipolynomials

To the memory of Professor Jorge Mujica

1. Introduction

Linear Functional Analysis emerged in the 30’s after the publication of Banach’s monograph. The investigation of polynomials and multilinear operators between normed spaces is, of course, the first natural step when moving from linear to nonlinear Functional Analysis. The theory of polynomials between normed spaces is a basic tool for the investigation of holomorphic mappings in Banach spaces. We recall that if $E,F$ are normed spaces, a map $P:E\rightarrow F$ is called an $m$-homogeneous polynomial when there is an $m$-linear operator

[TABLE]

such that

[TABLE]

for all $x$ in $E.$ Continuity is defined as usual when dealing with metric spaces, and it is well known that $P$ is continuous if and only if

[TABLE]

The basics of the theory of polynomials and multilinear operators between Banach spaces can be found in the classical books [18, 21]. Polynomials and multilinear operators have been exhaustively investigated by quite different viewpoints. While polynomials are suitable to investigation of the holomorphic mappings, multilinear operators are commonly explored in the context of the extension of the operators ideals theory to the nonlinear setting. The notion of ideals of polynomials between Banach spaces is due to Pietsch [26]. The natural extension to multilinear operators and polynomials was designed by Pietsch some years later in [25]. Nowadays, ideals of polynomials and multilinear operators are explored by several authors in different directions (see, for instance, [1, 5, 6, 8, 9, 10, 11, 12, 13, 14, 20]). In this paper we are mainly interested in the theory of ideals of polynomials and ideals of multilinear operators between Banach spaces. We propose an unified approach to the subject and some themes for future research.

2. Ideals of polynomials and multilinear operators: the classic

definitions

We first recall the classical definition of operator ideals.

Definition 2.1** (Operator ideal [26]).**

An *operator ideal *is a class $\mathcal{I}$ of continuous linear operators between Banach spaces such that for all Banach space $E$ and $F$, its components

[TABLE]

satisfy:

**(Oa): **

$\mathcal{I}\left(E;F\right)$ is a linear subspace of $\mathcal{L}\left(E;F\right)$ which contains the finite rank operators;

**(Ob): **

the ideal property: if $u\in\mathcal{I}\left(E;F\right),~{}v\in\mathcal{L}\left(G;E\right)$ and $t\in\mathcal{L}\left(F;H\right)$, then

[TABLE]

Moreover, $\mathcal{I}$ is said to be a (quasi-) normed operator ideal if there exists a map $\left\|\cdot\right\|_{\mathcal{I}}:\mathcal{I}\rightarrow[0,\infty)$ satisfying:

**(O1): **

$\left\|\cdot\right\|_{\mathcal{I}}$ restricted to $\mathcal{I}\left(E;F\right)$ is a (quasi-) norm, for all Banach spaces $E$ and $F$;

**(O2): **

$\left\|id_{\mathbb{K}}:\mathbb{K}\rightarrow\mathbb{K}:id_{\mathbb{K}}\left(\lambda\right)=\lambda\right\|_{\mathcal{I}}=1$;

**(O3): **

if if $u\in\mathcal{I}\left(E;F\right),~{}v\in\mathcal{L}\left(G;E\right)$ and $t\in\mathcal{L}\left(F;H\right)$, then

[TABLE]

When all the components $\mathcal{I}\left(E;F\right)$ are complete under the (quasi-) norm $\left\|\cdot\right\|_{\mathcal{I}}$ above, then $\mathcal{I}$ is called a (quasi-) Banach operator ideal.

For the multilinear operators we have the following concepts:

Definition 2.2**.**

Let $E_{1},\ldots,E_{m},F$ be normed spaces. A multilinear mapping $T\in$ $\mathcal{L}\left(E_{1},\ldots,E_{m};F\right)$ is said to be of finite type if there exist $k\in\mathbb{N},~{}\varphi_{i}^{\left(j\right)}\in E_{j}^{\prime}$ and $b_{i}\in F$ for $i=1,\ldots,k$ and $j=1,\ldots,m$, such that

[TABLE]

The subspace of all finite-type members of $\mathcal{L}\left(E_{1},\ldots,E_{m};F\right)$ is denoted by $\mathcal{L}_{f}\left(E_{1},\ldots,E_{m};F\right)$.

Definition 2.3** (Ideal of multilinear mappings [20]).**

For each positive integer $m$, let $\mathcal{L}_{m}$ denote the class of all continuous $m$-linear operators between Banach spaces. An ideal of multilinear mappings $\mathcal{M}$ is a subclass of the class $\mathcal{L}=\underset{m=1}{\overset{\infty}{\bigcup}}\mathcal{L}_{m}$ of all continuous multilinear operators between Banach spaces such that for a positive integer $m$, Banach spaces $E_{1},\ldots,E_{m}$ and $F$, the components

[TABLE]

satisfy:

**(Ma): **

$\mathcal{M}_{m}\left(E_{1},\ldots,E_{m};F\right)$ is a linear subspace of $\mathcal{L}_{m}\left(E_{1},\ldots,E_{m};F\right)$ which contains the $m$-linear mappings of finite type;

**(Mb): **

the ideal property: if $T\in\mathcal{M}_{m}\left(E_{1},\ldots,E_{m};F\right),~{}u_{j}\in\mathcal{L}_{1}\left(G_{j};E_{j}\right)$ for $j=1,\ldots,m$, and $t\in\mathcal{L}_{1}\left(F;H\right)$, then

[TABLE]

Moreover, $\mathcal{M}$ is said to be a (quasi-) normed ideal of multilinear mappings if there exists a map $\left\|\cdot\right\|_{\mathcal{M}}:\mathcal{M}\rightarrow[0,\infty)$ satisfying:

**(M1): **

$\left\|\cdot\right\|_{\mathcal{M}}$ restricted to $\mathcal{M}_{m}\left(E_{1},\ldots,E_{m};F\right)$ is a (quasi-) norm, for all $m\in\mathbb{N}$ and Banach spaces $E_{1},\ldots,E_{m}$ and $F$;

**(M2): **

$\left\|id_{m}:\mathbb{K}^{m}\rightarrow\mathbb{K}:id_{m}\left(\lambda_{1},\ldots,\lambda_{m}\right)=\lambda_{1}\cdots\lambda_{m}\right\|_{\mathcal{M}}=1$, for all $m\in\mathbb{N}$;

**(M3): **

If $T\in\mathcal{M}_{m}\left(E_{1},\ldots,E_{m};F\right),~{}u_{j}\in\mathcal{L}_{1}\left(G_{j};E_{j}\right)$ for $j=1,\ldots,m$, and $t\in\mathcal{L}_{1}\left(F;H\right)$, then

[TABLE]

When all the components $\mathcal{M}_{m}\left(E_{1},\ldots,E_{m};F\right)$ are complete under the (quasi-) norm $\left\|\cdot\right\|_{\mathcal{M}}$ above, $\mathcal{M}$ is said to be a *(quasi-) Banach ideal of multilinear mappings. *For a fixed ideal of multilinear mappings $\mathcal{M}$ and a positive integer $m\in\mathbb{N}$, the class

[TABLE]

is called an ideal of $m$-linear mappings.

For the homogeneous polynomials we have the following concepts:

Definition 2.4**.**

Let $E,F$ be normed spaces. A polynomial $P\in$ $\mathcal{P}\left({}^{m}E;F\right)$ is said to be of finite type if there exists $k\in\mathbb{N},~{}\varphi_{i}\in E^{\prime}$ and $b_{i}\in F$ for $i=1,\ldots,k$, such that

[TABLE]

The subspace of all finite-type members of $\mathcal{P}\left({}^{m}E;F\right)$ is denoted by $\mathcal{P}_{f}\left({}^{m}E;F\right)$.

Definition 2.5** (Polynomial ideal [25]).**

For each positive integers $m$, let $\mathcal{P}_{m}$ denote the class of all continuous $m$-homogeneous polynomials between Banach spaces. A polynomial ideal $\mathcal{Q}$ (or ideal of homogeneous polynomials) is a subclass of the class $\mathcal{P}=\underset{m=1}{\overset{\infty}{\bigcup}}\mathcal{P}_{m}$ of all continuous homogeneous polynomials between Banach spaces such that for all $m\in\mathbb{N}$ and all Banach spaces $E$ and $F$, the components

[TABLE]

satisfy:

**(Pa): **

$\mathcal{Q}_{m}\left({}^{m}E;F\right)$ is a linear subspace of $\mathcal{P}_{m}\left({}^{m}E;F\right)$ which contains the finity-type $m$-homogeneous polynomials;

**(Pb): **

the ideal property: if $u\in\mathcal{L}_{1}\left(G;E\right)$ $P\in\mathcal{Q}_{m}\left({}^{m}E;F\right)$ and $t\in\mathcal{L}_{1}\left(F;H\right)$, then

[TABLE]

Moreover, $\mathcal{Q}$ is said to be a (quasi-) normed polynomial ideal if there exists a map $\left\|\cdot\right\|_{\mathcal{Q}}:\mathcal{Q}\rightarrow[0,\infty)$ satisfying:

**(P1): **

$\left\|\cdot\right\|_{\mathcal{Q}}$ restricted to $\mathcal{Q}_{m}\left({}^{m}E;F\right)$ is a (quasi-) norm, for all $m\in\mathbb{N}$ and all Banach spaces $E$ and $F$;

**(P2): **

$\left\|id_{m}:\mathbb{K}\rightarrow\mathbb{K}:id_{m}\left(\lambda\right)=\lambda^{m}\right\|_{\mathcal{Q}}=1$, for all $m\in\mathbb{N}$;

**(P3): **

If $u\in\mathcal{L}_{1}\left(G;E\right),~{}P\in\mathcal{Q}_{m}\left({}^{m}E;F\right)$ and $t\in\mathcal{L}_{1}\left(F;H\right)$, then

[TABLE]

When all the components $\mathcal{Q}_{m}\left({}^{m}E;F\right)$ are complete under the (quasi-) norm $\left\|\cdot\right\|_{\mathcal{Q}}$ above, then $\mathcal{Q}$ is called a *(quasi-) Banach polynomial ideal. *For a fixed polynomial ideal $\mathcal{Q}$ and a positive integer $m\in\mathbb{N}$, the class

[TABLE]

is called an ideal of $m$-homogeneous polynomials.

3. Basic results

A fact apparently overloked in the literature is that every $m$-linear operator is in fact a polynomial (we thank Prof. Pilar Rueda and R. Aron for important conversations about it). More precisely, if

[TABLE]

is an $m$-linear operator then, denoting $E:=E_{1}\times\cdots\times E_{m},$ the map

[TABLE]

is an $m$-homogeneous polynomial. This fact can be easily proved by using tensor products. So, one can wonder why to define separately ideals of polynomials and ideals of multilinear operators, having in mind that every $m$-linear operator is in fact an $m$-homogeneous polynomial. Well, we can give a couple of reasons for that. A more obvious reason is that when considering a multilinear operator as a polynomial we have

[TABLE]

and this estimate is less precise than

[TABLE]

So, one cannot unify the theory of polynomial ideals and multilinear ideals just by realizing that multilinear operators are in fact homogeneous polynomials. In this section we unify the theory of polynomials and multilinear operators in a more careful analytic viewpoint. More precisely, if $m$ is a given positive integer and $n_{1},...,n_{m}$ are positive integers such that $n_{1}+\cdots+n_{j}=$ $m$ we introduce the notion of $\left(n_{1},...,n_{j}\right)$-homogeneous polynomial for $j\in\{1,...,m\}$. When $j=1$ we have an $m$-homogeneous polynomial and when $j=m$ then we have an $m$-linear operator. This kind of maps will be called multipolynomials.

Definition 3.1**.**

Let $m\in\mathbb{N}$ and $\left(n_{1},\ldots,n_{m}\right)\in\mathbb{N}^{m}$. A mapping $P:E_{1}\times\cdots\times E_{m}\rightarrow F$ is said to be an $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomial if, for each $i=1,\ldots,m$, the mapping $P\left(x_{1},\ldots,x_{i-1},\cdot,x_{i+1},\ldots,x_{m}\right):E_{i}\rightarrow F$ is an $n_{i}$-homogeneous polynomial for every $x_{1}\in E_{1},\ldots,x_{i-1}\in E_{i-1},x_{i+1}\in E_{i+1},\ldots,x_{m}\in E_{m}$ fixed.

We shall denote by $\mathcal{P}_{a}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ the vector space of all $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomials from the cartesian product $E_{1}\times\cdots\times E_{m}$ into $F$. We shall represent by $\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ the subspace of all continuous members of $\mathcal{P}_{a}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$. For each $P\in\mathcal{P}_{a}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ we shall set

[TABLE]

Our first step is to present a comprehensive list conditions which characterize the continuous multipolynomials similarly as in (1).

Henceforth $\mathcal{L}_{a}^{s}\left({}^{m}E;F\right)$ denotes the space of all $m$-linear forms from $E\times\cdots\times E$ to $F$ which are symmetric and for each $A\in\mathcal{L}_{a}\left({}^{m}E;F\right)$ the $m$-homogeneous polynomial $\widehat{A}\in\mathcal{P}_{a}\left({}^{m}E;F\right)$ is defined by $\widehat{A}\left(x\right)=Ax^{m}$ for every $x\in E$.

We recall some results from the theory of homogeneous polynomials between Banach spaces that will be useful in this paper (these results can be found, for instance, in [21, Theorem 1.10], [21, Theorem 2.2] and [21, Corollary 2.3]).

• •

(Polarization Formula) If $A\in\mathcal{L}_{a}^{s}\left({}^{m}E;F\right)$, then for $x_{0},\ldots,x_{m}\in E$ we have

[TABLE]

• •

The mapping $A\mapsto\widehat{A}$ induces a vector space isomorphism between $\mathcal{L}_{a}^{s}\left({}^{m}E;F\right)$ and $\mathcal{P}_{a}\left({}^{m}E;F\right)$.

• •

We have the inequalities

[TABLE]

for every $A\in\mathcal{L}_{a}^{s}\left({}^{m}E;F\right)$.

• •

A polynomial $P\in\mathcal{P}_{a}\left({}^{m}E;F\right)$ is continuous if and only if $\left\|P\right\|<\infty$.

• •

$\mathcal{P}\left({}^{m}E;F\right)$ is a Banach space under the norm $P\mapsto\left\|P\right\|$.

• •

The mapping $A\mapsto\widehat{A}$ induces a topological isomorphism between $\mathcal{L}^{s}\left({}^{m}E;F\right)$ and $\mathcal{P}\left({}^{m}E;F\right)$.

We begin with an useful lemma:

Lemma 3.2**.**

Let $E_{1},\ldots,E_{m},F$ be normed spaces and $P\in\mathcal{P}_{a}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$. If $P$ is bounded by $C$ on an open ball $B_{E_{1}\times\cdots\times E_{m}}\left(\left(a_{1},\ldots,a_{m}\right);r\right)$ then $P$ is bounded by $C\frac{n_{1}^{n_{1}}}{n_{1}!}\cdots\frac{n_{m}^{n_{m}}}{n_{m}!}$ on the ball $B_{E_{1}\times\cdots\times E_{m}}\left(\left(0,\ldots,0\right);r\right)$.

Proof.

Let $\left(x_{1},\ldots,x_{m}\right)\in B_{E_{1}\times\cdots\times E_{m}}\left(\left(0,\ldots,0\right);r\right)$. We prove this by induction on $m$. When $m=1$ it is just the already well-known result from the theory of homogeneous polynomials between Banach spaces (see [21, Lemma 2.5]). If $m>1$ then the multipolynomial $P\left(\underset{m-1}{\underbrace{\cdot,\ldots,\cdot}},y\right)\in\mathcal{P}_{a}\left({}^{n_{1}}E_{1},\ldots,^{n_{m-1}}E_{m-1};F\right)$ is bounded by $C$ on the ball $B_{E_{1}\times\cdots\times E_{m-1}}\left(\left(a_{1},\ldots,a_{m-1}\right);r\right)$, for all $y\in B_{E_{m}}\left(a_{m};r\right)$. The induction hypothesis implies that $P\left(\underset{m-1}{\underbrace{\cdot,\ldots,\cdot}},y\right)$ is bounded by $C\frac{n_{1}^{n_{1}}}{n_{1}!}\cdots\frac{n_{m-1}^{n_{m-1}}}{n_{m-1}!}$ on the ball $B_{E_{1}\times\cdots\times E_{m-1}}\left(\left(0,\ldots,0\right);r\right)$, whenever $y\in B_{E_{m}}\left(a_{m};r\right)$. We also have from [21, Theorem 2.2] that the $n_{m}$-linear mapping, denoted by $A\left(x_{1},\ldots,x_{m-1}\right),$ associated to the polynomial $P\left(x_{1},\ldots,x_{m-1},\cdot\right)\in\mathcal{P}_{a}\left({}^{n_{m}}E_{m};F\right)$ can be taken symmetrical. Now applying the Polarization Formula [21, Theorem 1.10] to $A\left(x_{1},\ldots,x_{m-1}\right)$ with $x_{0}=a_{m}$ and $x_{1}=\cdots=x_{n_{m}}=\frac{x_{m}}{n_{m}}$ we get

[TABLE]

Then it follows that $P$ is bounded by $C\frac{n_{1}^{n_{1}}}{n_{1}!}\cdots\frac{n_{m}^{n_{m}}}{n_{m}!}$ on the ball $B_{E_{1}\times\cdots\times E_{m}}\left(\left(0,\ldots,0\right);r\right)$, and the proof is complete.

Now we are ready to characterize continuous multipolynomials.

Theorem 3.3**.**

Let $E_{1},\ldots,E_{m},F$ be normed spaces and $P\in\mathcal{P}_{a}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$. The following conditions are equivalent:

**(i): **

$P$* is continuous;*

**(ii): **

$P$* is continuous at the origin;*

**(iii): **

There exists a constant $C\geq 0$ such that

[TABLE]

for all $\left(x_{1},\ldots,x_{m}\right)\in E_{1}\times\cdots\times E_{m}$;

**(iv): **

$\left\|P\right\|<\infty$;

**(v): **

$P$* is uniformly continuous on bounded subsets of $E_{1}\times\cdots\times E_{m}$;*

**(vi): **

$P$* is bounded on every ball with finite radius;*

**(vii): **

$P$* is bounded on some ball;*

**(viii): **

$P$* is bounded on some ball with center at the origin.*

Proof.

The implications $\left(i\right)\Rightarrow(ii)$ and $\left(vi\right)\Rightarrow\left(vii\right)$ are obvious.

$\left(ii\right)\Rightarrow\left(iii\right)$: Suppose $P$ continuous at the origin. Then, there exist $\delta>0$ such that

[TABLE]

The inequality in $\left(iii\right)$ is obvious if $x_{i}=0$ for some $i=1,\ldots,m$. So we can assume $x_{i}\neq 0$ for all $i=1,\ldots,m$. Then,

[TABLE]

and thus

[TABLE]

This give us $\left(iii\right)$ with $C=\left(\frac{2}{\delta}\right)^{n_{1}+\cdots+n_{m}}$.

$\left(iii\right)\Rightarrow\left(iv\right)$: If $\left(iii\right)$ is true then we have in particular,

[TABLE]

for all $x_{1}\in E_{1},\ldots,x_{m}\in E_{m}$, with $\left\|x_{1}\right\|,\ldots,\left\|x_{m}\right\|\leq 1$. This shows that $\left\|P\right\|\leq C$.

$\left(iv\right)\Rightarrow\left(v\right)$: Let $a=\left(a_{1},\ldots,a_{m}\right),~{}x=\left(x_{1},\ldots,x_{m}\right)\in E_{1}\times\cdots\times E_{m}$ with

[TABLE]

and

[TABLE]

Then

[TABLE]

Let $A\left(a_{1},\ldots,a_{i-1},x_{i+1},\ldots,x_{m}\right)$ be the symmetric $n_{i}$-linear form associated to $P\left(a_{1},\ldots,a_{i-1},\cdot,x_{i+1},\ldots,x_{m}\right).$ From (2) we get

[TABLE]

for every $i=1,\ldots,m$. Since $\left\|P\right\|<\infty$ the inequalities (3) and (3) show us that both the $i$-th ordinary polynomial $P\left(a_{1},\ldots,a_{i-1},\cdot,x_{i+1},\ldots,x_{m}\right)\in$ $\mathcal{P}_{a}\left({}^{n_{i}}E_{i};F\right)$ as its associated multilinear mapping $A\left(a_{1},\ldots,a_{i-1},x_{i+1},\ldots,x_{m}\right)\in\mathcal{L}_{a}\left({}^{n_{i}}E_{i};F\right)$ are continuous for each $i=1,\ldots,m$. Now we can write

[TABLE]

and the uniform continuity of $P$ on bounded subsets of $E_{1}\times\cdots\times E_{m}$ follows.

$\left(v\right)\Rightarrow\left(i\right)$: Let us show that $P$ is continuous at an arbitrary point $a=\left(a_{1},\ldots,a_{m}\right)\in E=E_{1}\times\cdots\times E_{m}$. Given $\varepsilon>0$ it follows from the uniform continuity of $P$ on bounded subsets that there exist $\delta_{0}>0$ such that, for every $x=\left(x_{1},\ldots,x_{m}\right),y=\left(y_{1},\ldots,y_{m}\right)\in B_{E}\left(0;\left\|a\right\|+1\right)$,

[TABLE]

Defining $\delta=\min\left\{\delta_{0},1\right\}$ we get

[TABLE]

and thus $P$ is continuous at the point $a$.

$\left(iii\right)\Rightarrow\left(vi\right)$: Let $B$ be a ball with center at $\left(a_{1},\ldots,a_{m}\right)\in E_{1}\times\cdots\times E_{m}$ and radius $r>0$. For every $\left(x_{1},\ldots,x_{m}\right)\in B$ the hypothesis $\left(iii\right)$ give us a constant $C\geq 0$ such that

[TABLE]

and so $P$ is bounded on $B$.

$\left(vii\right)\Rightarrow\left(viii\right)$: It follows immediately from the Lemma 3.2.

$\left(viii\right)\Rightarrow\left(iv\right)$: Suppose that there exist $r>0$  and  $C\geq 0$ such that

[TABLE]

Thus, given $x_{1}\in E_{1},\ldots,x_{m}\in E_{m}$, with $\left\|x_{1}\right\|,\ldots,\left\|x_{m}\right\|\leq 1$, we have $\left(\frac{r}{2}x_{1},\ldots,\frac{r}{2}x_{m}\right)\in B_{E_{1}\times\cdots\times E_{m}}\left(\left(0,\ldots,0\right);r\right)$ and hence

[TABLE]

Corollary 3.4**.**

Let $E_{1},\ldots,E_{m}$ and $F$ be normed spaces and let $P$ be a continuous multipolynomial in $\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$. Then:

**(i): **

$\left\|P\left(x_{1},\ldots,x_{m}\right)\right\|\leq\left\|P\right\|\left\|x_{1}\right\|^{n_{1}}\cdots\left\|x_{m}\right\|^{n_{m}}$, for all $\left(x_{1},\ldots,x_{m}\right)\in E_{1}\times\cdots\times E_{m}$.

**(ii): **

$\left\|P\right\|=\inf\left\{C\geq 0;\left\|P\left(x_{1},\ldots,x_{m}\right)\right\|\leq C\left\|x_{1}\right\|^{n_{1}}\cdots\left\|x_{m}\right\|^{n_{m}}\text{ for all }x_{j}\in E_{j},j=1,\ldots,m\right\}$.

Remark 3.5**.**

The following results are also ensured:

• •

Still in the context of normed spaces, the pointwise convergence of a sequence $\left(P_{j}\right)$ in $\mathcal{P}_{a}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ implies that the limit mapping $P$ is also a multipolynomial in that same space. More can be said if we take Banach spaces into account.

• •

The mapping $P\mapsto\left\|P\right\|$ defines a norm on the vector space $\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ which turns it into a Banach space, provided that $F$ is complete.

• •

If $F$ is just a normed space and all the factor spaces of the domain of $P$ are Banach spaces, then continuity and separated continuity are the same. Besides, the Uniform Boundedness Principle (UBP) and the Banach–Steinhaus Theorem (BST) hold, as we see next.

Theorem 3.6** (Uniform Boundedness Principle).**

Let $E_{1},\ldots,E_{m}$ be Banach spaces, $F$ be a normed space and let $\left\{P_{i}\right\}_{i\in I}$ be a family in $\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$. The following conditions are equivalent:

**(i): **

For every $x=\left(x_{1},\ldots,x_{m}\right)\in E_{1}\times\cdots\times E_{m}$ there exists $C_{x}<\infty$ such that

[TABLE]

**(ii): **

The family $\left\{P_{i}\right\}_{i\in I}$ is norm bounded, that is,

[TABLE]

Corollary 3.7** (Banach-Steinhaus Theorem).**

Let $\left(P_{j}\right)_{j=1}^{\infty}$ be a sequence in $\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ such that $\left(P_{j}\left(x_{1},\ldots,x_{m}\right)\right)_{j=1}^{\infty}$ is convergent in $F$ for all $\left(x_{1},\ldots,x_{m}\right)\in E_{1}\times\cdots\times E_{m}$. If we define

[TABLE]

by

[TABLE]

then $P\in\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$.

Under the same hypothesis of the BST presented above, we can also conclude that $P_{j}$ converges to $P$ uniformly on compact subsets of $E_{1}\times\cdots\times E_{m}$.

It is worth noting that the extreme cases $n_{1}=\ldots=n_{m}=1$ or $m=1$ recover the corresponding classical theorems for multilinear mappings (see, for instance, [29, 4]) and homogeneous polynomials. The special case $m=n_{1}=1$ recovers the classicals UBP and BST for linear operators.

4. The unified approach: multipolynomial ideals

In this section, we present a more general and unified version for the ideals of polynomials and multilinear operators.

Definition 4.1**.**

Let $E_{1},\ldots,E_{m},F$ be normed spaces. A multipolynomial $P\in\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ is said to be of finite type if there exists $k\in\mathbb{N},~{}\varphi_{i}^{\left(j\right)}\in E_{j}^{\prime}$ and $b_{i}\in F$ with $i=1,\ldots,k$ and $j=1,\ldots,m$, such that

[TABLE]

We shall represent by $\mathfrak{F}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ the subspace of all finite type members of $\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$. The so-called approximable multipolynomials are defined to be the members of

[TABLE]

Definition 4.2** (Multipolynomial ideal).**

For each $m\in\mathbb{N}$ and multi-degree $\left(n_{1},\ldots,n_{m}\right)\in\mathbb{N}^{m}$, let $\mathcal{P}_{m}^{\left(n_{1},\ldots,n_{m}\right)}$ denote the class of all continuous $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomials between Banach spaces. A* multipolynomial ideal* $\mathfrak{U}$ is a subclass of the class $\mathfrak{P}:\mathcal{=}\underset{m=1}{\overset{\infty}{\bigcup}}\left(\underset{\left(n_{1},\ldots,n_{m}\right)\in\mathbb{N}^{m}}{\bigcup}\mathcal{P}_{m}^{\left(n_{1},\ldots,n_{m}\right)}\right)$ of all continuous multipolynomials between Banach spaces such that for all $m\in\mathbb{N}$, multi-degree $\left(n_{1},\ldots,n_{m}\right)\in\mathbb{N}^{m}$ and all Banach spaces $E_{1},\ldots,E_{m}$ and $F$, the components

[TABLE]

satisfy:

**(Ua): **

$\mathfrak{U}_{m}^{\left(n_{1},\ldots,n_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ is a linear subspace of $\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ which contains the $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomials of finite type;

**(Ub): **

the ideal property: if $P\in\mathfrak{U}_{m}^{\left(n_{1},\ldots,n_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right),~{}u_{j}\in\mathcal{L}_{1}\left(G_{j};E_{j}\right)$ for $j=1,\ldots,m$, and $t\in\mathcal{L}_{1}\left(F;H\right)$, then

[TABLE]

Moreover, $\mathfrak{U}$ is said to be a (quasi-) normed multipolynomial ideal if there exists a map $\left\|\cdot\right\|_{\mathfrak{U}}:\mathfrak{U}\rightarrow[0,\infty)$ satisfying:

**(U1): **

$\left\|\cdot\right\|_{\mathfrak{U}}$ restricted to $\mathfrak{U}_{m}^{\left(n_{1},\ldots,n_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ is a (quasi-) norm, for all $m\in\mathbb{N}$, multi-degree $\left(n_{1},\ldots,n_{m}\right)\in$ $\mathbb{N}^{m}$ and all Banach spaces $E_{1},\ldots,E_{m}$ and $F$;

**(U2): **

$\left\|id_{m}^{\left(n_{1},\ldots,n_{m}\right)}:\mathbb{K}^{m}\rightarrow\mathbb{K}:id_{m}^{\left(n_{1},\ldots,n_{m}\right)}\left(\lambda_{1},\ldots,\lambda_{m}\right)=\lambda_{1}^{n_{1}}\cdots\lambda_{m}^{n_{m}}\right\|_{\mathfrak{U}}=1$, for all $m\in\mathbb{N}$ and $\left(n_{1},\ldots,n_{m}\right)\in$ $\mathbb{N}^{m}$;

**(U3): **

If $P\in\mathfrak{U}_{m}^{\left(n_{1},\ldots,n_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right),~{}u_{j}\in\mathcal{L}_{1}\left(G_{j};E_{j}\right)$ for $j=1,\ldots,m$, and $t\in\mathcal{L}_{1}\left(F;H\right)$, then

[TABLE]

When all the components $\mathfrak{U}_{m}^{\left(n_{1},\ldots,n_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ are complete under the (quasi-) norm $\left\|\cdot\right\|_{\mathfrak{U}}$ above, then $\mathfrak{U}$ is called a *(quasi-) Banach multipolynomial ideal. *For a fixed multipolynomial ideal $\mathfrak{U}$, a positive integer $m\in\mathbb{N}$ and a multi-degree $\left(n_{1},\ldots,n_{m}\right)\in\mathbb{N}^{m}$, the class

[TABLE]

is called an ideal of $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomials.

A multipolynomial ideal $\mathfrak{U}$ is said to be closed if all components* $\mathfrak{U}_{m}^{\left(n_{1},\ldots,n_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$* are closed subspace of $\left(\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right),\left\|\cdot\right\|\right)$.

Natural examples can be found.

Definition 4.3**.**

Given a multipolynomial ideal $\mathfrak{U}$, we shall define

[TABLE]

for all $m\in\mathbb{N}$, multi-degree $\left(n_{1},\ldots,n_{m}\right)\in$ $\mathbb{N}^{m}$ and all Banach spaces $E_{1},\ldots,E_{m}$ and $F$.

Definition 4.4**.**

A multipolynomial $P\in\mathcal{P}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ is said to be compact (resp. weakly compact) if

[TABLE]

is relatively compact (resp. weakly relatively compact). In that case we write $\mathfrak{K}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ (resp. $\mathfrak{W}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$).

Definition 4.5**.**

Let $0<p,q_{1},\ldots,q_{m}$ and let $E_{1},\ldots,E_{m}$ and $F$ be Banach spaces. A continuous $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomial $P:E_{1}\times\cdots\times E_{m}\rightarrow F$ is said to be absolutely $\left(p;q_{1},\ldots,q_{m}\right)$-*summing *if there exists $C\geq 0$ such that

[TABLE]

for all $\left(x_{j}^{\left(k\right)}\right)_{j=1}^{\infty}\in\ell_{q_{k}}^{w}\left(E_{k}\right)$, with$~{}k=1,\ldots,m$. In this case we write $\mathfrak{P}_{\text{\emph{as}}\left(p,q_{1},\ldots,q_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$.

In a quite more demanding way, $P$ is said to be fully $\left(p;q_{1},\ldots,q_{m}\right)$-summing if there exists $C\geq 0$ such that**

[TABLE]

for all $\left(x_{j}^{\left(k\right)}\right)_{j=1}^{\infty}\in\ell_{q_{k}}^{w}\left(E_{k}\right)$, with$~{}k=1,\ldots,m$. In this case we write $\mathfrak{P}_{\text{\emph{fs}}\left(p,q_{1},\ldots,q_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$.

The infimum of the $C>0$ for which inequality (5) (resp. (6)) always holds defines a norm for the case $p\geq 1$ and a $p$-norm for $p<1$ on the space $\mathfrak{P}_{as\left(p,q_{1},\ldots,q_{m}\right)}^{\left(n_{1},\ldots,n_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$ (resp. $\mathfrak{P}_{\text{\emph{fs}}\left(p,q_{1},\ldots,q_{m}\right)}\left({}^{n_{1}}E_{1},\ldots,^{n_{m}}E_{m};F\right)$). In any case, we thus obtain complete topological metrizable spaces.

We now give a list of several examples.

$\mathfrak{P}$**: **

Ideal of continuous multipolynomials;

$\mathfrak{F}$**: **

Ideal of finite-type multipolynomials;

$\overline{\mathfrak{U}}$**: **

The closure of a multipolynomial ideal $\mathfrak{U}$;

$\mathfrak{P}_{A}$**: **

Ideal of approximable multipolynomials;

$\mathfrak{K}$**: **

Ideal of compact multipolynomials;

$\mathfrak{W}$**: **

Ideal of weakly compact multipolynomials;

$\mathfrak{P}_{\text{\emph{as}}}$**: **

Ideal of absolutely summing multipolynomials;

$\mathfrak{P}_{\text{\emph{fs}}}$**: **

Ideal of fully summing multipolynomials.

The multipolynomial ideals $\mathfrak{P}$, $\overline{\mathfrak{U}}$, $\mathfrak{P}_{A}$, $\mathfrak{K}$, $\mathfrak{W}$ are closed. If $\frac{1}{p}\leq\frac{n_{1}}{q_{1}}+\cdots+\frac{n_{m}}{q_{m}}$ then $\mathfrak{P}_{\emph{as}}$ and $\mathfrak{P}_{\emph{fs}}$ are Banach multipolynomial ideals for $p\geq 1$ and $p$-Banach multipolynomial ideals for $p<1$. We have null-spaces when $\frac{1}{p}>\frac{n_{1}}{q_{1}}+\cdots+\frac{n_{m}}{q_{m}}$.

Remark 4.6**.**

We shall recall that as particular cases or, more precisely, as extreme cases ($n_{1}=\cdots=n_{m}=1$ or $m=1$), every ideal of multilinear mappings (which includes the linear operator ideals) as well as every polynomial ideal already established in the literature is a multipolynomial ideal. They will be called extreme multipolynomial ideals.

5. Multipolynomial hyper-ideals

Recently in papers [11] and [12] the authors introduced and developed the respective notions of hyper-ideals of multilinear operators and homogeneous polynomials between Banach spaces. While the well studied notions of ideals of multilinear operators (multi-ideals) as well as polynomial ideals relies on the composition with linear operators (the so-called ideal property), the notion proposed by the authors, called now as hyper-ideal property, considers in [11] the compositions with multilinear operators and, under the polynomial viewpoint, considers in [12] the compositions with homogeneous polynomials. Historically speaking, the hyper-ideal property has already been studied individually for some specific classes, see, e.g, [17, 27, 28], and then [11, 12] started the systematic study of the classes satisfying this stronger condition. The aim of this section is to invoke the multipolynomials again, as we have done before, to generalize and propose a unified approach for all these isolated notions of hyper-ideals of operators (multilinear and polynomial) which have been studied separately so far.

Definition 5.1** (Hyper-ideal of multilinear operators [11]).**

A hyper-ideal of multilinear operators is a subclass $\mathcal{H}$ of the class of all continuous multilinear operators between Banach spaces such that for all $n\in\mathbb{N}$ and all Banach spaces $E_{1},\ldots,E_{n}$ and $F$, the components

[TABLE]

satisfy:

**(ha): **

$\mathcal{H}\left(E_{1},\ldots,E_{n};F\right)$ is a linear subspace of $\mathcal{L}\left(E_{1},\ldots,E_{n};F\right)$ which contains the $n$-linear operators of finite type ;

**(hb): **

The hyper-ideal property: given natural numbers $n$ and $1\leq m_{1}<\cdots<m_{n}$ and Banach spaces $G_{1},\ldots,G_{m_{n}},E_{1},\ldots,E_{n},F$ and $H$, if B_{1}\in\mathcal{L}\left(G_{1},\ldots,G_{m_{1}};E_{1}\right),\ldots,$$B_{n}\in\mathcal{L}\left(G_{m_{n-1}+1},\ldots,G_{m_{n}};E_{n}\right),A\in\mathcal{H}\left(E_{1},\ldots,E_{n};F\right) and $t\in\mathcal{L}\left(F;H\right)$, then

[TABLE]

Moreover, $\mathcal{H}$ is said to be a (quasi-) normed hyper-ideal of multilinear operators if there exists a map $\left\|\cdot\right\|_{\mathcal{H}}:\mathcal{H}\rightarrow[0,\infty)$ satisfying:

**(h1): **

$\left\|\cdot\right\|_{\mathcal{H}}$ restricted to $\mathcal{H}\left(E_{1},\ldots,E_{n};F\right)$ is a (quasi-) norm, for all $n\in\mathbb{N}$ and all Banach spaces $E_{1},\ldots,E_{n}$ and $F$;

**(h2): **

$\left\|I_{n}:\mathbb{K}^{n}\rightarrow\mathbb{K},I_{n}\left(\lambda_{1},\ldots,\lambda_{n}\right)=\lambda_{1}\cdots\lambda_{n}\right\|_{\mathcal{H}}=1$, for all $n\in\mathbb{N}$;

**(h3): **

The hyper-ideal inequality: if B_{1}\in\mathcal{L}\left(G_{1},\ldots,G_{m_{1}};E_{1}\right),\ldots,$$B_{n}\in\mathcal{L}\left(G_{m_{n-1}+1},\ldots,G_{m_{n}};E_{n}\right),A\in\mathcal{H}\left(E_{1},\ldots,E_{n};F\right) and $t\in\mathcal{L}\left(F;H\right)$, then

[TABLE]

When all the components $\mathcal{H}\left(E_{1},\ldots,E_{n};F\right)$ are complete under the (quasi-) norm $\left\|\cdot\right\|_{\mathcal{H}}$ above, then $\left(\mathcal{H},\left\|\cdot\right\|_{\mathcal{H}}\right)$ is called a (quasi-) Banach hyper-ideal of multilinear operators.

It is plain that every (normed, quasi-normed, Banach, quasi-Banach) hyper-ideal is a (normed, quasi-normed, Banach, quasi-Banach) multi-ideal.

Definition 5.2** (Polynomial hyper-ideal [12]).**

A polynomial hyper-ideal is a subclass $\mathcal{Q}$ of the class of all continuous homogeneous polynomials between Banach spaces such that for all $n\in\mathbb{N}$ and all Banach spaces $E$ and $F$, the components

[TABLE]

satisfy:

**(pa): **

$\mathcal{Q}\left({}^{n}E;F\right)$ is a linear subspace of $\mathcal{P}\left({}^{n}E;F\right)$ which contains the $n$-homogeneous polynomials of finite type;

**(pb): **

The hyper-ideal property: given $m,n\in\mathbb{N}$ and Banach spaces $E,F,G$ and $H$, if $Q\in\mathcal{P}\left({}^{m}G;E\right),P\in\mathcal{Q}\left({}^{n}E;F\right)$ and $t\in\mathcal{L}\left(F;H\right)$, then

[TABLE]

If there exist a map $\left\|\cdot\right\|_{\mathcal{Q}}:\mathcal{Q}\rightarrow[0,\infty)$ and a sequence $\left(C_{j}\right)_{j=1}^{\infty}$ of real numbers with $C_{j}\geq 1$ for every $j\in\mathbb{N}$ and $C_{1}=1$, such that:

**(p1): **

$\left\|\cdot\right\|_{\mathcal{Q}}$ restricted to $\mathcal{Q}\left({}^{n}E;F\right)$ is a (quasi-) norm, for all $n\in\mathbb{N}$ and all Banach spaces $E$ and $F$;

**(p2): **

$\left\|I_{n}:\mathbb{K}\rightarrow\mathbb{K},I_{n}\left(\lambda\right)=\lambda^{n}\right\|_{\mathcal{Q}}=1$, for all $n\in\mathbb{N}$;

**(p3): **

The hyper-ideal inequality: if $Q\in\mathcal{P}\left({}^{m}G;E\right),P\in\mathcal{Q}\left({}^{n}E;F\right)$ and $t\in\mathcal{L}\left(F;H\right)$, then

[TABLE]

then $\left(\mathcal{Q},\left\|\cdot\right\|_{\mathcal{Q}}\right)$ is called a (quasi-) normed polynomial $\left(C_{j}\right)_{j=1}^{\infty}$-hyper-ideal. When all the components $\mathcal{Q}\left({}^{n}E;F\right)$ are complete under the (quasi-) norm $\left\|\cdot\right\|_{\mathcal{Q}}$ above, then $\left(\mathcal{Q},\left\|\cdot\right\|_{\mathcal{Q}}\right)$ is called a (quasi-) Banach polynomial $\left(C_{j}\right)_{j=1}^{\infty}$-hyper-ideal.

When $C_{j}=1$ for every $j\in\mathbb{N}$, we simply say that $\mathcal{Q}$ is a (quasi-) normed/(quasi-) Banach polynomial hyper-ideal. When the hyper-ideal property (and inequality) holds for every $n\in\mathbb{N}$, but only for $m=1$, we say that $\mathcal{Q}$ is a *(quasi-) normed/(quasi-) Banach polynomial ideal *(remember that $C_{1}=1$).

Definition 5.3** (Polynomial two-sided ideal [12]).**

A polynomial two-sided ideal is a subclass $\mathcal{Q}$ of the class of all continuous homogeneous polynomials between Banach spaces such that for all $n\in\mathbb{N}$ and all Banach spaces $E$ and $F$, the components

[TABLE]

satisfy:

**(ts-a): **

$\mathcal{Q}\left({}^{n}E;F\right)$ is a linear subspace of $\mathcal{P}\left({}^{n}E;F\right)$ which contains the $n$-homogeneous polynomials of finite type;

**(ts-b): **

The two-sided ideal property: given $m,n,r\in\mathbb{N}$ and Banach spaces $E,F,G$ and $H$, if $Q\in\mathcal{P}\left({}^{m}G;E\right),P\in\mathcal{Q}\left({}^{n}E;F\right)$ and $R\in\mathcal{P}\left({}^{r}F;H\right)$, then

[TABLE]

If there exist a map $\left\|\cdot\right\|_{\mathcal{Q}}:\mathcal{Q}\rightarrow[0,\infty)$ and a sequence $\left(C_{j},K_{j}\right)_{j=1}^{\infty}$ of pairs of real numbers with $C_{j},K_{j}\geq 1$ for every $j\in\mathbb{N}$ and $C_{1}=K_{1}=1$, such that:

**(ts-1): **

$\left\|\cdot\right\|_{\mathcal{Q}}$ restricted to $\mathcal{Q}\left({}^{n}E;F\right)$ is a (quasi-) norm, for all $n\in\mathbb{N}$ and all Banach spaces $E$ and $F$;

**(ts-2): **

$\left\|I_{n}:\mathbb{K}\rightarrow\mathbb{K},I_{n}\left(\lambda\right)=\lambda^{n}\right\|_{\mathcal{Q}}=1$, for all $n\in\mathbb{N}$;

**(ts-3): **

The two-sided ideal inequality: if $Q\in\mathcal{P}\left({}^{m}G;E\right),P\in\mathcal{Q}\left({}^{n}E;F\right)$ and $R\in\mathcal{P}\left({}^{r}F;H\right)$, then

[TABLE]

then $\left(\mathcal{Q},\left\|\cdot\right\|_{\mathcal{Q}}\right)$ is called a (quasi-) normed polynomial $\left(C_{j},K_{j}\right)_{j=1}^{\infty}$-two-sided ideal. When all the components $\mathcal{Q}\left({}^{n}E;F\right)$ are complete under the (quasi-) norm $\left\|\cdot\right\|_{\mathcal{Q}}$ above, then $\left(\mathcal{Q},\left\|\cdot\right\|_{\mathcal{Q}}\right)$ is called a (quasi-) Banach polynomial $\left(C_{j},K_{j}\right)_{j=1}^{\infty}$-two-sided ideal.

When $C_{j}=K_{j}=1$ for every $j\in\mathbb{N}$, we simply say that $\mathcal{Q}$ is a (quasi-) normed/(quasi-) Banach polynomial two-sided ideal.

Remark 5.4**.**

The condition $C_{1}=K_{1}=1$ guarantees that every (normed, quasi-normed, Banach, quasi-Banach) polynomial $\left(C_{j},K_{j}\right)_{j=1}^{\infty}$-two-sided ideal is a (normed, quasi-normed, Banach, quasi-Banach) polynomial $\left(C_{j}\right)_{j=1}^{\infty}$-hyper-ideal; and that, as we mentioned before, every (normed, quasi-normed, Banach, quasi-Banach) polynomial $\left(C_{j}\right)_{j=1}^{\infty}$-hyper-ideal is a (normed, quasi-normed, Banach, quasi-Banach) polynomial ideal.

Next we extend the above notions to the multipolynomials.

Definition 5.5** (Multipolynomial hyper-ideal).**

A hyper-ideal of multipolynomials (or multipolynomial hyper-ideals) is a subclass $\mathfrak{H}$ of the class of all continuous multipolynomials between Banach spaces such that for all $n\in\mathbb{N}$, multi-degree $\left(k_{1},\ldots,k_{n}\right)\in\mathbb{N}^{n}$ and all Banach spaces $E_{1},\ldots,E_{n}$ and $F$, the components

[TABLE]

satisfy:

**(Ha): **

$\mathfrak{H}_{n}^{\left(k_{1},\ldots,k_{n}\right)}\left({}^{k_{1}}E_{1},\ldots,^{k_{n}}E_{n};F\right)$ is a linear subspace of $\mathcal{P}\left({}^{k_{1}}E_{1},\ldots,^{k_{n}}E_{n};F\right)$ which contains the $\left(k_{1},\ldots,k_{n}\right)$-homogeneous polynomials of finite type ;

**(Hb): **

The hyper-ideal property: given natural numbers $n,~{}~{}$ $1\leq m_{1}<\cdots<m_{n}$ and $r_{1},\ldots,r_{m_{n}},k_{1},\ldots,k_{n}$ and $r$ and Banach spaces $G_{1},\ldots,G_{m_{n}},E_{1},\ldots,E_{n},F$ and $H$, if Q_{1}\in\mathcal{P}\left({}^{r_{1}}G_{1},\ldots,^{r_{m_{1}}}G_{m_{1}};E_{1}\right),\ldots,Q_{n}\in\mathcal{P}\left({}^{r_{m_{n-1}+1}}G_{m_{n-1}+1},\ldots,^{r_{m_{n}}}G_{m_{n}};E_{n}\right),$$P\in\mathfrak{H}_{n}^{\left(k_{1},\ldots,k_{n}\right)}\left({}^{k_{1}}E_{1},\ldots,^{k_{n}}E_{n};F\right) and $R\in\mathcal{P}\left({}^{r}F;H\right)$, then

[TABLE]

If there exist a map $\left\|\cdot\right\|_{\mathfrak{H}}:\mathfrak{H}\rightarrow[0,\infty)$ and a sequence $\left(C_{j},K_{j}\right)_{j=1}^{\infty}$ of pairs of real numbers with $C_{j},K_{j}\geq 1$ for every $j\in\mathbb{N}$ and $C_{1}=K_{1}=1$, such that:

**(H1): **

$\left\|\cdot\right\|_{\mathfrak{H}}$ restricted to $\mathfrak{H}_{n}^{\left(k_{1},\ldots,k_{n}\right)}\left({}^{k_{1}}E_{1},\ldots,^{k_{n}}E_{n};F\right)$ is a (quasi-) norm, for all $n\in\mathbb{N}$, multi-degree $\left(k_{1},\ldots,k_{n}\right)\in\mathbb{N}^{n}$ and all Banach spaces $E_{1},\ldots,E_{n}$ and $F$;

**(H2): **

$\left\|I_{n}^{\left(k_{1},\ldots,k_{n}\right)}:\mathbb{K}^{n}\rightarrow\mathbb{K},I_{n}^{\left(k_{1},\ldots,k_{n}\right)}\left(\lambda_{1},\ldots,\lambda_{n}\right)=\lambda_{1}^{k_{1}}\cdots\lambda_{n}^{k_{n}}\right\|_{\mathfrak{H}}=1$, for all $n\in\mathbb{N}$ and $\left(k_{1},\ldots,k_{n}\right)\in\mathbb{N}^{n}$;

**(H3): **

The hyper-ideal inequality: if Q_{1}\in\mathcal{P}\left({}^{r_{1}}G_{1},\ldots,^{r_{m_{1}}}G_{m_{1}};E_{1}\right),\ldots,$$Q_{n}\in\mathcal{P}\left({}^{r_{m_{n-1}+1}}G_{m_{n-1}+1},\ldots,^{r_{m_{n}}}G_{m_{n}};E_{n}\right),P\in\mathfrak{H}_{n}^{\left(k_{1},\ldots,k_{n}\right)}\left({}^{k_{1}}E_{1},\ldots,^{k_{n}}E_{n};F\right) and $R\in\mathcal{P}\left({}^{r}F;H\right)$, then

[TABLE]

then $\left(\mathfrak{H},\left\|\cdot\right\|_{\mathfrak{H}}\right)$ is called a (quasi-) normed multipolynomial $\left(C_{j},K_{j}\right)_{j=1}^{\infty}$-*hyper-ideal. *When all the components $\mathfrak{H}_{n}^{\left(k_{1},\ldots,k_{n}\right)}\left({}^{k_{1}}E_{1},\ldots,^{k_{n}}E_{n};F\right)$ are complete under the(quasi-) norm $\left\|\cdot\right\|_{\mathfrak{H}}$ above, then $\left(\mathfrak{H},\left\|\cdot\right\|_{\mathfrak{H}}\right)$ is called a (quasi-) Banach multipolynomial $\left(C_{j},K_{j}\right)_{j=1}^{\infty}$-hyper-ideal.

When $C_{j}=K_{j}=1$ for every $j\in\mathbb{N}$, we simply say that $\mathfrak{H}$ is a (quasi-) normed/(quasi-) Banach multipolynomial hyper-ideal.

Remark 5.6**.**

Our multipolynomial hyper-ideals definition is more general and unifying in the sense that it recovers the multilinear and polynomial cases. Indeed, setting $n=1=m_{1}$ we get Definition 5.3 if $r>1$ and Definition 5.2 if $r=1$. In the other end, setting $k_{1}=\cdots=k_{n}=r_{1}=\cdots=r_{m_{n}}=r=1$ we recover Definition 5.1. Finally, it is plain that every (normed, quasi-normed, Banach, quasi-Banach) multipolynomial hyper-ideal is a (normed, quasi-normed, Banach, quasi-Banach) multipolynomial ideal.

6. A Bohnenblust–Hille inequality for multipolynomials

In this section we present a unified version to the Bohnenblust–Hille inequalities [7] for homogeneous polynomials and for multilinear forms. The theory of Bohnenblust–Hille inequalities has been exhaustively investigated in recent years (see, for instance [2, 3, 15, 16, 19, 22, 23, 24, 30], and the references therein).

Let $\alpha=(\alpha_{j})_{j=1}^{\infty}$ be a sequence in $\mathbb{N}\cup\{0\}$ and, as usual, define $|\alpha|=\sum\limits_{j=1}^{\infty}\alpha_{j};$ in this case we also denote $\mathbf{x}^{\alpha}:=\prod\nolimits_{j}x_{j}^{\alpha_{j}}.$ An $m$-homogeneous polynomial $P:c_{0}\rightarrow\mathbb{K}$ $\left(\mathbb{K=R}\text{ or }\mathbb{C}\right)$ is denoted by

[TABLE]

We recall that the norm of $P$ is given by $\|P\|:=\sup_{\left\|x\right\|\leq 1}|P(x)|$.

The Bohnenblust–Hille inequality for homogeneous polynomials [7] asserts that

Theorem 6.1** (Polynomial Bohnenblust–Hille inequality).**

Let $m$ be a positive fixed integer. The following assertions are equivalent:

**(i): **

There exists a constant $C_{\mathbb{K},m}\geq 1$ such that

[TABLE]

for all continuous $m$-homogeneous polynomial $P:c_{0}\rightarrow\mathbb{K}$;

**(ii): **

[TABLE]

We also have the Bohnenblust–Hille inequality for multilinear forms [7]:

Theorem 6.2** (Multilinear Bohnenblust–Hille inequality).**

Let $m$ be a positive fixed integer. The following assertions are equivalent:

**(i): **

There exists a constant $C_{\mathbb{K},m}\geq 1$ such that

[TABLE]

for all continuous $m$-linear forms $T:c_{0}\times\cdots\times c_{0}\rightarrow\mathbb{K}$;

**(ii): **

[TABLE]

Our next step is to unify these results above. We shall observe that a $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomial $P:c_{0}\times\cdots\times c_{0}\rightarrow\mathbb{K}$ can be allways written as

[TABLE]

where, as we have previously defined, $\left(\alpha_{j}^{\left(i\right)}\right)_{j=1}^{\infty}$ is a sequence in $\mathbb{N\cup}\left\{0\right\}$, $\left|\alpha^{\left(i\right)}\right|=\sum\limits_{j=1}^{\infty}\alpha_{j}^{\left(i\right)}$ and $\left(x^{\left(i\right)}\right)^{\alpha^{\left(i\right)}}=\prod\nolimits_{j}\left(x_{j}^{\left(i\right)}\right)^{\alpha_{j}^{\left(i\right)}}$, for $i=1,\ldots,m$.

Next we invoke the notion of multipolynomials to unify Theorems 6.1 and 6.2. In fact, these are respectively the particular cases $m=1$ and $n_{1}=\cdots=n_{m}=1$ of the following theorem:

Theorem 6.3** (Multipolynomial Bohnenblust–Hille inequality).**

Let $n_{1},\ldots,n_{m}$ and $m$ be fixed positive integers. The following assertions are equivalent:

**(i): **

There is a constant $C_{n_{1},\ldots,n_{m}}\geq 1$ such that

[TABLE]

for all $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomial $P:c_{0}\times\cdots\times c_{0}\rightarrow\mathbb{K}$.

**(ii): **

[TABLE]

Proof.

$\left(ii\right)\Rightarrow\left(i\right)$: It suffices to prove the assertion for

[TABLE]

Let $Q:c_{0}\rightarrow\mathbb{K}$ be the $\left(n_{1}+\cdots+n_{m}\right)$-homogeneous polynomial given by

[TABLE]

where $\mathbb{N=N}_{1}\bigcup\cdots\bigcup\mathbb{N}_{m}$ is a disjoint union with $card\left(\mathbb{N}_{j}\right)=card\left(\mathbb{N}\right)$, for $j=1,\dots,m$. Note that since we are dealing with the $\sup$ norm we have

[TABLE]

and

[TABLE]

for all $p$. By the Polynomial Bohnenblust-Hille Inequality there exists a constant $C_{n_{1}+\cdots+n_{m}}\geq 1$ such that

[TABLE]

$\left(i\right)\Rightarrow\left(ii\right)$: Let

[TABLE]

be the $M$-linear operator given by the Kahane-Salem-Zygmund inequality (see, [2, Lemma 6.1]) with

[TABLE]

Define $P_{r}:\overset{m}{\overbrace{c_{0}\times\cdots\times c_{0}}}\rightarrow\mathbb{K}$ by

[TABLE]

where

[TABLE]

are disjoint unions with $card\left(\mathbb{N}_{k}^{\left(i\right)}\right)=card\left(\mathbb{N}\right)$, for $i=1,\dots,m$ and $k=1,\ldots,n_{i}$. Note that $P_{r}$ is an $\left(n_{1},\ldots,n_{m}\right)$-homogeneous polynomial and $\left\|P_{r}\right\|\leq\left\|T_{r}\right\|$. Moreover,

[TABLE]

for all $p$. Since

[TABLE]

by (i) we conclude that

[TABLE]

for all positive integers $r$. Thus

[TABLE]

and the proof is done.