---
abstract: |
    A famous result due to Grothendieck asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $(1,1)$-summing. If $n\geq2,$ however, it is very simple to prove that every continuous $n$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left(  1;1,...,1\right)  $-summing, and even absolutely $\left(  \frac{2}{n};1,...,1\right)  $-summing$.$ In this note we deal with the following problem:

    Given a positive integer $n\geq2$, what is the best constant $g_{n}>0$ so that every $n$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left(  g_{n};1,...,1\right)  $-summing?

    We prove that $g_{n}\leq\frac{2}{n+1}$ and also obtain an optimal improvement of previous recent results (due to Heinz Juenk $\mathit{et}$ $\mathit{al}$, Geraldo Botelho $\mathit{et}$ $\mathit{al}$ and Dumitru Popa) on inclusion theorems for absolutely summing multilinear operators.
address: 'UFRN/CERES - Centro de Ensino Superior do Seridó, Rua Joaquim Gregório, S/N, 59300-000, Caicó- RN, Brazil'
author:
- 'A. T. Bernardino'
title: 'On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators'
---

Introduction
============

Grothendieck’s theorem for absolutely summing operators asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $(1;1)$-summing (and hence absolutely $(p;p)$-summing for every $p\geq1$). For the linear theory of absolutely summing operators we refer to [@df; @djt] (see also [@bpr; @ku2; @seo] for recent developments).

In the multilinear setting, D. Pérez-García, in his PhD thesis [@dav] (see also [@bom] and [@botp] for a different proof), proved that every continuous $n$-linear operator from $\ell_{1}\times\cdots\times
\ell_{1}$ to $\ell_{2}$ is multiple $(1;1,...,1)$-summing (in fact, multiple $(p;p,...,p)$-summing for every $1\leq p\leq2$)$.$ This result can be regarded as the multilinear version of Grothendieck’s theorem.

Let us recall the notions.

The letters $X_{1},...,X_{n},X,Y$ will always denote Banach spaces over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $X^{\ast}$ represents the topological dual of $X$.

For any $s>0,$ we denote the conjugate of $s$ by $s^{\ast}$. Given a positive integer $n$, the space of all continuous $n$-linear operators from $X_{1}\times\cdots\times X_{n}$ to $Y$  endowed with the $\sup$ norm is denoted by $\mathcal{L}(X_{1},...,X_{n};Y).$ For $p>0$, the vector space of all sequences $\left(  x_{j}\right)  _{j=1}^{\infty}$ in $X$ such that$$\left\Vert \left(  x_{j}\right)  _{j=1}^{\infty}\right\Vert _{p}=\left(
\sum_{j=1}^{\infty}\left\Vert x_{j}\right\Vert ^{p}\right)  ^{\frac{1}{p}}<\infty$$ is denoted by $\ell_{p}\left(  X\right)  .$ We represent by $\ell_{p}^{w}\left(  X\right)  $ the linear space of the sequences $\left(
x_{j}\right)  _{j=1}^{\infty}$ in $X$ such that $\left(  \varphi\left(
x_{j}\right)  \right)  _{j=1}^{\infty}\in\ell_{p}\left(  \mathbb{K}\right)  $ for every $\varphi\in X^{\ast}$.

If $0<p,q_{1},...,q_{n}<\infty$ and $\frac{1}{p}\leq\frac{1}{q_{1}}+\cdots+\frac{1}{q_{n}},$ a multilinear operator $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is absolutely ** $(p;q_{1},...,q_{n})$-summing if $(T(x_{j}^{(1)},...,x_{j}^{(n)}))_{j=1}^{\infty}\in\ell_{p}(Y)$ for every $(x_{j}^{(k)})_{j=1}^{\infty}\in\ell_{q_{k}}^{w}(X_{k}),k=1,...,n.$ In this case we write $T\in\Pi_{\left(  p;q_{1},...,q_{n}\right)  }^{n}\left(
X_{1},...,X_{n};Y\right)  $. For details we refer to [@am].

When $1\leq q_{1},...,q_{n}\leq p<\infty$ a multilinear operator $T\in\mathcal{L}(X_{1},...,X_{n};Y)$ is multiple ** $(p;q_{1},...,q_{n})$-summing if $(T(x_{j_{1}}^{(1)},...,x_{j_{n}}^{(n)}))_{j_{1},..,j_{n}=1}^{\infty}\in\ell_{p}(Y)$ for every $(x_{j}^{(k)})_{j=1}^{\infty}\in
\ell_{q_{k}}^{w}(X_{k}),k=1,...,n.$ In this case we write $T\in\Pi_{m\left(
p;q_{1},...,q_{n}\right)  }^{n}\left(  X_{1},...,X_{n};Y\right)  $. For details we mention [@bom; @collec] and for recent developments and applications related to the multilinear and polynomial theory we refer to [@ag; @bbb; @bh; @an; @dgm; @se; @lit; @mat; @ppp] and references therein. For $n=1$ we write $\Pi$ instead of $\Pi^{1}$ and we recover the classical theory of absolutely summing linear operators.

For $1\leq q_{1},...,q_{n}\leq p<\infty,$ the inclusion $$\Pi_{m\left(  p;q_{1},...,q_{n}\right)  }^{n}\left(  X_{1},...,X_{n};Y\right)
\subseteqq\Pi_{\left(  p;q_{1},...,q_{n}\right)  }^{n}\left(  X_{1},...,X_{n};Y\right)$$ is obvious. So, the following coincidence result is an immediate consequence of Pérez-García multilinear version of Grothendieck’s theorem:

For every positive integer $n$, $$\Pi_{\left(  1;1,...,1\right)  }^{n}\left(  \ell_{1},...,\ell_{1};\ell
_{2}\right)  =\mathcal{L}\left(  \ell_{1},...,\ell_{1};\ell_{2}\right)  .$$

However, using that $\ell_{1}$ has cotype $2$ it is easy to prove that the above result is far from being optimal. In fact, we have the following improvement (see [@irish; @irishd]):

For every positive integer $n\geq2$, $$\Pi_{\left(  \frac{2}{n};1,...,1\right)  }^{n}\left(  \ell_{1},...,\ell
_{1};\ell_{2}\right)  =\mathcal{L}\left(  \ell_{1},...,\ell_{1};\ell
_{2}\right)  . \label{dda}$$

So, the following problem is quite natural:

\[xc\]Given a positive integer $n\geq2$, what is the best constant $g_{n}>0$ so that$$\Pi_{\left(  g_{n};1,...,1\right)  }^{n}\left(  \ell_{1},...,\ell_{1};\ell
_{2}\right)  =\mathcal{L}\left(  \ell_{1},...,\ell_{1};\ell_{2}\right)  ?$$

If we test $n=1$ in (\[dda\]) we obtain $$\Pi_{(2;1)}\left(  \ell_{1};\ell_{2}\right)  =\mathcal{L}\left(  \ell_{1};\ell_{2}\right)$$ which is not surprising at all, in view of Grothendieck’s Theorem. So, in some sense, we feel that the estimate $g_{n}\leq\frac{2}{n}$ for $n\geq2$ is probably not optimal. The optimistic reader will probably hope for an estimate for $g_{n}$ so that in the case $n=1$ we recover Grothendieck’s Theorem. Fortunately, in the last section we will precisely obtain such an estimate.

The problem of estimating $g_{n}$ is related to the generalization of certain results involving cotype and absolutely summing multilinear operators. The following result is a combination of [@junek Theorem $3$ and Remark $2$], [@pop Corollary $4.6$] and [@michels Theorem $3.8$ (ii)]:

\[Inclusion Theorem\]\[tt\] Let $X_{1},...,X_{n}$ be Banach spaces with cotype $s$ and $n\geq2$ be a positive integer:

\(i) If $s=$ $2,$ then $$\Pi_{\left(  q;q,...,q\right)  }^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{\left(
p;p,...,p\right)  }^{n}(X_{1},...,X_{n};Y) \label{uu}$$ holds true for $1\leq p\leq q\leq2$ and every $Y$.

\(ii) If $s>2,$ then $$\Pi_{\left(  q;q,...,q\right)  }^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{\left(
p;p,...,p\right)  }^{n}(X_{1},...,X_{n};Y) \label{vv}$$ holds true for $1\leq p\leq q<s^{\ast}$ and every $Y$.

The results above are clearly not always optimal since, for example,$$\Pi_{\left(  2;2,2,2\right)  }^{3}(\ell_{2},\ell_{2},\ell_{2};\mathbb{K})\neq\mathcal{L}(\ell_{2},\ell_{2},\ell_{2};\mathbb{K})=\Pi_{\left(  \frac
{2}{3};1,1,1\right)  }^{3}(\ell_{2},\ell_{2},\ell_{2};\mathbb{K}).$$

So, another natural problem is:

\[xz\] Given $1\leq p\leq q<\infty$ and a positive integer $n\geq2$, what are the optimal $\alpha:=\alpha_{p,q,n}>0$ so that, under the same circumstances of (\[uu\]) and (\[vv\]), we have$$\Pi_{\left(  q;q,...,q\right)  }^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{\left(
\alpha;p,...,p\right)  }^{n}(X_{1},...,X_{n};Y) \label{ii}$$ for all Banach spaces $X_{1},...,X_{n},Y$ $?$

In this direction we extend Theorem \[tt\] and also recent results from [@4; @3] by showing that $$\alpha\leq\frac{qp}{n\left(  q-p\right)  +p}$$ and, in some sense, this constant is optimal, since for this value of $\alpha$ we have an equality in (\[ii\]).

An estimate for $\alpha$
========================

Let $1\leq k\leq n,$ where $n\geq2$ is a positive integer. If $X_{i}$ has cotype $s_{i}\geq2,i=1,...,k$ and $$1\leq p\leq q<\min_{1\leq i\leq k}s_{i}^{\ast}\text{ if }s_{i}>2\text{ for
some }i=1,...,k$$ or $$1\leq p\leq q\leq2\text{ if }s_{i}=2\text{ for all }i=1,...,k,$$ then $$\Pi_{(z;q,...,q,t,...,t)}^{n}(X_{1},...,X_{n};Y)=\Pi_{(\frac{zqp}{zk\left(
q-p\right)  +qp};p,...,p,t,...,t)}^{n}(X_{1},...,X_{n};Y),$$ for all $X_{k+1},...,X_{n},Y$ and all $z,t\geq1$ (here $q$ and $p$ are repeated $k$ times). In particular, if $k=n$,$$\Pi_{(z;q,...,q)}^{n}(X_{1},...,X_{n};Y)=\Pi_{(\frac{zqp}{zk\left(
q-p\right)  +qp};p,...,p)}^{n}(X_{1},...,X_{n};Y)$$

Since $X_{i}$ has finite cotype $s_{i}\geq2,i=1,...,k$, then we have$$\ell_{p}^{w}(X_{i})=\ell_{qp/\left(  q-p\right)  }\ell_{q}^{w}(X_{i})$$

for all $i=1,...,k$ with $$1\leq p\leq q<\min_{1\leq i\leq k}s_{i}^{\ast}\text{ if }s_{i}>2\text{ for
some }i=1,...,k$$ or $$1\leq p\leq q\leq2\text{ if }s_{i}=2\text{ for all }i=1,...,k.$$

Let $(x_{j}^{(i)})_{j=1}^{\infty}\in\ell_{p}^{w}(X_{i}),i=1,...,k$ and $(x_{j}^{(i)})_{j=1}^{\infty}\in\ell_{t}^{w}(X_{i})$ for $i=k+1,...,n$. So $x_{j}^{(i)}=\alpha_{j}^{\left(  i\right)  }y_{j}^{\left(  i\right)  },$ with $\left(  \alpha_{j}^{\left(  i\right)  }\right)  _{j=1}^{\infty}\in
\ell_{qp/\left(  q-p\right)  }$ and $\left(  y_{j}^{\left(  i\right)
}\right)  _{j=1}^{\infty}\in\ell_{q}^{w}\left(  X_{i}\right)  ,$ for all $j$ and $i=1,...,k$. If $A\in\Pi_{(z;q...,q,t,...,t)}^{n}(X_{1},...,X_{n};Y)$, then$$\begin{aligned}
&  \left(  \sum_{j=1}^{\infty}\left\Vert A\left(  x_{j}^{(1)},...,x_{j}^{\left(  n\right)  }\right)  \right\Vert ^{\frac{zqp}{zk\left(  q-p\right)
+qp}}\right)  ^{\frac{zk\left(  q-p\right)  +qp}{zqp}}\\
&  =\left(  \sum_{j=1}^{\infty}\left(  \left\vert \alpha_{j}^{\left(
1\right)  }\cdots\alpha_{j}^{\left(  k\right)  }\right\vert \left\Vert
A\left(  y_{j}^{(1)},...,y_{j}^{\left(  k\right)  },x_{j}^{(k+1)},...,x_{j}^{\left(  n\right)  }\right)  \right\Vert \right)  ^{\frac
{zqp}{zk\left(  q-p\right)  +qp}}\right)  ^{\frac{zk\left(  q-p\right)
+qp}{zqp}}\\
&  \leq\left(  \sum_{j=1}^{\infty}\left\Vert A\left(  y_{j}^{(1)},...,y_{j}^{\left(  k\right)  },x_{j}^{(k+1)},...,x_{j}^{\left(  n\right)
}\right)  \right\Vert ^{z}\right)  ^{\frac{1}{z}}\left(  \sum_{j=1}^{\infty
}\left\vert \alpha_{j}^{\left(  1\right)  }\cdots\alpha_{j}^{\left(  k\right)
}\right\vert ^{\frac{qp}{k\left(  q-p\right)  }}\right)  ^{k\left(  \frac
{q-p}{qp}\right)  }\\
&  \leq\left(  \sum_{j=1}^{\infty}\left\Vert A\left(  y_{j}^{(1)},...,y_{j}^{\left(  k\right)  },x_{j}^{(k+1)},...,x_{j}^{\left(  n\right)
}\right)  \right\Vert ^{z}\right)  ^{\frac{1}{z}}{\displaystyle\prod\limits_{i=1}^{k}}
\left(  \sum_{j=1}^{\infty}\left\vert \alpha_{j}^{\left(  i\right)
}\right\vert ^{\frac{qp}{\left(  q-p\right)  }}\right)  ^{\frac{q-p}{qp}}<\infty\end{aligned}$$ and we conclude that$$\Pi_{(z;q,...,q,t,...,t)}^{n}(X_{1},...,X_{n};Y)\subseteqq\Pi_{(\frac
{zqp}{zk\left(  q-p\right)  +qp};p,...,p,t,...,t)}^{n}(X_{1},...,X_{n};Y).$$ The other inclusion is a consequence of the inclusion theorem for absolutely summing multilinear operators.

A similar result holds if $X_{j_{1}},...,X_{j_{k}},\left\{  j_{1},...,j_{k}\right\}  \subseteqq\left\{  1,...,n\right\}  $ (instead of $X_{1},...,X_{k}$) have cotype $s_{j_{i}}\geq2,i=1,...,k.$

The following immediate corollary is an optimal (in the sense that we have an equality instead of an inclusion) generalization of Theorem \[tt\]:

\[ut\]If $n\geq2$ and $X_{1},...,X_{n}$ have finite cotype $s$ and $$1\leq p\leq q<s^{\ast}\text{ if }s>2$$ or$$1\leq p\leq q\leq2\text{ if }s=2,$$ then$$\Pi_{(q;q,...,q)}^{n}(X_{1},...,X_{n};Y)=\Pi_{(\frac{qp}{n\left(  q-p\right)
+p};p,...,p)}^{n}(X_{1},...,X_{n};Y)$$ for every Banach space $Y$ and$$\alpha\leq\frac{qp}{n\left(  q-p\right)  +p}.$$

The above results were independently proved in [@Bla].

An estimate for $g_{n}$
=======================

From Corollary \[ut\] we know that $$\Pi_{(2;2,...,2)}^{n}\left(  \ell_{1},...,\ell_{1};\ell_{2}\right)
=\Pi_{(\frac{2}{n+1};1,...,1)}^{n}\left(  \ell_{1},...,\ell_{1};\ell
_{2}\right)$$ for all $n\geq2$. But, since$$\mathcal{L}\left(  \ell_{1},...,\ell_{1};\ell_{2}\right)  =\Pi_{m(2;2,...,2)}^{n}\left(  \ell_{1},...,\ell_{1};\ell_{2}\right)  \subseteqq\Pi
_{(2;2,...,2)}^{n}\left(  \ell_{1},...,\ell_{1};\ell_{2}\right)$$ it readily follows that $$\Pi_{(\frac{2}{n+1};1,...,1)}^{n}\left(  \ell_{1},...,\ell_{1};\ell
_{2}\right)  =\mathcal{L}\left(  \ell_{1},...,\ell_{1};\ell_{2}\right)$$ for all $n\geq2$. So we have:

If $n\geq2$, then$$g_{n}\leq\frac{2}{n+1}.$$

Note that Grothendieck’s Theorem asserts that $g_{1}=1$ and $1=\frac{2}{1+1};$ hence we conjecture that $\frac{2}{n+1}$ is in fact the optimal estimate for $g_{n}$.

This paper is a part of the doctoral thesis of the author which is being written under supervision of Prof. Daniel Pellegrino. The author thanks Prof. Pellegrino for introducing him to the subject and the main problem from this note and also for several suggestions and important insights.

[99]{}

R. Alencar and M. C. Matos, Some classes of multilinear mappings between Banach spaces, Publicaciones del Departamento de Análisis Matemático 12, Universidad Complutense Madrid, (1989).

R. Aron and J. Globevnik, Analytic Functions on $c_{0}$, Revista Matematica Universidad Complutense de Madrid **2** (1989), 27-33.

G. Badea, On nuclear and multiple summing bilinear operators, Quaest. Math. **33** (2010), 253–261.

A. T. Bernardino, Remarks on cotype and absolutely summing multilinear operators, to appear in Cubo A Mathematical Journal (and arXiv:1101.5119v1).

A. T. Bernardino and D. Pellegrino, Some remarks on absolutely summing multilinear mappings, arXiv:1101.2119v3.

O. Blasco, G. Botelho, D. Pellegrino and P. Rueda, Lifting coincidence results to multilinear operators, preprint.

H. F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) **32** (1931), 600-622.

F. Bombal, D. Pérez–García, and I. Villanueva, Multilinear extensions of Grothendieck’s theorem, Quart. J. Math. 55, 441–450 (2004).

G. Botelho, Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proc. Roy. Irish Acad Sect. A **97** (1997), 145-153.

G. Botelho, C. Michels and D. Pellegrino, Complex interpolation and summability properties  of multilinear operators, Rev. Matem. Complut. **23** (2010), 139-161.

G. Botelho and D. Pellegrino, When every multilinear mapping is multiple summing, Math. Nachr. **282** (2009), 1414-1422.

G. Botelho, D. Pellegrino and P. Rueda, Cotype and absolutely summing linear operators, Mathematische Zeitschrift **267** (2011), 1-7.

A. Defant, K. Floret, Tensor norms and operator ideals, North-Holland (1993).

A. Defant, L. Frerick, J. Ortega-Cerdá, M. Ounaïes and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2), to appear.

A. Defant, D. García, M. Maestre and D. Pérez-García, Bohr’s strip for vector valued Dirichlet series, Math. Ann. **342** (2008), 533-555.

A. Defant and P. Sevilla-Peris, A new multilinear insight on Littlewood’s 4/3-inequality, J. Funct. Anal. **256** (2009), 1642–1664.

J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge University Press, 1995.

H. Junek, M.C. Matos and D. Pellegrino, Inclusion theorems for absolutely summing holomorphic mappings, Proc. Amer. Math. Soc. **136** (2008), 3983-3991.

T. Kühn and M. Mastyło, Weyl numbers and eigenvalues of abstract summing operators, J. Math. Anal. Appl. **369** (2010), 408-422.

J. Littlewood, On bounded bilinear forms in an infinite number of variables, Q. J. Math. **2** (1930), 167-171.

M.C. Matos. Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collectanea Math. **54** (2003), 111-136.

M.C. Matos, Nonlinear absolutely summing mappings, Math. Nachr. **258** (2003), 71-89.

D. Pellegrino, Cotype and nonlinear absolutely summing mappings, Proceedings of the Royal Irish Academy Section A-Mathematical and Physical Sciences **105(A)** (2005), 75-91.

D. Pellegrino and J. Santos, Absolutely summing multilinear operators: a panorama, to appear in Quaest. Math.

D. Pérez-García, Operadores multilineales absolutamente sumantes, Thesis, Univ. Complutense de Madrid (2003).

D. Popa, Multilinear variants of Maurey and Pietsch theorems and applications, J. Math. Anal. Appl. **368** (2010) 157–168.

D. Puglisi and J. B. Seoane-Sepúlveda, Bounded linear non-absolutely summing operators, J. Math. Anal. Appl. **338** (2008), 292-298.
