## Abstract

A famous theorem due to Maurey and Pisier asserts that for an infinite dimensional Banach space *E*, the infumum of the *q* such that the identity map \(id_{E}\) is absolutely \(\left( q,1\right) \)-summing is precisely \(\cot E\). In the same direction, the Dvoretzky–Rogers Theorem asserts \(id_{E}\) fails to be absolutely \(\left( p,p\right) \)-summing, for all \(p\ge 1\). In this note, among other results, we unify both theorems by charactering the parameters *q* and *p* for which the identity map is absolutely \(\left( q,p\right) \)-summing. We also provide a result that we call *strings of coincidences* that characterize a family of coincidences between classes of summing operators. We illustrate the usefulness of this result by extending a classical result of Diestel, Jarchow and Tonge and the coincidence result of Kwapień.

## Introduction and Background

Let \(2\le q<\infty \). A Banach space *E* has cotype *q* (see Diestel et al. 1995, p 218) if there is a constant \(C>0\) such that, no matter how we select finitely many vectors \(x_{1},\dots ,x_{n}\in E\),

where \(r_{k}\) denotes the *k*-th Rademacher function. In this context we also define

Recall that if \(1\le p\le q\le \infty \), for Banach spaces *E*, *F*, a linear operator \(u:E\rightarrow F\) is absolutely \(\left( q,p\right) \)-summing if \(\left( u(x_{j})\right) _{j=1}^{\infty }\in \ell _{q}(F)\) whenever \(\left( x_{j}\right) _{j=1}^{\infty }\in \ell _{p}^{w}(E)\). We recall that \(\ell _{p}^{w}(E)\) is the linear space of the sequences \(\left( x_{j}\right) _{j=1}^{\infty }\) in *E* such that \(\left( \varphi \left( x_{j}\right) \right) _{j=1}^{\infty }\in \ell _{p}\) for every continuous linear functional \(\varphi :E\rightarrow \mathbb {K}\); the expression

defines a norm on \(\ell _{p}^{w}(E)\). The class of all absolutely (*q*, *p*)-summing operators from *E* to *F* will be denoted by \(\varPi _{(q,p)}(E;F)\). We denote \(\varPi _{(p,p)}(E;F)\) by \(\varPi _{p}(E;F)\). From now on, for any \(p>1\) the symbol \(p^{*}\) denotes the conjugate of *p*, i.e., \(p^{*}=p/\left( p-1\right) .\) Cotype and absolutely summing operators are closely related by the famous Maurey–Pisier Theorem (see also Botelho et al. 2001 for further relations between cotype and absolutely summing operators):

### Theorem 1

(Maurey–Pisier) For every infinite dimensional Banach space *E*, we have \(\cot E=\inf \){\(a:id_{E}\) is absolutely \(\left( a,1\right) \)-summing}.

In the same direction, the Dvoretzky–Rogers Theorem Dvoretzky and Rogers (1950) tells us that \(id_{E}\) is not absolutely (*p*, *p*)-summing, regardless of the \(p\ge 1.\) In a more general version, as stated in Diestel et al. (1995, Theorem 10.5), it reads as follows:

### Theorem 2

(Dvoretzky–Rogers) If *E* is an infinite dimensional Banach space, the identity map \(id_{E}\) is not absolutely (*q*, *p*)-summing whenever

Our first main result revisits and unifies both theorems. Our second main result provides a family of coincidences for the classes of absolutely summing operators that encompasses the following classical results:

### Theorem 3

(Diestel–Jarchow–Tonge) Let *E* and *F* be Banach spaces.

- (a)
If

*E*has cotype 2, then \(\varPi _{2 }\left( E;F\right) =\varPi _{1}\left( E;F\right) \). - (b)
If

*E*has cotype \(2<q<\infty \), then \(\varPi _{r }\left( E;F\right) =\varPi _{1 }\left( E;F\right) \) for all \(1<r<q^*\).

### Theorem 4

(Kwapień (1968)) Let \(1\le p\le \infty .\) Then

Moreover, if \(r<r(p)\), then \(\varPi _{\left( r,1\right) }\left( \ell _1;\ell _p\right) \ne \mathcal {L}\left( \ell _1;\ell _p\right) \).

## Main Results

The next definition was introduced by Matos (2003) and, independently, by Pérez-García (2003).

### Definition 1

Let \(1\le p\le q\le \infty \). A multilinear operator \(T:E_{1}\times \cdots \times E_{m}\rightarrow F\) is multiple (*q*, *p*)-summing if there exist a constant \(C>0\) such that

for all \((x_{j}^{(k)})_{j=1}^{\infty }\in \ell _{p}^{w}\left( E_{k}\right) \), with \(k=1,\ldots ,m\). We represent the class of all multiple (*q*, *p*)-summing operators by \(\varPi _{(q,p)}^{m}\left( E_{1},\dots ,E_{m};F\right) \). When \(E_{1}=\cdots =E_{m}=E\), we denote simply by \(\varPi _{(q,p)}^{m}\left( ^{m}E;F\right) \).

Following Arregui and Blasco (2002), when we write \((x_{j})_{j=1}^{\infty }\in \ell _{r}\ell _{s}^{\omega }(E)\), it means that there are \(\left( a_{j}\right) _{j=1}^{\infty }\in \ell _{r}\) and \(\left( z_{j}\right) _{j=1}^{\infty }\in \ell _{s}^{w}(E)\) such that \(x_{j}=a_{j}z_{j}\) for all *j*.

The following result of Arregui and Blasco (2002, Lemma 3 and Proposition 6) is crucial for us:

### Proposition 1

Let \(1<r<\infty \). Then \(\ell ^{\omega }_{1}(E)=\ell _{r}\ell ^{\omega }_{r^{*}}(E)\) if and only if \(\mathcal {L}\left( c_{0};E\right) =\varPi _{r}\left( c_{0};E\right) \). In particular,

- (a)
If

*E*has cotype 2, then \(\ell _{1}^{\omega }(E)=\ell _{2}\ell _{2}^{\omega }(E)\). - (b)
If

*E*has cotype \(q>2\), then \(\ell _{1}^{\omega }(E)=\ell _{r}\ell _{r^{*}}^{\omega }(E)\) for any \(r>q\).

Note that from the previous result it is immediate that if *E* has cotype 2 then \(\ell _{1}^{\omega }(E)=\ell _{r} \ell ^{\omega }_{r^{*}}(E)\) for any \(r\ge 2\).

Let us present now an inclusion theorem for multiple summing multilinear operators that, when restricted to the linear case, will be important in the proof of our main result. Before that, let us to recall the classical inclusion theorem for absolutely summing operators Diestel et al. (1995, Theorem 10.4), which will be very useful to our propose.

### Theorem 5

If \(1\le a_{j}\le b_{j}<\infty \) for \(j=1,2\), \(a_1\le a_2\), \(b_1\le b_2\) and

then every absolutely \(\left( a_{1},b_{1}\right) \)-summing operator is also absolutely \(\left( a_{2},b_{2}\right) \)-summing.

### Lemma 1

Let \(E_{1},\ldots ,E_{m}\) and *F* be infinite dimensional Banach spaces.

- (a)
If \(E_{1},\ldots ,E_{m}\) has cotype 2 and \(r\le 2\), then

$$\begin{aligned} \varPi _{\left( a,r\right) }\left( E_{1},\ldots , E_{m};F\right) \subset \varPi _{\left( s,1\right) }\left( E_{1},\ldots ,E_{m};F\right) \ \text {whenever} \ \frac{1}{r}-\frac{1}{a}\le 1-\frac{1}{s}. \end{aligned}$$ - (b)
If \(E_{i}\) has cotype \(q_{i}>2\), \(q=\max \{q_{i}:i=1,\ldots ,m\}\) and \(1<r<q^{*}\), then

$$\begin{aligned} \varPi _{\left( a,r\right) }\left( E_{1},\ldots , E_{m};F\right) \subset \varPi _{\left( s,1\right) }\left( E_{1},\ldots , E_{m};F\right) \ \text {whenever} \ \frac{1}{r}-\frac{1}{a}\le 1-\frac{1}{s}. \end{aligned}$$

### Proof

(a) If \(r\le 2\), then \(r^*\ge 2\) and, from Theorem 5, \(\varPi _2(E;F)\subseteq \varPi _{r^*}(E;F)\) for all Banach spaces *E* and *F*. Thus, since \(E_i\) has cotype 2, it follows that \(\mathcal {L}(c_0;E_i)=\varPi _2(c_0;E_i)\subseteq \varPi _{r^*}(c_0;E_i)\), that is, \(\mathcal {L}(c_0;E_i)=\varPi _{r^*}(c_0;E_i)\). From Proposition 1 we thus have \(\ell _1^w(E_i)=\ell _{r^*}\ell _r^w(E_i)\). Then, if \((x_{j}^{(i)})_{j=1}^\infty \in \ell _{1}^{w}(E_{i})\),

with \(( a_{j}^{(i)}) _{j}\in \ell _{r^{*}}\) and \(( z_{j}^{(i)})_{j}\in \ell _{r}^{w}(E_{i})\). Since

from Hölder’s inequality we have

(b) Follows as in (a) using Proposition 1(b). \(\square \)

### Remark 1

This theorem is a generalization of Popa (2009, Theorem 10).

### Theorem 6

Let \(a,b\in [1,\infty )\) and *E* be an infinite dimensional Banach space.

- (i)
If \(b\ge \left( \cot E\right) ^{*}\), then

$$\begin{aligned} \inf \left\{ a \ : \ id_{E}\text { is absolutely }(a,b){\text {-}}{\text {summing}}\right\} =\infty . \end{aligned}$$ - (ii)
If \(b<\left( \cot E\right) ^{*}\), then

$$\begin{aligned} \inf \left\{ a \ : \ id_{E} \ \text {is absolutely} \ (a,b)\text {-}{\text {summing}}\right\} =\frac{b\cot E}{b+\cot E-b\cot E}. \end{aligned}$$

### Proof

(i) Let us prove that \(id_{E}\) fails to be \(\left( a,\left( \cot E\right) ^{*}\right) \)-summing; this result seems to have been overlooked in the literature; it appears, in a more general form, in the preprint Bayart et al. (2018) and we sketch the proof below. A famous result of Maurey and Pisier asserts that \(\ell _{\cot E}\) is finitely representable in *E*, i.e., for all \(n\ge 1\), there exists \(E_{n}\subset E\) and an isomorphism \(S_{n}:\ell _{\cot E}^{n}\rightarrow E_{n}\) such that \(\Vert S_{n}\Vert \cdot \Vert S_{n}^{-1} \Vert \le 2\). There is no loss of generality in assuming \(\Vert S_{n}\Vert \le 1\). For \(1\le i\le n\), let

Note that

and thus \(\left\| y_{i}\right\| \ge 1/2.\) Moreover

We thus conclude that for any \(a\ge 1\) we have

which means that \(id_{E}\) is not absolutely \(\left( a,\left( \cot E\right) ^{*}\right) \)-summing. The proof of (i) is done.

Now let us prove (ii). Consider

If \(r<\lambda \), then

Thus, there is \(0<\varepsilon <\cot E- \frac{rb}{rb+b-r}\) such that

From Lemma 1, with \(m=1\) and \(s=\cot E-\varepsilon \), we conclude that

and thus \(id_{E}\) fails to be \(\left( r,b\right) \)-summing. So

Now, if \(r>\lambda \) then

Therefore, there exist \(\varepsilon >0\) such that

Thus \(r>\cot E+\varepsilon \) and

The inclusion theorem (Theorem 5) asserts that every absolutely \(\left( \cot E+\varepsilon ,1\right) \)-summing operator is also absolutely \(\left( r,b\right) \)-summing. Since \(id_{E}\) is absolutely \(\left( \cot E+\varepsilon ,1\right) \)-summing, it follows that \(id_{E}\) is also absolutely \(\left( r,b\right) \)-summing. Consequently,

and the proof is done. \(\square \)

### Remark 2

Note that this result completes the information of the Dvoretzky–Rogers Theorem (Theorem 2) and, when \(b=1\), we recover the classical result of Maurey and Pisier (Theorem 1).

In the previous theorem we prove that for \(b<\left( \cot E\right) ^{*}\) we can fully characterize the classes of \(\left( a,b\right) \)-summing operators that coincide. The Fig. 1 illustrates this result and motivates us to name the next result as “strings of coincidences” (see Fig. 2), which generalizes Theorem 3.

### Theorem 7

(Strings of coincidences) Let *E* and *F* be infinite dimensional Banach spaces.

- (a)
If

*E*has cotype 2, \(a\le a_{1}\), \(r\le r_{1}\le 2\) and \(\frac{1}{r}-\frac{1}{a}=\frac{1}{r_{1}}-\frac{1}{a_{1}}\), then$$\begin{aligned} \varPi _{\left( a,r\right) }\left( E;F\right) =\varPi _{\left( a_{1},r_{1}\right) }\left( E;F\right) . \end{aligned}$$ - (b)
If \(a\le a_{1}\), \(1\le r\le r_{1}<(\cot E)^{*}\) and \(\frac{1}{r}-\frac{1}{a}=\frac{1}{r_{1}}-\frac{1}{a_{1}}\), then

$$\begin{aligned} \varPi _{\left( a,r\right) }\left( E;F\right) =\varPi _{\left( a_{1},r_{1}\right) }\left( E;F\right) . \end{aligned}$$

### Proof

Note that

where

In fact, the two equalities in (2) are immediate consequences of the previous theorem and the inclusion theorem for absolutely summing operators (Theorem 5). \(\square \)

### Remark 3

The strings of coincidence can also be obtained as a consequence of Bernardino (2011, Theorem 2.1).

We finish this section by illustrating how the strings of coincidence can be useful to extend the classical result of Kwapień (Theorem 4).

### Theorem 8

Let \(1\le p\le \infty \) and \(a,b\ge 1.\) If \(b\le 2\), then \( \varPi _{\left( a,b\right) }\left( \ell _1;\ell _p\right) =\mathcal {L}\left( \ell _1;\ell _p\right) \) if, and only if,

where

### Proof

For \(b\le 2\) the string of coincidence associated to \(\left( r(p),1\right) \), where \(\frac{1}{r(p)}=1- \left| \frac{1}{p}-\frac{1}{2}\right| \), is composed by the pairs \(\left( a,b\right) \) such that

Thus

In view of the optimality of \(\frac{br(p)}{r(p)+b-br(p)}\) when \(b=1\), we can conclude that the above estimate of *a* is sharp. \(\square \)

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J. Santos was supported by Conselho Nacional de Desenvolvimento Cientfico e Tecnolgico (CNPq Grant 309466/2018-0).

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Araújo, G., Santos, J. On the Maurey–Pisier and Dvoretzky–Rogers Theorems.
*Bull Braz Math Soc, New Series* **51, **1–9 (2020). https://doi.org/10.1007/s00574-019-00140-5

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### Keywords

- Absolutely summing operators
- Maurey–Pisier theorem
- Dvoretzky–Rogers theorem

### Mathematics Subject Classification

- 46A32
- 47H60