Some Generalized Versions of Chevet–Saphar Tensor Norms

Abstract

The paper is concerned with some generalized versions $g_E$ and $w_E$ of classical tensor norms. We find a Banach space E for which $g_E$ and $w_E$ are finitely generated tensor norms, and show that $g_E$ and $w_E$ are associated with the ideals of some E-nuclear operators. We also initiate the study of some theories of our tensor norms.

1. Introduction

One of the important theories in the study of Banach spaces is the theory of tensor norms (see Section 2 for the definition of tensor norm). It provides not only new examples of Banach spaces but also a powerful tool in the study of Banach operator ideals. One may refer to [1,2,3,4,5] and the references therein for various information and content about tensor norms. Throughout this paper, Banach spaces will be denoted by X and Y over $\mathbb{R}$ or $\mathbb{C}$, with dual spaces $X^{*}$ and $Y^{*}$, and the closed unit ball of X will be denoted by $B_X$. We will denote by $X\otimes Y$ the algebraic tensor product of X and Y. The most classical two tensor norms are the injective norm $\epsilon$ and the projective norm$\pi$, which were systematically investigated by Grothendieck [6,7]. For $u\in X\otimes Y$,

$$\epsilon (u;X,Y):=sup\left\{|\sum _{n=1}^l{x}^{*}\left(x_n\right)y^{*}\left(y_n\right)|:x^{*}\in B_{X^{*}},y^{*}\in B_{Y^{*}}\right\},$$

where ${\sum }_{n=1}^l{x}_n\otimes y_n$ is any representation of u, and

$$\pi (u;X,Y):=inf\left\{\sum _{n=1}^l\parallel x_n\parallel \parallel y_n\parallel :u=\sum _{n=1}^l{x}_n\otimes y_n,l\in \mathbb{N}\right\}.$$

More recently, the author [8] introduced a tensor norm related with the injective norm. Lapresté [9] introduced the most generalized version ${\alpha }_{p,q}$ of the projective norm, and its some related topics were studied by Díaz, López-Molina, Rivera [10] and the author [11]. Many of the interesting tensor norms can be obtained from the tensor norm ${\alpha }_{p,q}$ ($1\le p$, $q\le \infty ,1/p+1/q\ge 1$), which is defined as follows. Let $1\le r\le \infty$ with $1/r=1/p+1/q-1$. For $u\in X\otimes Y$, let

$$\begin{array}{c}{\alpha }_{p,q}\left(u\right):=inf\{\parallel {\left({\lambda }_n\right)}_{n=1}^l{\parallel }_r\underset{x^{*}\in B_{X^{*}}}{sup}\parallel {\left(x^{*}\left(x_n\right)\right)}_{n=1}^l{\parallel }_{q^{*}}\underset{y^{*}\in B_{Y^{*}}}{sup}{\parallel {\left(y^{*}\left(y_n\right)\right)}_{n=1}^l\parallel }_{p^{*}}:u\hfill \\ =\sum _{n=1}^l{\lambda }_n{x}_n\otimes y_n,l\in \mathbb{N}\},\hfill \end{array}$$

where $p^{*}$ is the conjugate index of p and ${\parallel ·\parallel }_p$ means the ${\ell }_p$-norm. Then, we see that

$$g_p\left(u\right):=inf\left\{\parallel (\parallel x_n{\parallel )}_{n=1}^l{\parallel }_p\underset{y^{*}\in B_{Y^{*}}}{sup}{\parallel {\left(y^{*}\left(y_n\right)\right)}_{n=1}^l\parallel }_{p^{*}}:u=\sum _{n=1}^l{x}_n\otimes y_n,l\in \mathbb{N}\right\}={\alpha }_{p,1}\left(u\right),$$

$$\begin{array}{c}{w}_p\left(u\right):=inf\{\underset{x^{*}\in B_{X^{*}}}{sup}\parallel {\left(x^{*}\left(x_n\right)\right)}_{n=1}^l{\parallel }_p\underset{y^{*}\in B_{Y^{*}}}{sup}{\parallel {\left(y^{*}\left(y_n\right)\right)}_{n=1}^l\parallel }_{p^{*}}:u\hfill \\ =\sum _{n=1}^l{x}_n\otimes y_n,l\in \mathbb{N}\}={\alpha }_{p,p^{*}}\left(u\right)\hfill \end{array}$$

and $\pi \left(u\right)={\alpha }_{1,1}\left(u\right)$. The tensor norms $g_p$ and $w_p$ were introduced and studied by Chevet and Saphar [12,13]; see [10,11,14,15,16,17,18,19,20,21,22,23,24] and the references therein for the investigation on related topics.

In this paper, we consider another generalization of $g_p$ and $w_p$. These tensor norms are somehow determined by the Banach space ${\ell }_p$. Naturally, one may extend these notions by replacing ${\ell }_p$ by a general Banach space with a Schauder basis. Throughout this paper, E is a Banach space having the 1-unconditional Schauder basis ${\left(e_n\right)}_n$, ${\left(e_n^{*}\right)}_n$ is the sequence of coordinate functionals for ${\left(e_n\right)}_n$ and $E_{*}:=\overline{\mathrm{span}}{\left\{e_n^{*}\right\}}_{n=1}^{\infty }$. For a finite subset F of $\mathbb{N}$ and ${\left\{x_n\right\}}_{n\in F}\subset X$, let

$$\parallel {\left(x_n\right)}_{n\in F}{\parallel }_{E\left(X\right)}:=\parallel \sum _{n\in F}\parallel x_n\parallel e_n{\parallel }_E\mathrm{and}{\parallel {\left(x_n\right)}_{n\in F}\parallel }_{E^w\left(X\right)}:=\underset{x^{*}\in B_{X^{*}}}{sup}\parallel \sum _{n\in F}{x}^{*}\left(x_n\right)e_n{\parallel }_E.$$

We are now ready to introduce the main notion in this paper.

Definition 1.

For $u\in X\otimes Y$, let

$$g_E(u;X,Y):=inf\left\{\parallel {\left(x_n\right)}_{n\in F}{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y\right)}:u=\sum _{n\in F}{x}_n\otimes y_n,F\subset \mathbb{N}\right\},$$

$$w_E(u;X,Y):=inf\left\{\parallel {\left(x_n\right)}_{n\in F}{\parallel }_{E^w\left(X\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y\right)}:u=\sum _{n\in F}{x}_n\otimes y_n,F\subset \mathbb{N}\right\}.$$

For instance, $g_{{\ell }_p}=g_p$ and $w_{{\ell }_p}=w_p$$(1\le p<\infty )$, and $g_{c_0}=w_{c_0}=g_{\infty }=w_{\infty }$.

Tensor norms are closely related with normed operator ideals. Actually, in view of the monograph of Defant and Floret [2], there is a one-to-one correspondence between maximal Banach operator ideals and finitely generated tensor norms. A tensor norm $\alpha$ is said to be associated with a normed operator ideal $[\mathcal{A},\parallel ·{\parallel }_{\mathcal{A}}]$ if the canonical map from $\mathcal{A}(M,N)$ to $M^{*}{\otimes }_{\alpha }N$ equipped with the norm $\alpha$ is an isometry for every finite-dimensional normed spaces M and N. It is well known that $g_p$ is associated with the ideal of p-nuclear operators. The starting point of this paper comes from [25], where the E-nuclear operators (see Section 2 for the definition of E-nuclear operators) were defined by replacing ${\ell }_p$ by E in the notion of p-nuclear operators. The main goal of this paper is to find a Banach space E for which $g_E$ and $w_E$ are tensor norms, and show that $g_E$ and $w_E$ are associated with the ideals of E-nuclear operators. Obtaining some results for $g_E$ and $w_E$, we provide a base for further investigations of the $g_E$- and $w_E$-tensor norms and E-operator ideals. We focus on the Banach space $E={(\sum {\ell }_q)}_p$$(1\le p,q\le \infty )$ of infinite ${\ell }_p$ direct sum of ${\ell }_q$s, which is a generalization of ${\ell }_p$. For this case, we extend some well known results for $g_p$ and $w_p$ as follows.

In Section 2, for $E={(\sum {\ell }_q)}_p$$(1\le p,q\le \infty )$, we prove that $g_E$ and $w_E$ are finitely generated tensor norms, and it is demonstrated that $g_E$ and $w_E$ are associated with the ideals of E-nuclear operators. In Section 3, we prove that $g_E$ is left projective and for every Banach space X, the injective tensor product $X{\otimes }_{\epsilon }E$ is isometric to $X{\otimes }_{w_E}E$; furthermore, if ${\left(e_n\right)}_n$ is shrinking, then $E^{*}{\otimes }_{\epsilon }X$ is isometric to $E^{*}{\otimes }_{w_E}X$. Additionally, we establish the completions of our E-tensor norms for $E={(\sum {\ell }_q)}_p$, and as an application, we represent E-nuclear operators acting on dual spaces. We refer to the book of Defant and Floret [2] as a reference to the main notions and formulas in the theory of tensor norms and (quasi) normed operator ideals.

2. The ${\mathbf{g}}_{\mathbf{E}}$- and ${\mathbf{w}}_{\mathbf{E}}$-Tensor Norms and Their Associated Operator Ideals

Let us recall that a tensor norm $\alpha$ is a norm on $X\otimes Y$ for each pair of Banach spaces X and Y such that

(TN1)

$\epsilon \le \alpha \le \pi$;

(TN2)

for all operators $T_1:X_1\to Y_1$ and $T_2:X_2\to Y_2$,

$$\parallel T_1\otimes T_2:X_1{\otimes }_{\alpha }{X}_2\to Y_1{\otimes }_{\alpha }{Y}_2\parallel \le \parallel T_1\parallel \parallel T_2\parallel .$$

A tensor norm $\alpha$ is said to be finitely generated if

$$\alpha (u;X,Y)=inf\left\{\alpha \right(u;M,N):u\in M\otimes N,dimM,dimN<\infty \}$$

for every $u\in X\otimes Y$.

Proposition 1.

Suppose that ${\left(e_n\right)}_n$ is normalized. If $g_E$ and $w_E$ satisfy the triangle inequality, then they are finitely generated tensor norms.

Proof. 

We only consider $g_E$. Let X and Y be Banach spaces. Let $c\in \mathbb{C}$ and let $u={\sum }_{n\in F}{x}_n\otimes y_n$ be an arbitrary representation in $X\otimes Y$. Then

$$g_E(cu;X,Y)\le \parallel {\left(c{x}_n\right)}_{n\in F}{\parallel }_{E\left(X\right)}\parallel {\left(y_n\right)}_{n\in F}{\parallel }_{E_{*}^w\left(Y\right)}=|c|\parallel {\left(x_n\right)}_{n\in F}{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y\right)}.$$

Thus $g_E(cu;X,Y)\le |c|g_E(u;X,Y)$. Since $g_E(u;X,Y)=g_E((1/c)\left(cu\right);X,Y)\le$$(1/|c|)g_E(cu;X,Y)$, $g_E(cu;X,Y)\ge |c|g_E(u;X,Y)$.

(TN1): Let $u={\sum }_{n\in F}{x}_n\otimes y_n$ be an arbitrary representation in $X\otimes Y$. Let $x^{*}\in B_{X^{*}}$ and $y^{*}\in B_{Y^{*}}$. Then

$$|\sum _{n\in F}{x}^{*}\left(x_n\right)y^{*}\left(y_n\right)|=|\left(\sum _{k\in F}{y}^{*}\left(y_k\right)e_k^{*}\right)\left(\sum _{n\in F}{x}^{*}\left(x_n\right)e_n\right)|\le \parallel {\left(x_n\right)}_{n\in F}{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y\right)}$$

and

$$g_E(u;X,Y)\le \sum _{n\in F}{g}_E(x_n\otimes y_n)\le \sum _{n\in F}\parallel x_n\parallel \parallel y_n\parallel .$$

It follows that $\epsilon (u;X,Y)\le g_E(u;X,Y)\le \pi (u;X,Y)$, and so

$$g_E(u;X,Y)=0\iff u=0$$

for $u\in X\otimes Y$.

(TN2): Let $T_1:X_1\to Y_1$ and $T_2:X_2\to Y_2$ be operators. Let $u\in X_1\otimes X_2$ and let $u={\sum }_{n\in F}{x}_n^1\otimes x_n^2$ be an arbitrary representation. Then

$$\begin{array}{cc}\hfill g_E((T_1\otimes T_2)\left(u\right);Y_1,Y_2)& =g_E\left(\sum _{n\in F}{T}_1{x}_n^1\otimes T_2{x}_n^2;Y_1,Y_2\right)\hfill \\ \hfill & \le \parallel {\left(T_1{x}_n^1\right)}_{n\in F}{\parallel }_{E\left(Y_1\right)}{\parallel {\left(T_2{x}_n^2\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y_2\right)}\hfill \\ \hfill & =\parallel T_1\parallel \parallel T_2\parallel \parallel \left(\right(1/\parallel T_1\parallel )T_1{x}_n^1{)}_{n\in F}{\parallel }_{E\left(Y_1\right)}\parallel \left(\right(1/\parallel T_2\parallel )T_2{x}_n^2{{)}_{n\in F}\parallel }_{E_{*}^w\left(Y_2\right)}\hfill \\ \hfill & \le \parallel T_1\parallel \parallel T_2\parallel \parallel {\left(x_n^1\right)}_{n\in F}{\parallel }_{E\left(X_1\right)}{\parallel {\left(x_n^2\right)}_{n\in F}\parallel }_{E_{*}^w\left(X_2\right)}.\hfill \end{array}$$

Hence

$$g_E((T_1\otimes T_2)\left(u\right);Y_1,Y_2)\le \parallel T_1\parallel \parallel T_2\parallel g_E(u;X_1,X_2).$$

To show that $g_E$ is finitely generated, let $u\in X\otimes Y$ and let $u={\sum }_{n\in F}{x}_n\otimes y_n$ be an arbitrary representation. Let $M_0:=\mathrm{span}{\left\{x_n\right\}}_{n\in F}$ and $N_0:=\mathrm{span}{\left\{y_n\right\}}_{n\in F}$. Using the Hahn–Banach extension theorem, we have

$$\begin{array}{cc}\hfill & inf\{g_E(u;M,N):u\in M\otimes N,dimM,dimN<\infty \}\hfill \\ \hfill & \le g_E(u;M_0,N_0)\hfill \\ \hfill & \le \parallel {\left(x_n\right)}_{n\in F}{\parallel }_{E\left(M_0\right)}\underset{z^{*}\in B_{N_0^{*}}}{sup}\parallel \sum _{n\in F}{z}^{*}\left(y_n\right)e_n^{*}{\parallel }_{E^{*}}\hfill \\ \hfill & =\parallel {\left(x_n\right)}_{n\in F}{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y\right)}.\hfill \end{array}$$

Hence,

$$inf\{g_E(u;M,N):u\in M\otimes N,dimM,dimN<\infty \}\le g_E(u;X,Y).$$

□

We can now prove:

Theorem 1.

If $E={(\sum {\ell }_q)}_p$$(1\le p,q<\infty )$, $E={(\sum c_0)}_p$$(1\le p<\infty )$ or $E={(\sum {\ell }_q)}_{c_0}$$(1\le q<\infty )$, then $g_E$ and $w_E$ are finitely generated tensor norms.

Proof. 

We only consider $g_E$. Let X and Y be Banach spaces. By Proposition 1, we only need to show the triangle inequality of $g_E$.

For the he case $E={(\sum {\ell }_q)}_p$$(1<p,q<\infty )$, let $u,v\in X\otimes Y$ and let $\delta >0$ be given. We can find representations

$$u=\sum _{n=1}^l\sum _{k=1}^l{x}_{nk}^1\otimes y_{nk}^1\mathrm{and}v=\sum _{n=1}^l\sum _{k=1}^l{x}_{nk}^2\otimes y_{nk}^2$$

such that

$$\parallel {\left({\left(x_{nk}^1\right)}_{k=1}^l\right)}_{n=1}^l{\parallel }_{E\left(X\right)}{\parallel {\left({\left(y_{nk}^1\right)}_{k=1}^l\right)}_{n=1}^l\parallel }_{E_{*}^w\left(Y\right)}\le (1+\delta )g_E(u;X,Y),$$

$$\parallel {\left({\left(x_{nk}^2\right)}_{k=1}^l\right)}_{n=1}^l{\parallel }_{E\left(X\right)}{\parallel {\left({\left(y_{nk}^2\right)}_{k=1}^l\right)}_{n=1}^l\parallel }_{E_{*}^w\left(Y\right)}\le (1+\delta )g_E(v;X,Y).$$

We may assume that

$$\begin{array}{cc}\hfill & \parallel {({(x_{nk}^1)}_{k=1}^l)}_{n=1}^l{\parallel }_{E\left(X\right)}\le {((1+\delta )g_E(u;X,Y))}^{1/p},\hfill \\ \hfill & \parallel {({(y_{nk}^1)}_{k=1}^l)}_{n=1}^l{\parallel }_{E_{*}^w\left(Y\right)}\le {((1+\delta )g_E(u;X,Y))}^{1/p^{*}},\hfill \\ \hfill & \parallel {({(x_{nk}^2)}_{k=1}^l)}_{n=1}^l{\parallel }_{E\left(X\right)}\le {((1+\delta )g_E(v;X,Y))}^{1/p},\hfill \\ \hfill & \parallel {({\left(y_{nk}^2\right)}_{k=1}^l)}_{n=1}^l{\parallel }_{E_{*}^w\left(Y\right)}\le {((1+\delta )g_E(v;X,Y))}^{1/p^{*}}.\hfill \end{array}$$

Since

$$u+v=\sum _{i=1}^2\sum _{n=1}^l\sum _{k=1}^l{x}_{nk}^i\otimes y_{nk}^i,$$

$$\begin{array}{cc}\hfill & g_E(u+v;X,Y)\hfill \\ \hfill & \le {\left(\sum _{i=1}^2\sum _{n=1}^l{\left(\sum _{k=1}^l{\parallel x_{nk}^i\parallel }^q\right)}^{p/q}\right)}^{1/p}\underset{y^{*}\in B_{Y^{*}}}{sup}{\left(\sum _{i=1}^2\sum _{n=1}^l{\left(\sum _{k=1}^l{|y^{*}\left(y_{nk}^i\right)|}^{q^{*}}\right)}^{p^{*}/q^{*}}\right)}^{1/p^{*}}\hfill \\ \hfill & \le {\left(\sum _{n=1}^l{\left(\sum _{k=1}^l{\parallel x_{nk}^1\parallel }^q\right)}^{p/q}+\sum _{n=1}^l{\left(\sum _{k=1}^l{\parallel x_{nk}^2\parallel }^q\right)}^{p/q}\right)}^{1/p}\hfill \\ \hfill & {\left(\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l{\left(\sum _{k=1}^l{|y^{*}\left(y_{nk}^1\right)|}^{q^{*}}\right)}^{p^{*}/q^{*}}+\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l{\left(\sum _{k=1}^l{|y^{*}\left(y_{nk}^2\right)|}^{q^{*}}\right)}^{p^{*}/q^{*}}\right)}^{1/p^{*}}\hfill \\ \hfill & \le {\left((1+\delta )(g_E(u;X,Y)+g_E(v;X,Y))\right)}^{1/p}{\left((1+\delta )(g_E(u;X,Y)+g_E(v;X,Y))\right)}^{1/p^{*}}\hfill \\ \hfill & =(1+\delta )(g_E(u;X,Y)+g_E(v;X,Y)).\hfill \end{array}$$

Since $\delta >0$ was arbitrary,

$$g_E(u+v;X,Y)\le g_E(u;X,Y)+g_E(v;X,Y).$$

For the case $E={(\sum {\ell }_1)}_p$$(1<p<\infty )$:

$$\begin{array}{cc}\hfill & g_E(u+v;X,Y)\hfill \\ \hfill & \le {\left(\sum _{i=1}^2\sum _{n=1}^l{\left(\sum _{k=1}^l\parallel x_{nk}^i\parallel \right)}^p\right)}^{1/p}\underset{y^{*}\in B_{Y^{*}}}{sup}{\left(\sum _{i=1}^2\sum _{n=1}^l(\underset{1\le k\le l}{sup}|y^{*}\left(y_{nk}^i\right){|)}^{p^{*}}\right)}^{1/p^{*}}\hfill \\ \hfill & \le {\left(\sum _{n=1}^l{\left(\sum _{k=1}^l\parallel x_{nk}^1\parallel \right)}^p+\sum _{n=1}^l{\left(\sum _{k=1}^l\parallel x_{nk}^2\parallel \right)}^p\right)}^{1/p}\hfill \\ \hfill & {\left(\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l(\underset{1\le k\le l}{sup}|y^{*}\left(y_{nk}^1\right){|)}^{p^{*}}+\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l(\underset{1\le k\le l}{sup}|y^{*}\left(y_{nk}^2\right){|)}^{p^{*}}\right)}^{1/p^{*}}\hfill \\ \hfill & \le {\left((1+\delta )(g_E(u;X,Y)+g_E(v;X,Y))\right)}^{1/p}{\left((1+\delta )(g_E(u;X,Y)+g_E(v;X,Y))\right)}^{1/p^{*}}\hfill \\ \hfill & =(1+\delta )(g_E(u;X,Y)+g_E(v;X,Y)).\hfill \end{array}$$

For the case $E={(\sum {\ell }_q)}_1$$(1<q<\infty )$: We may assume that

$$\parallel {\left({\left(x_{nk}^1\right)}_{k=1}^l\right)}_{n=1}^l{\parallel }_{E\left(X\right)}\le (1+\delta )g_E(u;X,Y),{\parallel {\left({\left(y_{nk}^1\right)}_{k=1}^l\right)}_{n=1}^l\parallel }_{E_{*}^w\left(Y\right)}\le 1,$$

$$\parallel {\left({\left(x_{nk}^2\right)}_{k=1}^l\right)}_{n=1}^l{\parallel }_{E\left(X\right)}\le (1+\delta )g_E(v;X,Y),{\parallel {\left({\left(y_{nk}^2\right)}_{k=1}^l\right)}_{n=1}^l\parallel }_{E_{*}^w\left(Y\right)}\le 1.$$

Then

$$\begin{array}{cc}\hfill & g_E(u+v;X,Y)\hfill \\ \hfill & \le \sum _{i=1}^2\sum _{n=1}^l{\left(\sum _{k=1}^l{\parallel x_{nk}^i\parallel }^q\right)}^{1/q}\underset{y^{*}\in B_{Y^{*}}}{sup}\underset{i=1,2,1\le n\le l}{sup}{\left(\sum _{k=1}^l{|y^{*}\left(y_{nk}^i\right)|}^{q^{*}}\right)}^{1/q^{*}}\hfill \\ \hfill & \le \sum _{n=1}^l{\left(\sum _{k=1}^l{\parallel x_{nk}^1\parallel }^q\right)}^{1/q}+\sum _{n=1}^l{\left(\sum _{k=1}^l{\parallel x_{nk}^2\parallel }^q\right)}^{1/q}\hfill \\ \hfill & \le (1+\delta )(g_E(u;X,Y)+g_E(v;X,Y)).\hfill \end{array}$$

For the case $E={(\sum c_0)}_p$$(1<p<\infty )$:

$$\begin{array}{cc}\hfill & g_E(u+v;X,Y)\hfill \\ \hfill & \le {\left(\sum _{i=1}^2\sum _{n=1}^l(\underset{1\le k\le l}{sup}\parallel x_{nk}^i{\parallel )}^p\right)}^{1/p}\underset{y^{*}\in B_{Y^{*}}}{sup}{\left(\sum _{i=1}^2\sum _{n=1}^l{\left(\sum _{k=1}^l|y^{*}\left(y_{nk}^i\right)|\right)}^{p^{*}}\right)}^{1/p^{*}}\hfill \\ \hfill & \le {\left(\sum _{n=1}^l(\underset{1\le k\le l}{sup}\parallel x_{nk}^1{\parallel )}^p+\sum _{n=1}^l(\underset{1\le k\le l}{sup}\parallel x_{nk}^2{\parallel )}^p\right)}^{1/p}\hfill \\ \hfill & {\left(\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l{\left(\sum _{k=1}^l|y^{*}\left(y_{nk}^1\right)|\right)}^{p^{*}}+\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l{\left(\sum _{k=1}^l|y^{*}\left(y_{nk}^2\right)|\right)}^{p^{*}}\right)}^{1/p^{*}}\hfill \\ \hfill & \le {\left((1+\delta )(g_E(u;X,Y)+g_E(v;X,Y))\right)}^{1/p}{\left((1+\delta )(g_E(u;X,Y)+g_E(v;X,Y))\right)}^{1/p^{*}}\hfill \\ \hfill & =(1+\delta )(g_E(u;X,Y)+g_E(v;X,Y)).\hfill \end{array}$$

For the case $E={(\sum {\ell }_q)}_{c_0}$$(1<q<\infty )$: We may assume that

$$\parallel {\left({\left(x_{nk}^1\right)}_{k=1}^l\right)}_{n=1}^l{\parallel }_{E\left(X\right)}\le 1,{\parallel {\left({\left(y_{nk}^1\right)}_{k=1}^l\right)}_{n=1}^l\parallel }_{E_{*}^w\left(Y\right)}\le (1+\delta )g_E(u;X,Y),$$

$$\parallel {\left({\left(x_{nk}^2\right)}_{k=1}^l\right)}_{n=1}^l{\parallel }_{E\left(X\right)}\le 1,{\parallel {\left({\left(y_{nk}^2\right)}_{k=1}^l\right)}_{n=1}^l\parallel }_{E_{*}^w\left(Y\right)}\le (1+\delta )g_E(v;X,Y).$$

Then

$$\begin{array}{cc}\hfill & g_E(u+v;X,Y)\hfill \\ \hfill & \le \underset{i=1,2,1\le n\le l}{sup}{\left(\sum _{k=1}^l{\parallel x_{nk}^i\parallel }^q\right)}^{1/q}\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{i=1}^2\sum _{n=1}^l{\left(\sum _{k=1}^l{|y^{*}\left(y_{nk}^i\right)|}^{q^{*}}\right)}^{1/q^{*}}\hfill \\ \hfill & \le \underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l{\left(\sum _{k=1}^l{|y^{*}\left(y_{nk}^1\right)|}^{q^{*}}\right)}^{1/q^{*}}+\underset{y^{*}\in B_{Y^{*}}}{sup}\sum _{n=1}^l{\left(\sum _{k=1}^l{|y^{*}\left(y_{nk}^2\right)|}^{q^{*}}\right)}^{1/q^{*}}\hfill \\ \hfill & \le (1+\delta )(g_E(u;X,Y)+g_E(v;X,Y)).\hfill \end{array}$$

The cases $E={(\sum {\ell }_1)}_{c_0}$ and $E={(\sum c_0)}_1$ also follow from similar proofs. □

Throughout the remainder of this paper, we will assume that $g_E$ and $w_E$ are finitely generated tensor norms. For a Banach space X, let us consider the Banach spaces

$$E\left(X\right):=\left\{{\left(x_n\right)}_n\mathrm{in}X:\sum _{n=1}^{\infty }\parallel x_n\parallel e_n\mathrm{converges}\mathrm{in}E\right\}$$

equipped with the norm $\parallel {\left(x_n\right)}_n{\parallel }_{E\left(X\right)}:=\parallel {\sum }_{n=1}^{\infty }\parallel x_n{\parallel e_n\parallel }_E$,

$$E^w\left(X\right):=\left\{{\left(x_n\right)}_n\mathrm{in}X:\sum _{n=1}^{\infty }{x}^{*}\left(x_n\right)e_n\mathrm{converges}\mathrm{in}E\mathrm{for}\mathrm{each}{x}^{*}\in X^{*}\right\}$$

equipped with the norm $\parallel {\left(x_n\right)}_n{\parallel }_{E^w\left(X\right)}:={sup}_{x^{*}\in B_{X^{*}}}{\parallel {\sum }_{n=1}^{\infty }{x}^{*}\left(x_n\right)e_n\parallel }_E$ and

$$E^u\left(X\right):=\left\{{\left(x_n\right)}_n\mathrm{in}X:\underset{l\to \infty }{lim}\underset{x^{*}\in B_{X^{*}}}{sup}\parallel \sum _{n\ge l}{x}^{*}\left(x_n\right)e_n{\parallel }_E=0\right\}$$

equipped with the norm $\parallel {\left(x_n\right)}_n{\parallel }_{E^w\left(X\right)}$.

Let X and Y be Banach spaces, and let $T:X\to Y$ be an operator such that

$$T=\sum _{n=1}^{\infty }{x}_n^{*}\underline{\otimes }{y}_n,$$

where $x_n^{*}\underline{\otimes }{y}_n\left(x\right)=x_n^{*}\left(x\right)y_n$. The following operators were introduced in [25]. We say that T is E-nuclear (respectively, dual E-nuclear) if ${\left(x_n^{*}\right)}_n\in E\left(X^{*}\right)$ (respectively, $E_{*}^w\left(X^{*}\right)$) and ${\left(y_n\right)}_n\in E_{*}^w\left(Y\right)$ (respectively, $E\left(Y\right)$). The collection of all E-nuclear (respectively, dual E-nuclear) operators from X to Y is denoted by ${\mathcal{N}}_E(X,Y)$ (respectively, ${\mathcal{N}}^E(X,Y)$), and for $T\in {\mathcal{N}}_E(X,Y)$ (respectively, ${\mathcal{N}}^E(X,Y)$), let

$${\parallel T\parallel }_{{\mathcal{N}}_E}:=inf\parallel {\left(x_n^{*}\right)}_n{\parallel }_{E\left(X^{*}\right)}{\parallel {\left(y_n\right)}_n\parallel }_{E_{*}^w\left(Y\right)}$$

$$(\mathrm{respectively},\parallel T{\parallel }_{{\mathcal{N}}^E}:=inf\parallel {\left(x_n^{*}\right)}_n{\parallel }_{E_{*}^w\left(X^{*}\right)}\parallel {\left(y_n\right)}_n{\parallel }_{E\left(Y\right)}),$$

where the infimum is taken over all such representations. We say that T is uniform E-nuclear (respectively, dual uniform E-nuclear) if ${\left(x_n^{*}\right)}_n\in E^u\left(X^{*}\right)$ (respectively, $E_{*}^w\left(X^{*}\right)$) and ${\left(y_n\right)}_n\in E_{*}^w\left(Y\right)$ (respectively, $E^u\left(Y\right)$). The collection of all uniform E-nuclear (respectively, dual uniform E-nuclear) operators from X to Y is denoted by ${}_u{\mathcal{N}}_E(X,Y)$ (respectively, ${}_u{\mathcal{N}}^E(X,Y)$) and for $T\in {}_u{\mathcal{N}}_E(X,Y)$ (respectively, ${}_u{\mathcal{N}}^E(X,Y)$), let

$${\parallel T\parallel }_{{}_u{\mathcal{N}}_E}:=inf\parallel {\left(x_n^{*}\right)}_n{\parallel }_{E^w\left(X^{*}\right)}{\parallel {\left(y_n\right)}_n\parallel }_{E_{*}^w\left(Y\right)}$$

$$(\mathrm{respectively},\parallel T{\parallel }_{{}_u{\mathcal{N}}^E}:=inf\parallel {\left(x_n^{*}\right)}_n{\parallel }_{E_{*}^w\left(X^{*}\right)}\parallel {\left(y_n\right)}_n{\parallel }_{E^w\left(Y\right)}),$$

where the infimum is taken over all such representations. For instance, ${\mathcal{N}}_{{\ell }_p}$ is the ideal of p-nuclear operators, and ${}_u{\mathcal{N}}_{{\ell }_p}$ is the ideal of p-compact operators (cf. [2,15]).

Let $\mathcal{F}$ be the ideal of finite rank operators and let X and Y be Banach spaces. For $T\in \mathcal{F}(X,Y)$, let

$$\begin{array}{cc}\hfill & {\parallel T\parallel }_{{\mathcal{N}}_E^0}:=inf\left\{\parallel {\left(x_n^{*}\right)}_{n\in F}{\parallel }_{E\left(X^{*}\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y\right)}:T=\sum _{n\in F}{x}_n^{*}\underline{\otimes }{y}_n,\mathrm{finite}F\subset \mathbb{N}\right\},\hfill \\ \hfill & {\parallel T\parallel }_{{\mathcal{N}}_0^E}:=inf\left\{\parallel {\left(x_n^{*}\right)}_{n\in F}{\parallel }_{E_{*}^w\left(X^{*}\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E\left(Y\right)}:T=\sum _{n\in F}{x}_n^{*}\underline{\otimes }{y}_n,\mathrm{finite}F\subset \mathbb{N}\right\},\hfill \\ \hfill & {\parallel T\parallel }_{{}_u{\mathcal{N}}_E^0}:=inf\left\{\parallel {\left(x_n^{*}\right)}_{n\in F}{\parallel }_{E^w\left(X^{*}\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E_{*}^w\left(Y\right)}:T=\sum _{n\in F}{x}_n^{*}\underline{\otimes }{y}_n,\mathrm{finite}F\subset \mathbb{N}\right\},\hfill \\ \hfill & {\parallel T\parallel }_{{}_u{\mathcal{N}}_0^E}:=inf\left\{\parallel {\left(x_n^{*}\right)}_{n\in F}{\parallel }_{E_{*}^w\left(X^{*}\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{E^w\left(Y\right)}:T=\sum _{n\in F}{x}_n^{*}\underline{\otimes }{y}_n,\mathrm{finite}F\subset \mathbb{N}\right\}.\hfill \end{array}$$

Let ${\alpha }^t$ be the transposed tensor norm (see [2]) of a tensor norm $\alpha$. Let X and Y be Banach spaces. For $T={\sum }_{n\in F}{x}_n^{*}\underline{\otimes }{y}_n\in \mathcal{F}(X,Y)$, let $u_T:={\sum }_{n\in F}{x}_n^{*}\otimes y_n\in X^{*}\otimes Y$. Then we see that

$${\parallel T\parallel }_{{\mathcal{N}}_E^0}=g_E(u_T;X^{*},Y),{\parallel T\parallel }_{{}_u{\mathcal{N}}_E^0}=w_E(u_T;X^{*},Y),$$

$${\parallel T\parallel }_{{\mathcal{N}}_0^E}=g_E^t(u_T;X^{*},Y),{\parallel T\parallel }_{{}_u{\mathcal{N}}_0^E}=w_E^t(u_T;X^{*},Y).$$

Proposition 2.

If X or Y is a finite-dimensional normed space, then

$${\parallel T\parallel }_{{\mathcal{N}}_E^0}={\parallel T\parallel }_{{\mathcal{N}}_E}{,\parallel T\parallel }_{{\mathcal{N}}_0^E}={\parallel T\parallel }_{{\mathcal{N}}^E}{,\parallel T\parallel }_{{}_u{\mathcal{N}}_E^0}={\parallel T\parallel }_{{}_u{\mathcal{N}}_E}{,\parallel T\parallel }_{{}_u{\mathcal{N}}_0^E}={\parallel T\parallel }_{{}_u{\mathcal{N}}^E}$$

for every operartor T from X to Y.

Proof. 

We only consider ${\mathcal{N}}_0^E$. Let $T:X\to Y$ be an operator, and let $\delta >0$ be given. Let

$$T=\sum _{n=1}^{\infty }{x}_n^{*}\underline{\otimes }{y}_n$$

be a dual E-nuclear representation such that

$$\parallel {\left(x_n^{*}\right)}_n{\parallel }_{E_{*}^w\left(X^{*}\right)}\parallel {\left(y_n\right)}_n{\parallel }_{E\left(Y\right)}\le (1+\delta ){\parallel T\parallel }_{{\mathcal{N}}^E}.$$

If X is finite-dimensional, then there exists an $l\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill \parallel \sum _{n\ge l+1}{x}_n^{*}\underline{\otimes }{y}_n\parallel & \le \underset{x\in B_X}{sup}\sum _{n\ge l+1}|x_n^{*}\left(x\right)|\parallel y_n\parallel \hfill \\ \hfill & =\underset{x\in B_X}{sup}\left(\sum _{n\ge l+1}|x_n^{*}\left(x\right)|e_n^{*}\right)\left(\sum _{n\ge l+1}\parallel y_n\parallel e_n\right)\hfill \\ \hfill & \le \parallel {\left(x_n^{*}\right)}_n{\parallel }_{E_{*}^w\left(X^{*}\right)}\parallel \sum _{n\ge l+1}\parallel y_n\parallel e_n{\parallel }_E\hfill \\ \hfill & \le {\delta \parallel T\parallel }_{{\mathcal{N}}^E}/{\parallel i{d}_X\parallel }_{{\mathcal{N}}_0^E},\hfill \end{array}$$

where $i{d}_X$ is the identity map on X. We have

$$\begin{array}{cc}\hfill {\parallel T\parallel }_{{\mathcal{N}}_0^E}& \le \parallel \sum _{n=1}^l{x}_n^{*}\underline{\otimes }{y}_n{\parallel }_{{\mathcal{N}}_0^E}+\parallel \sum _{n\ge l+1}{x}_n^{*}\underline{\otimes }{y}_n{\parallel }_{{\mathcal{N}}_0^E}\hfill \\ \hfill & \le \parallel {\left(x_n^{*}\right)}_n{\parallel }_{E_{*}^w\left(X^{*}\right)}\parallel {\left(y_n\right)}_n{\parallel }_{E^w\left(Y\right)}+\parallel \sum _{n\ge l+1}{x}_n^{*}\underline{\otimes }{y}_n\parallel {\parallel i{d}_X\parallel }_{{\mathcal{N}}_0^E}\hfill \\ \hfill & \le {(1+2\delta )\parallel T\parallel }_{{\mathcal{N}}^E}.\hfill \end{array}$$

If Y is finite-dimensional, then $i{d}_X$ can be replaced by $i{d}_Y$ in the above proof. □

From Proposition 2, we have:

Corollary 1.

The tensor norms $g_E$, $g_E^t$, $w_E$ and $w_E^t$, respectively, are associated with $[{\mathcal{N}}_E,\parallel ·{\parallel }_{{\mathcal{N}}_E}]$, $[{\mathcal{N}}^E,\parallel ·{\parallel }_{{\mathcal{N}}^E}]$, ${[}_u{\mathcal{N}}_E,\parallel ·{\parallel }_{{}_u{\mathcal{N}}_E}]$ and ${[}_u{\mathcal{N}}^E,\parallel ·{\parallel }_{{}_u{\mathcal{N}}^E}]$.

3. Some Results of the ${\mathit{g}}_{\mathit{E}}$- and ${\mathit{w}}_{\mathit{E}}$-Tensor Norms

A tensor norm $\alpha$ is called left-projective if, for every quotient operator $q:Z\to X$, the operator

$$q\otimes i{d}_Y:Z{\otimes }_{\alpha }Y\to X{\otimes }_{\alpha }Y$$

is a quotient operator for all Banach spaces $X,Y$ and Z. If the transposed ${\alpha }^t$ of $\alpha$ is left-projective, then $\alpha$ is called right-projective.

Proposition 3.

The tensor norm $g_E$ is left-projective.

Proof. 

Let $q:Z\to X$ be a quotient operator. To show that the map

$$q\otimes i{d}_Y:Z{\otimes }_{g_E}Y\to X{\otimes }_{g_E}Y$$

is a quotient operator, let $u={\sum }_{n\in F}{x}_n\otimes y_n\in X{\otimes }_{g_E}Y$. We should show that

$$g_E(u;X,Y)\ge inf\{g_E(v;Z,Y):v\in Z{\otimes }_{g_E}Y,q\otimes i{d}_Y\left(v\right)=u\}.$$

Let $\delta >0$ be given. Since q is a quotient operator, there exists ${\left\{z_n\right\}}_{n\in F}\subset Z$ such that

$$q{z}_n=x_n,\parallel z_n\parallel \le (1+\delta )\parallel x_n\parallel$$

for every $n\in F$. Then we have

$$\begin{array}{cc}\hfill & inf\{g_E(v;Z,Y):v\in Z{\otimes }_{g_E}Y,q\otimes i{d}_Y\left(v\right)=u\}\hfill \\ \hfill & \le g_E\left(\sum _{n\in F}{z}_n\otimes y_n;Z,Y\right)\hfill \\ \hfill & \le \parallel {\left(z_n\right)}_{n\in F}{\parallel }_{E\left(Z\right)}{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{{\left(E_{*}\right)}^w\left(Y\right)}\hfill \\ \hfill & =\parallel \sum _{n\in F}\parallel z_n\parallel e_n{\parallel }_E{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{{\left(E_{*}\right)}^w\left(Y\right)}\hfill \\ \hfill & \le (1+\delta )\parallel \sum _{n\in F}\parallel x_n\parallel e_n{\parallel }_E{\parallel {\left(y_n\right)}_{n\in F}\parallel }_{{\left(E_{*}\right)}^w\left(Y\right)}.\hfill \end{array}$$

Since $u={\sum }_{n\in F}{x}_n\otimes y_n$ was an arbitrary representation,

$$inf\{g_E(v;Z,Y):v\in Z{\otimes }_{g_E}Y,q\otimes i{d}_Y\left(v\right)=u\}\le (1+\delta )g_E(u;X,Y).$$

Since $\delta >0$ was also arbitrary, we complete the proof. □

For a tensor norm $\alpha$, we will denote by $X{\widehat{\otimes }}_{\alpha }Y$ the completion of the normed space $X{\otimes }_{\alpha }Y$.

Lemma 1

([2], Proposition 21.7(1)). For a finitely generated tensor norm α, if a Banach space X has the approximation property, then for every Banach space Y, the natural map

$$I_{\alpha }:Y{\widehat{\otimes }}_{\alpha }X⟶Y{\widehat{\otimes }}_{\epsilon }X$$

is injective.

Theorem 2.

For every Banach space X,

$$X{\otimes }_{\epsilon }E=X{\otimes }_{w_E}E$$

holds isometrically, and if ${\left(e_n\right)}_n$ is shrinking, then

$$E^{*}{\otimes }_{\epsilon }X=E^{*}{\otimes }_{w_E}X$$

holds isometrically.

Proof. 

In order to prove the first statement, let $u\in X\otimes E$, and let $U:X^{*}\to E$ be the corresponding finite rank operator for u. Then, $U^{*}\left(E^{*}\right)$ can be viewed with a subset of X. Thus, for every $x^{*}\in X^{*}$,

$$U{x}^{*}=\sum _{i=1}^{\infty }\left(e_i^{*}U{x}^{*}\right)e_i=\sum _{i=1}^{\infty }{x}^{*}\left(U^{*}{e}_i^{*}\right)e_i.$$

Since $U\left(B_{X^{*}}\right)$ is a relatively compact subset of E,

$$\begin{array}{cc}\hfill \underset{l\to \infty }{lim}\epsilon \left(\sum _{i=1}^l{U}^{*}{e}_i^{*}\otimes e_i-u;X,E\right)& =\underset{l\to \infty }{lim}\parallel \sum _{i=1}^l{U}^{*}{e}_i^{*}\underline{\otimes }{e}_i-U\parallel \hfill \\ \hfill & =\underset{l\to \infty }{lim}\underset{x^{*}\in B_{X^{*}}}{sup}\parallel \sum _{i=1}^l\left(e_i^{*}U{x}^{*}\right)e_i-U{x}^{*}{\parallel }_E=0.\hfill \end{array}$$

Consequently,

$$u=\sum _{i=1}^{\infty }{U}^{*}{e}_i^{*}\otimes e_i$$

converges in $X{\widehat{\otimes }}_{\epsilon }E$.

To show that the above series unconditionally converges in $X{\widehat{\otimes }}_{w_E}E$, let $\delta >0$ be given. Let ${\left\{U{x}_k^{*}\right\}}_{k=1}^m$ be a $\delta /2$-net for $U\left(B_{X^{*}}\right)$. Choose an $l_{\delta }\in \mathbb{N}$ so that

$$\parallel \sum _{i\ge l_{\delta }}\left(e_i^{*}U{x}_k^{*}\right)e_i{\parallel }_E\le \frac{\delta }{2}$$

for every $k=1,...,m$. Now, let G be an arbitrary finite subset of $\mathbb{N}$ with $minG>l_{\delta }$. Let $x^{*}\in B_{X^{*}}$ and $e^{*}\in B_{E^{*}}$. Let $k_0\in \{1,...,m\}$ be such that

$$\parallel U{x}^{*}-U{x}_{k_0}^{*}{\parallel }_E\le \frac{\delta }{2}.$$

Then we have

$$\begin{array}{cc}\hfill \parallel \sum _{i\in G}{x}^{*}\left(U^{*}{e}_i^{*}\right)e_i{\parallel }_E\parallel \sum _{i\in G}\left(e^{*}{e}_i\right)e_i^{*}{\parallel }_{E^{*}}& =\parallel \sum _{i\in G}\left(e_i^{*}U{x}^{*}\right)e_i{\parallel }_E\underset{{\sum }_k{\alpha }_k{e}_k\in B_E}{sup}|\sum _{i\in G}{e}^{*}\left({\alpha }_i{e}_i\right)|\hfill \\ \hfill & \le \parallel \sum _{i\in G}\left(e_i^{*}U{x}^{*}\right)e_i{\parallel }_E\hfill \\ \hfill & \le \parallel \sum _{i\in G}\left(e_i^{*}U(x^{*}-x_{k_0}^{*})\right)e_i{\parallel }_E+\parallel \sum _{i\in G}\left(e_i^{*}U{x}_{k_0}^{*}\right)e_i{\parallel }_E\hfill \\ \hfill & \le \parallel \sum _{i=1}^{\infty }\left(e_i^{*}U(x^{*}-x_{k_0}^{*})\right)e_i{\parallel }_E+\parallel \sum _{i\ge l_{\delta }}\left(e_i^{*}U{x}_{k_0}^{*}\right)e_i{\parallel }_E\hfill \\ \hfill & \le \parallel U{x}^{*}-U{x}_{k_0}^{*}{\parallel }_E+\frac{\delta }{2}\le \delta .\hfill \end{array}$$

Consequently,

$$w_E\left(\sum _{i\in G}{U}^{*}{e}_i^{*}\otimes e_i;X,E\right)\le \parallel {\left(U^{*}{e}_i^{*}\right)}_{i\in G}{\parallel }_{E^w\left(X\right)}{\parallel {\left(e_i\right)}_{i\in G}\parallel }_{E_{*}^w\left(E\right)}\le \delta$$

and so

$$v:=\sum _{i=1}^{\infty }{U}^{*}{e}_i^{*}\otimes e_i$$

unconditionally converges in $X{\widehat{\otimes }}_{w_E}E$. Since a Banach space with a basis has the approximation property, by Lemma 1, $u=v$ in $X{\widehat{\otimes }}_{w_E}E$. Then, since for every $l\in \mathbb{N}$,

$$\begin{array}{cc}\hfill w_E\left(\sum _{i=1}^l{U}^{*}{e}_i^{*}\otimes e_i;X,E\right)& \le \parallel {\left(U^{*}{e}_i^{*}\right)}_{i=1}^l{\parallel }_{E^w\left(X\right)}{\parallel {\left(e_i\right)}_{i=1}^l\parallel }_{E_{*}^w\left(E\right)}\hfill \\ \hfill & \le \underset{x^{*}\in B_{X^{*}}}{sup}\parallel \sum _{i=1}^l\left(e_i^{*}U{x}^{*}\right)e_i{\parallel }_E,\hfill \end{array}$$

$$w_E(u;X,E)\le \parallel U\parallel =\epsilon (u;X,E).$$

In order to prove the second statement, let $v\in E^{*}\otimes X$ and let $V:E\to X$ be the corresponding finite rank operator for v. For every $e={\sum }_i{\alpha }_i{e}_i\in E$,

$$Ve=\sum _{i=1}^{\infty }{\alpha }_{i}V{e}_i=\sum _{i=1}^{\infty }\left(e_i^{*}e\right)V{e}_i.$$

Since ${\left(e_i^{*}\right)}_i$ is a basis for $E^{*}$, and $V^{*}\left(B_{X^{*}}\right)$ is a relatively compact subset of $E^{*}$,

$$\begin{array}{cc}\hfill \underset{l\to \infty }{lim}\epsilon \left(\sum _{i=1}^l{e}_i^{*}\otimes V{e}_i-v;E^{*},X\right)& =\underset{l\to \infty }{lim}\parallel \sum _{i=1}^l{e}_i^{*}\underline{\otimes }V{e}_i-V\parallel \hfill \\ \hfill & =\underset{l\to \infty }{lim}\parallel \sum _{i=1}^{l}V{e}_i\underline{\otimes }{e}_i^{*}-V^{*}\parallel \hfill \\ \hfill & =\underset{l\to \infty }{lim}\underset{x^{*}\in B_{X^{*}}}{sup}\parallel \sum _{i=1}^l\left(V^{*}{x}^{*}\right)\left(e_i\right)e_i^{*}-V^{*}{x}^{*}{\parallel }_{E^{*}}=0.\hfill \end{array}$$

Consequently,

$$v=\sum _{i=1}^{\infty }{e}_i^{*}\otimes V{e}_i$$

converges in $E^{*}{\widehat{\otimes }}_{\epsilon }X$.

To show that the above series unconditionally converges in $E^{*}{\widehat{\otimes }}_{w_E}X$, let $\delta >0$ be given. Let ${\left\{V^{*}{x}_k^{*}\right\}}_{k=1}^m$ be a $\delta /2$-net for $V^{*}\left(B_{X^{*}}\right)$. Choose an $l_{\delta }\in \mathbb{N}$ so that

$$\parallel \sum _{i\ge l_{\delta }}{V}^{*}{x}_k^{*}\left(e_i\right)e_i^{*}{\parallel }_{E^{*}}\le \frac{\delta }{2}$$

for every $k=1,...,m$. Now, let G be an arbitrary finite subset of $\mathbb{N}$ with $minG>l_{\delta }$. Let $x^{*}\in B_{X^{*}}$ and $e^{**}\in B_{E^{**}}$. Let $k_0\in \{1,...,m\}$ be such that

$$\parallel V^{*}{x}^{*}-V^{*}{x}_{k_0}^{*}{\parallel }_{E^{*}}\le \frac{\delta }{2}.$$

Then, we have

$$\begin{array}{cc}\hfill \parallel \sum _{i\in G}{e}^{**}\left(e_i^{*}\right)e_i{\parallel }_E\parallel \sum _{i\in G}\left(x^{*}V{e}_i\right)e_i^{*}{\parallel }_{E^{*}}& =\underset{{\sum }_k{\alpha }_k{e}_k^{*}\in B_{E^{*}}}{sup}|\sum _{i\in G}{e}^{**}\left({\alpha }_i{e}_i^{*}\right)|\parallel \sum _{i\in G}{V}^{*}{x}^{*}\left(e_i\right)e_i^{*}{\parallel }_{E^{*}}\hfill \\ \hfill & \le \parallel \sum _{i\in G}{V}^{*}{x}^{*}\left(e_i\right)e_i^{*}{\parallel }_{E^{*}}\hfill \\ \hfill & \le \parallel \sum _{i\in G}{V}^{*}(x^{*}-x_{k_0}^{*})\left(e_i\right)e_i^{*}{\parallel }_{E^{*}}+\parallel \sum _{i\in G}{V}^{*}{x}_{k_0}^{*}\left(e_i\right)e_i^{*}{\parallel }_{E^{*}}\hfill \\ \hfill & \le \parallel \sum _{i=1}^{\infty }{V}^{*}(x^{*}-x_{k_0}^{*})\left(e_i\right)e_i^{*}{\parallel }_{E^{*}}+\parallel \sum _{i\ge l_{\delta }}{V}^{*}{x}_{k_0}^{*}\left(e_i\right)e_i^{*}{\parallel }_{E^{*}}\hfill \\ \hfill & \le \parallel V^{*}(x^{*}-x_{k_0}^{*}){\parallel }_{E^{*}}+\frac{\delta }{2}\le \delta .\hfill \end{array}$$

Consequently,

$$w_E\left(\sum _{i\in G}{e}_i^{*}\otimes V{e}_i;E^{*},X\right)\le \parallel {\left(e_i^{*}\right)}_{i\in G}{\parallel }_{E^w\left(E^{*}\right)}{\parallel {\left(V{e}_i\right)}_{i\in G}\parallel }_{E_{*}^w\left(X\right)}\le \delta$$

and so

$$u:=\sum _{i=1}^{\infty }{e}_i^{*}\otimes V{e}_i$$

unconditionally converges in $E^{*}{\widehat{\otimes }}_{w_E}X$. By Lemma 1, since $v^t=u^t$ in $X{\widehat{\otimes }}_{\epsilon }{E}^{*}$, $v^t=u^t$ in $X{\widehat{\otimes }}_{w_E^t}{E}^{*}$, and so $v=u$ in $E^{*}{\widehat{\otimes }}_{w_E}X$. Since for every $l\in \mathbb{N}$,

$$\begin{array}{cc}\hfill w_E\left(\sum _{i=1}^l{e}_i^{*}\otimes V{e}_i;E^{*},X\right)& \le \parallel {\left(e_i^{*}\right)}_{i=1}^l{\parallel }_{E^w\left(E^{*}\right)}{\parallel {\left(V{e}_i\right)}_{i=1}^l\parallel }_{E_{*}^w\left(X\right)}\hfill \\ \hfill & \le \underset{x^{*}\in B_{X^{*}}}{sup}\parallel \sum _{i=1}^l{V}^{*}{x}^{*}\left(e_i\right)e_i^{*}{\parallel }_{E^{*}},\hfill \end{array}$$

$$w_E(v;E^{*},X)\le \parallel V\parallel =\epsilon (v;E^{*},X).$$

□

Now, we consider the completions of our tensor norms. The following lemma is well known.

Lemma 2.

Let $(Z,\parallel ·\parallel )$ be a normed space, and let $(\widehat{Z},\parallel ·\parallel )$ be its completion. If $z\in \widehat{Z}$, then for every $\delta >0$, there exists a sequence ${\left(z_n\right)}_n$ in Z such that

$$\sum _{n=1}^{\infty }\parallel z_n\parallel \le (1+\delta )\parallel z\parallel$$

and $z={\sum }_{n=1}^{\infty }{z}_n$ converges in $\widehat{Z}$.

Proposition 4.

Suppose that $E={(\sum {\ell }_q)}_p$$(1\le p,q<\infty )$, $E={(\sum c_0)}_p$$(1\le p<\infty )$ or $E={(\sum {\ell }_q)}_{c_0}$$(1\le q<\infty )$. If $u\in X{\widehat{\otimes }}_{w_E}Y$, then there exist ${\left(x_n\right)}_n\in E^u\left(X\right)$ and ${\left(y_n\right)}_n\in E_{*}^u\left(Y\right)$ such that

$$u=\sum _{n=1}^{\infty }{x}_n\otimes y_n$$

unconditionally converges in $X{\widehat{\otimes }}_{w_E}Y$ and

$$w_E(u;X,Y)=inf\left\{\parallel {\left(x_n\right)}_n{\parallel }_{E^w\left(X\right)}{\parallel {\left(y_n\right)}_n\parallel }_{E_{*}^w\left(Y\right)}:u=\sum _{n=1}^{\infty }{x}_n\otimes y_n\right\}.$$

Proof. 

Let $u\in X{\widehat{\otimes }}_{w_E}Y$, and let $\delta >0$ be given. Then, by Lemma 2, there exists a sequence ${\left(u_n\right)}_n$ in $X\otimes Y$ such that

$$\sum _{n=1}^{\infty }{w}_E(u_n;X,Y)\le (1+\delta )w_E(u;X,Y)$$

and $u={\sum }_{n=1}^{\infty }{u}_n$ converges in $X{\widehat{\otimes }}_{w_E}Y$.

We only consider the case $E={(\sum {\ell }_q)}_p$$(1<p,q<\infty )$. The proofs of the other cases are similar. For every $n\in \mathbb{N}$, let

$$u_n=\sum _{i=1}^{m_n}\sum _{j=1}^{m_n}{x}_{ij}^n\otimes y_{ij}^n$$

be such that

$$\parallel {\left({\left(x_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}{\parallel }_{E^w\left(X\right)}{\parallel {\left({\left(y_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}\parallel }_{E_{*}^w\left(Y\right)}\le (1+\delta )w_E(u_n;X,Y).$$

We may assume that

$$\parallel {\left({\left(x_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}{\parallel }_{E^w\left(X\right)}\le {\left((1+\delta )w_E(u_n;X,Y)\right)}^{1/p},$$

$$\parallel {\left({\left(y_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}{\parallel }_{E_{*}^w\left(Y\right)}\le {\left((1+\delta )w_E(u_n;X,Y)\right)}^{1/p^{*}}.$$

In order to show that $u={\sum }_{n=1}^{\infty }{\sum }_{i=1}^{m_n}{\sum }_{j=1}^{m_n}{x}_{ij}^n\otimes y_{ij}^n$ unconditionally converges in $X{\widehat{\otimes }}_{w_E}Y$ and ${\left({\left({\left(x_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}\right)}_n\in E^u\left(X\right)$ and ${\left({\left({\left(y_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}\right)}_n\in E_{*}^u\left(Y\right)$, let $\gamma >0$ be given. Choose an $N_{\gamma }\in \mathbb{N}$ so that for all $l\ge N_{\gamma }$,

$$w_E\left(u-\sum _{n=1}^l{u}_n;X,Y\right)\le \gamma \mathrm{and}\sum _{n\ge l}{w}_E(u_n;X,Y)\le \gamma .$$

Then, for all $l\ge N_{\gamma }$ and $1\le a,b\le m_{l+1}$,

$$\begin{array}{cc}\hfill & w_E\left(u-\left(\sum _{n=1}^l{u}_n+\sum _{i=1}^a\sum _{j=1}^{m_{l+1}}{x}_{ij}^{l+1}\otimes y_{ij}^{l+1}+\sum _{j=1}^b{x}_{(a+1)j}^{l+1}\otimes y_{(a+1)j}^{l+1}\right);X,Y\right)\hfill \\ \hfill & \le \gamma +w_E\left(\sum _{i=1}^a\sum _{j=1}^{m_{l+1}}{x}_{ij}^{l+1}\otimes y_{ij}^{l+1}+\sum _{j=1}^b{x}_{(a+1)j}^{l+1}\otimes y_{(a+1)j}^{l+1};X,Y\right)\hfill \\ \hfill & \le \gamma +\parallel {\left({\left(x_{ij}^{l+1}\right)}_{j=1}^{m_{l+1}}\right)}_{i=1}^{m_{l+1}}{\parallel }_{E^w\left(X\right)}{\parallel {\left({\left(y_{ij}^{l+1}\right)}_{j=1}^{m_{l+1}}\right)}_{i=1}^{m_{l+1}}\parallel }_{E_{*}^w\left(Y\right)}\hfill \\ \hfill & \le \gamma +(1+\delta )w_E(u_{l+1};X,Y)\hfill \\ \hfill & \le \gamma +(1+\delta )\gamma .\hfill \end{array}$$

This shows that

$$u=\sum _{n=1}^{\infty }\sum _{i=1}^{m_n}\sum _{j=1}^{m_n}{x}_{ij}^n\otimes y_{ij}^n$$

converges in $X{\widehat{\otimes }}_{w_E}Y$. To show that the above series converges unconditionally, let F be an arbitrary finite subset of $\mathbb{N}$ with $minF>{\sum }_{n=1}^{N_{\gamma }}{m}_n^2$, and let ${\{s_k\otimes t_k\}}_{k\in F}$ be the set of corresponding tensors. Then, there exists $l_1,l_2>N_{\gamma }$ such that ${\{s_k\otimes t_k\}}_{k\in F}\subset {\{{\{x_{ij}^n\otimes y_{ij}^n\}}_{i,j=1}^{m_n}\}}_{n=l_1}^{l_2}$. We have

$$\begin{array}{cc}\hfill w_E\left(\sum _{k\in F}{s}_k\otimes t_k;X,Y\right)& \le \sum _{n=l_1}^{l_2}\parallel {\left({\left(x_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}{\parallel }_{E^w\left(X\right)}{\parallel {\left({\left(y_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}\parallel }_{E_{*}^w\left(Y\right)}\hfill \\ \hfill & \le \sum _{n=l_1}^{l_2}(1+\delta )w_E(u_n;X,Y)\hfill \\ \hfill & \le (1+\delta )\gamma .\hfill \end{array}$$

Since for all $l\ge N_{\gamma }$ and $1\le a,b\le m_l$,

$$\begin{array}{cc}\hfill & \underset{x^{*}\in B_{X^{*}}}{sup}{\left({\left(\sum _{j=b}^{m_l}{|x^{*}(x_{aj}^l)|}^q\right)}^{p/q}+\sum _{i=a+1}^{m_l}{\left(\sum _{j=1}^{m_l}{|x^{*}(x_{ij}^l)|}^q\right)}^{p/q}+\sum _{n\ge l+1}\sum _{i=1}^{m_n}{\left(\sum _{j=1}^{m_n}{|x^{*}(x_{ij}^n)|}^q\right)}^{p/q}\right)}^{1/p}\hfill \\ \hfill & \le \underset{x^{*}\in B_{X^{*}}}{sup}{\left(\sum _{n\ge l}\sum _{i=1}^{m_n}{\left(\sum _{j=1}^{m_n}{|x^{*}(x_{ij}^n)|}^q\right)}^{p/q}\right)}^{1/p}\hfill \\ \hfill & \le {\left(\sum _{n\ge l}\underset{x^{*}\in B_{X^{*}}}{sup}\sum _{i=1}^{m_n}{\left(\sum _{j=1}^{m_n}{|x^{*}(x_{ij}^n)|}^q\right)}^{p/q}\right)}^{1/p}\hfill \\ \hfill & \le {\left(\sum _{n\ge l}(1+\delta )w_E(u_n;X,Y)\right)}^{1/p}\le {((1+\delta )\gamma )}^{1/p},\hfill \end{array}$$

${\left({\left({\left(x_{ij}^n\right)}_{j=1}^{m_n}\right)}_{i=1}^{m_n}\right)}_n\in E^u\left(X\right)$ and we see that

$$\parallel {({({(x_{ij}^n)}_{j=1}^{m_n})}_{i=1}^{m_n})}_n{\parallel }_{E^w(X)}\le {\left((1+\delta )\sum _{n=1}^{\infty }{w}_E(u_n;X,Y)\right)}^{1/p}.$$

Similarly,

$${({({(y_{ij}^n)}_{j=1}^{m_n})}_{i=1}^{m_n})}_n\in E_{*}^u(Y)\mathrm{and}{\parallel {({({(y_{ij}^n)}_{j=1}^{m_n})}_{i=1}^{m_n})}_n\parallel }_{E_{*}^w(Y)}\le {\left((1+\delta )\sum _{n=1}^{\infty }{w}_E(u_n;X,Y)\right)}^{1/p^{*}}.$$

Consequently, the infimum

$$inf\{·\}\le \parallel {({({(x_{ij}^n)}_{j=1}^{m_n})}_{i=1}^{m_n})}_n{\parallel }_{E^w(X)}{\parallel {({({(y_{ij}^n)}_{j=1}^{m_n})}_{i=1}^{m_n})}_n\parallel }_{E_{*}^w(Y)}\le {(1+\delta )}^2{w}_E(u;X,Y).$$

Since $\delta >0$ was arbitrary, $inf\{·\}\le w_E(u;X,Y)$.

For every such representation

$$u=\sum _{n=1}^{\infty }{x}_n\otimes y_n$$

unconditionally converging in $X{\widehat{\otimes }}_{w_E}Y$,

$$\begin{array}{cc}\hfill w_E(u;X,Y)& =\underset{l\to \infty }{lim}{w}_E\left(\sum _{n=1}^l{x}_n\otimes y_n\right)\hfill \\ \hfill & \le \underset{l\to \infty }{lim}\parallel {\left(x_n\right)}_{n=1}^l{\parallel }_{E^w\left(X\right)}{\parallel {\left(y_n\right)}_{n=1}^l\parallel }_{E_{*}^w\left(Y\right)}\hfill \\ \hfill & =\parallel {\left(x_n\right)}_{n=1}^{\infty }{\parallel }_{E^w\left(X\right)}{\parallel {\left(y_n\right)}_{n=1}^{\infty }\parallel }_{E_{*}^w\left(Y\right)}.\hfill \end{array}$$

Thus, $w_E(u;X,Y)\le inf\{·\}$. □

As in the proof of Proposition 4, we have:

Proposition 5.

Suppose that $E={(\sum {\ell }_q)}_p$$(1\le p,q<\infty )$, $E={(\sum c_0)}_p$$(1\le p<\infty )$ or $E={(\sum {\ell }_q)}_{c_0}$$(1\le q<\infty )$. If $u\in X{\widehat{\otimes }}_{g_E}Y$, then there exist ${\left(x_n\right)}_n\in E\left(X\right)$ and ${\left(y_n\right)}_n\in E_{*}^u\left(Y\right)$ such that

$$u=\sum _{n=1}^{\infty }{x}_n\otimes y_n$$

unconditionally converges in $X{\widehat{\otimes }}_{g_E}Y$ and

$$g_E(u;X,Y)=inf\left\{\parallel {\left(x_n\right)}_n{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_n\parallel }_{E_{*}^w\left(Y\right)}:u=\sum _{n=1}^{\infty }{x}_n\otimes y_n\right\}.$$

Let $\alpha$ be a finitely generated tensor norm. Let $\mathcal{L}(X,Y)$ be the Banach space of all operators from X to Y. The operator $j_{\alpha }:X^{*}{\otimes }_{\alpha }Y\to \mathcal{L}(X,Y)$ is defined by $j_{\alpha }({\sum }_{n=1}^m{x}_n^{*}\otimes y_n)={\sum }_{n=1}^m{x}_n^{*}\underline{\otimes }{y}_n$, and let

$$J_{\alpha }:X^{*}{\widehat{\otimes }}_{\alpha }Y\to \mathcal{L}(X,Y)$$

be the coninuous extension of $j_{\alpha }$. We equip $J_{\alpha }\left(X^{*}{\widehat{\otimes }}_{\alpha }Y\right)$ with the quotient norm of $X^{*}{\widehat{\otimes }}_{\alpha }Y$/$\ker J_{\alpha }$, which will be denoted by ${\parallel ·\parallel }_{J_{\alpha }}$. According to a well-known result of Grothendieck [16] (cf. [10], Proposition 1.5.4), if $X^{*}$ or Y has the approximation property (AP), then $J_{\alpha }$ is injective; hence, $X^{*}{\widehat{\otimes }}_{\alpha }Y$ is isometric to $(J_{\alpha }\left(X^{*}{\widehat{\otimes }}_{\alpha }Y\right),\parallel ·{\parallel }_{J_{\alpha }})$.

Lemma 3

([21], Theorem 2.4). Assume that $X^{***}$ or Y has the AP.

If $T\in J_{\alpha }\left(X^{**}{\widehat{\otimes }}_{\alpha }Y\right)\subset \mathcal{L}(X^{*},Y)$ and $T^{*}\left(Y^{*}\right)\subset X$, then $T\in {\overline{J_{\alpha }(X\otimes Y)}}^{{\parallel ·\parallel }_{J_{\alpha }}}$.

The prototype of the following theorem is described in [21] (Theorem 3.1).

Theorem 3.

Suppose that $E={(\sum {\ell }_q)}_p$$(1\le p,q<\infty )$, $E={(\sum c_0)}_p$$(1\le p<\infty )$ or $E={(\sum {\ell }_q)}_{c_0}$$(1\le q<\infty )$. Assume that $X^{***}$ or Y has the AP. If $T\in {\mathcal{N}}_E(X^{*},Y)$ (respectively, ${}_u{\mathcal{N}}_E(X^{*},Y)$) and $T^{*}\left(Y^{*}\right)\subset X$, then there exist ${\left(x_n\right)}_n\in E\left(X\right)$ (respectively, $E^u\left(X\right)$) and ${\left(y_n\right)}_n\in E_{*}^u\left(Y\right)$ such that

$$T=\sum _{n=1}^{\infty }{x}_n\underline{\otimes }{y}_n$$

unconditionally converges in ${\mathcal{N}}_E(X^{*},Y)$ (respectively, ${}_u{\mathcal{N}}_E(X^{*},Y)$).

Proof. 

We only consider ${\mathcal{N}}_E$. The proof of the case ${}_u{\mathcal{N}}_E$ is similar. First, we show that $(J_{g_E}\left(X^{**}{\widehat{\otimes }}_{g_E}Y\right){,\parallel ·\parallel }_{J_{g_E}})=({\mathcal{N}}_E(X^{*},Y){,\parallel ·\parallel }_{{\mathcal{N}}_E})$. Let $J_{g_E}\left(u\right)\in J_{g_E}\left(X^{**}{\widehat{\otimes }}_{g_E}Y\right)$. Let $u={\sum }_{n=1}^{\infty }{x}_n^{**}\otimes y_n$ be an arbitrary representation in Proposition 5. Then

$$J_{g_E}\left(u\right)=\sum _{n=1}^{\infty }{x}_n^{**}\underline{\otimes }{y}_n\in {\mathcal{N}}_E(X^{*},Y)$$

and $\parallel J_{g_E}{\left(u\right)\parallel }_{{\mathcal{N}}_E}\le \parallel {\left(x_n^{**}\right)}_n{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_n\parallel }_{E_{*}^w\left(Y\right)}$. Since the representation of u was arbitrary, $\parallel J_{g_E}{\left(u\right)\parallel }_{{\mathcal{N}}_E}\le g_E(u;X^{**},Y)={\parallel J_{g_E}\left(u\right)\parallel }_{J_{g_E}}$.

Let $T\in {\mathcal{N}}_E(X^{*},Y)$ and let $\delta >0$ be given. Let $T={\sum }_{n=1}^{\infty }{x}_n^{**}\underline{\otimes }{y}_n$ be an arbitrary ${\mathcal{N}}_E$-representation. Since

$$\begin{array}{cc}\hfill g_E\left(\sum _{n=m}^l{x}_n^{**}\otimes y_n;X^{*},Y\right)& \le \parallel {\left(x_n^{**}\right)}_{n=m}^l{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_{n=m}^l\parallel }_{E_{*}^w\left(Y\right)}\hfill \\ \hfill & \le \parallel {\left(y_n\right)}_n{\parallel }_{E_{*}^w\left(Y\right)}\parallel \sum _{n=m}^l\parallel x_n^{**}\parallel e_n{\parallel }_E,\hfill \end{array}$$

${\sum }_{n=1}^{\infty }{x}_n^{**}\otimes y_n$ converges in $X^{**}{\widehat{\otimes }}_{g_E}Y$. Thus,

$$T=J_{g_E}\left(\sum _{n=1}^{\infty }{x}_n^{**}\otimes y_n\right)\in J_{g_E}\left(X^{**}{\widehat{\otimes }}_{g_E}Y\right).$$

Choose an $l\in \mathbb{N}$ so that $g_E({\sum }_{n>l}{x}_n^{**}\otimes y_n;X^{**},Y)\le \delta$. Then, we have

$$\begin{array}{cc}\hfill {\parallel T\parallel }_{J_{g_E}}& =g_E\left(\sum _{n=1}^{\infty }{x}_n^{**}\otimes y_n;X^{**},Y\right)\hfill \\ \hfill & \le g_E\left(\sum _{n=1}^l{x}_n^{**}\otimes y_n;X^{**},Y\right)+\delta \hfill \\ \hfill & \le \parallel {\left(x_n^{**}\right)}_n{\parallel }_{E\left(X\right)}{\parallel {\left(y_n\right)}_n\parallel }_{E_{*}^w\left(Y\right)}+\delta .\hfill \end{array}$$

Since the representation of T was arbitrary, ${\parallel T\parallel }_{J_{g_E}}\le {\parallel T\parallel }_{{\mathcal{N}}_E}$.

Now, let $T\in {\mathcal{N}}_E(X^{*},Y)$. Choose $u\in X^{**}{\widehat{\otimes }}_{g_E}Y$ so that $T=J_{g_E}\left(u\right)$. By Lemma 3, $J_{g_E}\left(u\right)\in {\overline{J_{g_E}(X\otimes Y)}}^{{\parallel ·\parallel }_{J_{g_E}}}$. Since $J_{g_E}$ is an isometry and $X{\widehat{\otimes }}_{g_E}Y$ is isometrically embeded in $X^{**}{\widehat{\otimes }}_{g_E}Y$ (cf. [3], Proposition 6.4), we see that $u\in X{\widehat{\otimes }}_{g_E}Y$. By Proposition 5, there exist ${\left(x_n\right)}_n\in E\left(X\right)$ and ${\left(y_n\right)}_n\in E_{*}^u\left(Y\right)$ such that $u={\sum }_{n=1}^{\infty }{x}_n\otimes y_n$ unconditionally converges in $X{\widehat{\otimes }}_{g_E}Y$. Hence,

$$T=J_{g_E}\left(u\right)=\sum _{n=1}^{\infty }{x}_n\underline{\otimes }{y}_n$$

unconditionally converges in ${\mathcal{N}}_E(X^{*},Y)$. □

4. Discussion

This work is the general and natural extension of some results about the tensor norms $g_p$ and $w_p$. There have been many more investigations about $g_p$ and $w_p$ since their introduction. We expect that several more results on $g_p$ and $w_p$, and the ideals of p-nuclear and p-compact operators, can be developed. For instance, for a finitely generated tensor norm $\alpha$, a Banach space X is said to have the α-approximation property ($\alpha$-AP) if for every Banach space Y, the natural map

$$J_{\alpha }:Y{\widehat{\otimes }}_{\alpha }X⟶Y{\widehat{\otimes }}_{\epsilon }X$$

is injective (cf. [2]), Section 21.7. The $g_p$-AP and the $w_p$-AP were well studied, and the $g_p$-AP (respectively, $w_p$-AP) is closely related with an approximation property of the ideal of p-summing operators (respectively, ideal of p-dominated operators) (cf. [11]). We can consider the $g_E$-AP and the $w_E$-AP as the following subjects:

• An investigation of the ideals of E-summing operators and E-dominated operators;
• Some relationships of the ideals of E-summing operators and E-dominated operators, respectively, between the $g_E$-AP and the $w_E$-AP, respectively.