The K_p,q-Compactness and K_p,q-Null Sequences, and the ${\mathcal{K}}_{{\mathit{K}}_{\mathit{p},\mathit{q}}}$-Approximation Property for Banach Spaces

Abstract

Let $K_{p,q}$ ($1\le p,q\le \infty$ with $1/p+1/q\ge 1$) be the ideal of $(p,q)$-compact operators. This paper investigates the compactness and null sequences via $K_{p,q}$, and an approximation property of the ideal of $K_{p,q}$-compact operators.

1. Introduction

One of the most important theories in the study of Banach spaces is the theory of Banach operator ideals. One may refer to [1,2] for various informations and contents about Banach operator ideals. The main subjects of this paper come from a classical Banach operator ideal and a theory of compactness of Carl and Stephani [3]. Let $\mathcal{A}$ be a Banach operator ideal. A subset K of a Banach space X is said to be $\mathcal{A}$-compact if there exist a Banach space Z, $U\in \mathcal{A}(Z,X)$ and a relatively compact subset C of Z such that $K\subset U\left(C\right)$. In fact, this notion is an equivalent statement of the original definition of $\mathcal{A}$-compactness (see ([3], Definition 1.1 and Theorem 1.2)).

Lassalle and Turco [4] introduced a way to measure the size of $\mathcal{A}$-compact sets. For an $\mathcal{A}$-compact subset K of X, let:

$$m_{\mathcal{A}}(K;X):={inf\{\parallel U\parallel }_{\mathcal{A}}:U\in \mathcal{A}(Z,X),\mathrm{relatively}\mathrm{compact}C\subset B_Z,K\subset U\left(C\right)\},$$

where $B_Z$ is the unit ball of a Banach space Z.

A linear map $R:Y\to X$ is said to be $\mathcal{A}$-compact if $R\left(B_Y\right)$ is an $\mathcal{A}$-compact subset of X. Let ${\mathcal{K}}_{\mathcal{A}}(Y,X)$ be the space of all $\mathcal{A}$-compact operators from Y to X. For $R\in {\mathcal{K}}_{\mathcal{A}}(Y,X)$, let:

$${\parallel R\parallel }_{{\mathcal{K}}_{\mathcal{A}}}:=m_{\mathcal{A}}(R\left(B_Y\right);X).$$

Then $[{\mathcal{K}}_{\mathcal{A}},\parallel ·{\parallel }_{{\mathcal{K}}_{\mathcal{A}}}]$ is a Banach operator ideal (see ([4], Section 2)). Let $[\mathcal{K},\parallel ·\parallel ]$ be the ideal of classical compact operators. Then, from ([4], Remarks 1.3 and 1.7), the classical compactness coincides with the $\mathcal{K}$-compactness and for a relatively compact subset K of X, $m_{\mathcal{K}}(K;X)={sup}_{x\in K}\parallel x\parallel$. Consequently, $[{\mathcal{K}}_{\mathcal{K}}{,\parallel ·\parallel }_{{\mathcal{K}}_{\mathcal{K}}}]=[\mathcal{K},\parallel ·\parallel ]$.

Carl and Stephani [3] also introduced a general notion of null sequences. For a Banach operator ideal $\mathcal{A}$ and a Banach space X, a sequence ${\left(x_n\right)}_n$ in X is called $\mathcal{A}$-null if there exist a Banach space Z and $S\in \mathcal{A}(Z,X)$ such that for every $\epsilon >0$, there is a natural number $n_{\epsilon }$ so that:

$$x_n\in \epsilon S\left(B_Z\right)$$

for all $n\ge n_{\epsilon }$. Let $c_{0,\mathcal{A}}\left(X\right)$ be the space of all $\mathcal{A}$-null sequences in X. Since $\mathcal{A}$-null sequences are $\mathcal{A}$-compact (see [3]), Lassalle and Turco [5] defined a norm on $c_{0,\mathcal{A}}\left(X\right)$ by:

$$\parallel {\left(x_n\right)}_n{\parallel }_{c_{0,\mathcal{A}}}:=m_{\mathcal{A}}({\left\{x_n\right\}}_n;X).$$

Then $(c_{0,\mathcal{A}}\left(X\right),\parallel ·{\parallel }_{c_{0,\mathcal{A}}})$ is a Banach space (see [5]). Let $(c_0\left(X\right),\parallel ·{\parallel }_{\infty })$ be the Banach space of all null sequences in X. Then, from ([4], Proposition 1.4), null sequences coincide with $\mathcal{K}$-null sequences and so $(c_0{\left(X\right),\parallel ·\parallel }_{\infty })=(c_{0,\mathcal{K}}{\left(X\right),\parallel ·\parallel }_{c_{0,\mathcal{K}}})$. One of the well known operator ideals is the ideal $K_{p,q}$ of $(p,q)$-compact operators ($1\le p,q\le \infty$, $1/p+1/q\ge 1$) (see Section 2). This paper is devoted to the $K_{p,q}$-compactness, $K_{p,q}$-null sequences, and an approximation property of the ideal of $K_{p,q}$-compact operators. We refer to [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] for the related investigations.

Sinha and Karn [21] introduced the p-compactness ($1\le p\le \infty$) (see Section 3), which is an extension of the norm compactness, and for a p-compact subset K of a Banach space X, Galicer, Lassalle and Turco [12] introduced the notion $m_p(K;X)$ to measure the size of K. It was shown in [4] (Remarks 1.3 and 1.7) that K is p-compact if and only if K is $K_{1,p}$-compact, in this case, $m_p(K;X)=m_{K_{1,p}}(K;X)$. The author in [14] introduced the unconditional p-compactness (see Section 3), which is a weaker notion of p-compactness, and it was shown in [15] (Proposition 5.2) that a subset K of X is unconditionally p-compact if and only if K is $K_{p^{*},p}$-compact, in this case, a size of K is equal to $m_{K_{p^{*},p}}(K;X)$. In Section 3, we generalize those results. For every $1\le p,q\le \infty$ with $1/p+1/q\ge 1$, some characterizations and sizes of $K_{p,q}$-compact sets are established.

Piñeiro and Delgado [20] introduced the p-null sequence ($1\le p\le \infty$) (see Section 3), which is an exension of the norm null sequence, and Lassalle and Turco ([4], Proposition 1.5) showed that p-null sequences and $K_{1,p}$-null sequences coincide. The author [14] introduced the unconditionally p-null sequence (see Section 3), which is a weaker notion of p-null sequence, and Ain and Oja [7] introduced a more general concept of p-null and unconditionally p-null sequences. They showed that unconditionally p-null sequences and $K_{p^{*},p}$-null sequences coincide, and p-null sequences and $K_{1,p}$-null sequences coincide in a more general setting. In Section 3, we generalize those results. For every $1\le p,q\le \infty$ with $1/p+1/q\ge 1$, some characterizations of $K_{p,q}$-null sequences are established.

One of the important objects in the study of Banach spaces is the approximation property. For given a Banach operator ideal $\mathcal{A}$, Lassalle and Turco [4] introduced the ${\mathcal{K}}_{\mathcal{A}}$-approximation property (${\mathcal{K}}_{\mathcal{A}}$-AP) (see Section 4), which is an extension of the approximation property. Section 4 is concerned with the ${\mathcal{K}}_{K_{p,q}}$-AP. A characterization of the ${\mathcal{K}}_{K_{p,q}}$-AP is established and using this, it is shown that for every $1\le p,q\le \infty$ with $1/p+1/q\ge 1$, if a Banach space has the ${\mathcal{K}}_{K_{p,p^{*}}}$-AP, then it has the ${\mathcal{K}}_{K_{p,q}}$-AP.

2. Preliminaries

Let $[\mathcal{A},\parallel ·{\parallel }_{\mathcal{A}}]$ be a Banach operator ideal. Let X and Y be Banach spaces. We denote by $\mathcal{L}(X,Y)$ the space of all operators from X to Y. The surjective hull ${[\mathcal{A},\parallel ·\parallel }_{\mathcal{A}}{]}^{sur}$ is defined as follows:

$${\mathcal{A}}^{sur}(X,Y):=\{T\in \mathcal{L}(X,Y):T{q}_X\in \mathcal{A}({\ell }_1\left(B_X\right),Y)\},$$

where $q_X:{\ell }_1\left(B_X\right)\to X$ is the canonical quotient map, and ${\parallel T\parallel }_{{\mathcal{A}}^{sur}}:={\parallel T{q}_X\parallel }_{\mathcal{A}}$ for $T\in {\mathcal{A}}^{sur}(X,Y)$ (see ([1], p. 113 and [7], Section 8.5)). Note that ${\mathcal{A}}^{sur}({\ell }_1,Y)$ is isometrically equal to $\mathcal{A}({\ell }_1,Y)$ (cf. [1], Corollary 9.8).

It was shown in [4] (Proposition 1.8) that a subset K of X is $\mathcal{A}$-compact if and only if there exist $U\in \mathcal{A}({\ell }_1,X)$ and a relatively compact subset C of $B_{{\ell }_1}$ such that $K\subset U\left(C\right)$, in this case:

$$m_{\mathcal{A}}(K;X):={inf\{\parallel U\parallel }_{\mathcal{A}}:U\in \mathcal{A}({\ell }_1,X),\mathrm{relatively}\mathrm{compact}C\subset B_{{\ell }_1},K\subset U\left(C\right)\}.$$

Consequently, K is $\mathcal{A}$-compact if and only if K is ${\mathcal{A}}^{sur}$-compact, in this case:

$$m_{\mathcal{A}}(K;X)=m_{{\mathcal{A}}^{sur}}(K;X).$$

It was shown in [5] (Proposition 2.5) that ${\left(x_n\right)}_n\in c_{0,\mathcal{A}}\left(X\right)$ if and only if ${\sum }_{n=1}^{\infty }{e}_n\otimes x_n$ converges in ${\mathcal{K}}_{\mathcal{A}}({\ell }_1,X)$, where ${\left(e_n\right)}_n$ is the sequence of standard unit vectors in $c_0$, in this case:

$$\parallel {\left(x_n\right)}_n{\parallel }_{c_{0,\mathcal{A}}}=\parallel \sum _{n=1}^{\infty }{e}_n\otimes x_n{\parallel }_{{\mathcal{K}}_{\mathcal{A}}}.$$

Consequently, a sequence ${\left(x_n\right)}_n$ in X is $\mathcal{A}$-null if and only if ${\left(x_n\right)}_n$ is ${\mathcal{A}}^{sur}$-null, in this case:

$$\parallel {\left(x_n\right)}_n{\parallel }_{c_{0,\mathcal{A}}}={\parallel {\left(x_n\right)}_n\parallel }_{c_{0,{\mathcal{A}}^{sur}}}.$$

Recall the minimal kernel ${[\mathcal{A},\parallel ·\parallel }_{\mathcal{A}}{]}^{min}:=[\overline{\mathcal{F}}\circ \mathcal{A}\circ \overline{\mathcal{F}}{,\parallel ·\parallel }_{\overline{\mathcal{F}}\circ \mathcal{A}\circ \overline{\mathcal{F}}}]$, where $\overline{\mathcal{F}}$ is the ideal of approximable operators, namely, the operator norm closure of the ideal of finite rank operators. If ${[\mathcal{A},\parallel ·\parallel }_{\mathcal{A}}{]}^{min}={[\mathcal{A},\parallel ·\parallel }_{\mathcal{A}}]$, then we call $[\mathcal{A},\parallel ·{\parallel }_{\mathcal{A}}]$ minimal. It was shown in [4] (Proposition 2.1) that $[{\mathcal{K}}_{\mathcal{A}}{,\parallel ·\parallel }_{{\mathcal{K}}_{\mathcal{A}}}]=[{(\mathcal{A}\circ \overline{\mathcal{F}})}^{sur}{,\parallel ·\parallel }_{{(\mathcal{A}\circ \overline{\mathcal{F}})}^{sur}}]$. Consequently, if a Banach operator ideal $[\mathcal{A},\parallel ·{\parallel }_{\mathcal{A}}]$ is minimal, then:

$${[\mathcal{A},\parallel ·\parallel }_{\mathcal{A}}{]}^{sur}=[{\mathcal{K}}_{\mathcal{A}}{,\parallel ·\parallel }_{{\mathcal{K}}_{\mathcal{A}}}].$$

The injective hull ${[\mathcal{A},\parallel ·\parallel }_{\mathcal{A}}{]}^{inj}$ is defined as follows:

$${\mathcal{A}}^{inj}(X,Y):=\{T\in \mathcal{L}(X,Y):I_{Y}T\in \mathcal{A}(X,{\ell }_{\infty }\left(B_{Y^{*}}\right))\},$$

where $I_Y:Y\to {\ell }_{\infty }\left(B_{Y^{*}}\right)$ is the canonical isometry, and ${\parallel T\parallel }_{{\mathcal{A}}^{inj}}:={\parallel I_{Y}T\parallel }_{\mathcal{A}}$ for $T\in {\mathcal{A}}^{inj}(X,Y)$ (see ([1], p. 112 and [2], Section 8.4)).

The dual ideal ${[\mathcal{A},\parallel ·\parallel }_{\mathcal{A}}{]}^{dual}$ is defined as follows:

$${\mathcal{A}}^{dual}(X,Y):=\{T\in \mathcal{L}(X,Y):T^{*}\in \mathcal{A}(Y^{*},X^{*})\},$$

where $T^{*}$ is the adjoint of T, and ${\parallel T\parallel }_{{\mathcal{A}}^{dual}}={\parallel T^{*}\parallel }_{\mathcal{A}}$.

Let $1\le p,q\le \infty$ with $1/p+1/q\ge 1$ and let $1\le r\le \infty$ with $1/r=1/p+1/q-1$. A linear map $T:X\to Y$ is called $(p,q)$-compact if there exist ${\left({\lambda }_n\right)}_n\in {\ell }_r$ ($c_0$ when $r=\infty$), ${\left(x_n^{*}\right)}_n\in {\ell }_{q^{*}}^w\left(X^{*}\right)$ ($1/q+1/q^{*}=1$) and ${\left(y_n\right)}_n\in {\ell }_{p^{*}}^w\left(Y\right)$ such that:

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

where ${\ell }_p^w\left(Z\right)$ is the Banach space with the norm ${\parallel ·\parallel }_p^w$ of all Z-valued weakly p-summable sequences for a Banach space Z. The space of all $(p,q)$-compact operators from X to Y is denoted by $K_{p,q}(X,Y)$ with the norm:

$${\parallel T\parallel }_{K_{p,q}}:=inf\left\{\parallel {\left({\lambda }_n\right)}_n{\parallel }_r\parallel {\left(x_n^{*}\right)}_n{\parallel }_{q^{*}}^w{\parallel {\left(y_n\right)}_n\parallel }_{p^{*}}^w:T=\sum _{n=1}^{\infty }{\lambda }_n{x}_n^{*}\otimes y_n\right\},$$

where the infimum is taken over all such representations. Then, $[K_{p,q},\parallel ·{\parallel }_{K_{p,q}}]$ is a minimal Banach operator ideal (cf. [1,2]). Consequently:

$$[K_{p,q}{,\parallel ·\parallel }_{K_{p,q}}{]}^{sur}=[{\mathcal{K}}_{K_{p,q}}{,\parallel ·\parallel }_{{\mathcal{K}}_{K_{p,q}}}].$$

where $K_{p,1}$ is well known as the ideal of p-nuclear operators and $K_{p,p^{*}}$ is the ideal of p-compact operators.

The closed subspace ${\ell }_p^u\left(Z\right)$ of ${\ell }_p^w\left(Z\right)$ consists of all sequences ${\left(z_n\right)}_n$ in Z satisfying that:

$$\underset{z^{*}\in B_{Z^{*}}}{sup}\sum _{n\ge m}{\left|z^{*}\left(z_n\right)\right|}^p⟶0$$

as $m\to \infty$ (cf. ([1], Section 8.2 and [23,24])). Note that ${\ell }_{\infty }^u\left(Z\right)=c_0\left(Z\right)$, and it is well known that a sequence is in ${\ell }_1^u\left(Z\right)$ if and only if it is unconditionally summable. We remark that the spaces ${\ell }_{q^{*}}^w\left(X^{*}\right)$ and ${\ell }_{p^{*}}^w\left(Y\right)$, respectively, in the definition of $(p,q)$-compact operator, can be replaced by ${\ell }_{q^{*}}^u\left(X^{*}\right)$ and ${\ell }_{p^{*}}^u\left(Y\right)$, respectively.

3. The ${\mathit{K}}_{\mathit{p},\mathit{q}}$-Compactness and ${\mathit{K}}_{\mathit{p},\mathit{q}}$-Null Sequences

Grothendieck [25] established a criterion of classical compactness. A subset K of a Banach space X is relatively compact if and only if for every $\epsilon >0$, there exists ${\left(x_n\right)}_n\in c_0\left(X\right)$ with $\parallel {\left(x_n\right)}_n{\parallel }_{\infty }\le (1+\epsilon ){sup}_{x\in K}\parallel x\parallel$ such that:

$$K\subset \left\{\sum _{n=1}^{\infty }{\alpha }_n{x}_n:{\left({\alpha }_n\right)}_n\in B_{{\ell }_1}\right\}.$$

The research by Sinha and Karn [21] was motivated from that criterion to introduce a new compactness. For $1\le p<\infty$, a subset K of a Banach space X is p-compact if there exists ${\left(x_n\right)}_n\in {\ell }_p\left(X\right)$ such that:

$$K\subset p-co{\left(x_n\right)}_n:=\left\{\sum _{n=1}^{\infty }{\alpha }_n{x}_n:\left({\alpha }_n\right)\in B_{{\ell }_{p^{*}}}\right\},$$

where ${\ell }_p\left(X\right)$ is the Banach space with the norm ${\parallel ·\parallel }_p$ of all X-valued absolutely p-summable sequences.

Galicer, Lassalle and Turco [12] introduced a notion to measure the size of p-compact sets. For a p-compact subset K of X:

$$m_p(K;X):=inf\{\parallel {\left(x_n\right)}_n{\parallel }_p:{\left(x_n\right)}_n\in {\ell }_p\left(X\right),K\subset p-co{\left(x_n\right)}_n\}.$$

According to ([4], Remarks 1.3 and 1.7), K is p-compact if and only if K is $K_{1,p}$-compact, in this case, $m_p(K;X)=m_{K_{1,p}}(K;X)$.

We say that a subset K of X is unconditionally p-compact [14] when ${\ell }_p\left(X\right)$ is replaced by ${\ell }_p^u\left(X\right)$ in the definition of p-compact set. It was shown in [15] (Proposition 5.2) that K is unconditionally p-compact if and only if K is $K_{p^{*},p}$-compact, in this case:

$$m_{K_{p^{*},p}}(K;X)=inf\{\parallel {\left(x_n\right)}_n{\parallel }_p^w:{\left(x_n\right)}_n\in {\ell }_p^u\left(X\right),K\subset p-co{\left(x_n\right)}_n\}.$$

We extend those results to the $K_{p,q}$-compactness.

Lemma 1

([15], Lemma 2.1). Let K be a collection of sequences of positive numbers.

If ${sup}_{{\left(k_j\right)}_j\in K}{\sum }_{j=1}^{\infty }{k}_j<\infty$ and ${lim}_{l\to \infty }{sup}_{{\left(k_j\right)}_j\in K}{\sum }_{j\ge l}{k}_j=0$, then for every $\epsilon >0$, there exists a sequence ${\left(b_j\right)}_j$ of real numbers with $b_j\nearrow \infty$ and $b_j>1$ for all j such that:

$$\underset{{\left(k_j\right)}_j\in K}{sup}\sum _{j=1}^{\infty }{k}_j{b}_j\le (1+\epsilon )\underset{{\left(k_j\right)}_j\in K}{sup}\sum _{j=1}^{\infty }{k}_{j}and\underset{l\to \infty }{lim}\underset{{\left(k_j\right)}_j\in K}{sup}\sum _{j\ge l}{k}_j{b}_j=0.$$

Theorem 1.

Let K be a subset of a Banach space X and let $1\le p,q,r\le \infty$ with $1/r=1/p+1/q-1$. Then, the following statements are equivalent:

(a) 

K is $K_{p,q}$-compact.

(b) 

There exist ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$ such that:

$$K\subset q\text{-}co{\left({\lambda }_n{x}_n\right)}_n.$$

(c) 

There exist a relatively compact $M\subset B_{{\ell }_{q^{*}}}$ ($B_{c_0}$ when q = 1) and an $S\in K_{p,q}({\ell }_{q^{*}},X)$ such that $K\subset S\left(M\right)$.

In this case:

$$m_{K_{p,q}}(K;X)$$

$$=inf\{\parallel {\left(x_n\right)}_n{\parallel }_{p^{*}}^w\parallel {\left({\lambda }_n\right)}_n{\parallel }_r:{\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right),{\left({\lambda }_n\right)}_n\in {\ell }_r,K\subset q-co{\left({\lambda }_n{x}_n\right)}_n\}$$

$$=inf\{\parallel S{\parallel }_{K_{p,q}}:relatively compactM\subset B_{{\ell }_{q^{*}}},S\in K_{p,q}({\ell }_{q^{*}},X),K\subset S\left(M\right)\}.$$

Proof. 

(a)⇒(b): Let $\epsilon >0$ be given. By (a), there exist a Banach space Z, $T\in K_{p,q}(Z,X)$ and a relatively compact subset M of $B_Z$ such that $K\subset T\left(M\right)$ and:

$${\parallel T\parallel }_{K_{p,q}}\le (1+\epsilon )m_{K_{p,q}}(K;X).$$

Since $T\in K_{p,q}(Z,X)$, there exist ${\left({\lambda }_n\right)}_n\in {\ell }_r$, ${\left(z_n^{*}\right)}_n\in {\ell }_{q^{*}}^u\left(Z^{*}\right)$ and ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ with $\parallel {\left({\lambda }_n\right)}_n{\parallel }_r\parallel {\left(z_n^{*}\right)}_n{\parallel }_{q^{*}}^w\parallel {\left(x_n\right)}_n{\parallel }_{p^{*}}^w\le (1+\epsilon ){\parallel T\parallel }_{K_{p,q}}$ such that:

$$T=\sum _{n=1}^{\infty }{\lambda }_n{z}_n^{*}\otimes x_n.$$

Then:

$$K\subset T\left(M\right)\subset T\left(B_Z\right)=\left\{\sum _{n=1}^{\infty }{\lambda }_n{z}_n^{*}\left(z\right)x_n:z\in B_Z\right\}\subset \mathit{q}\text{-}\mathit{co}({\lambda }_n\parallel {\left(z_k^{*}\right)}_k{{\parallel }_{q^{*}}^w{x}_n)}_n$$

and the infimum:

$${inf\parallel ·\parallel }_{p^{*}}^w{\parallel ·\parallel }_r\le \parallel {\left({\lambda }_n\right)}_n{\parallel }_r\parallel {\left(z_n^{*}\right)}_n{\parallel }_{q^{*}}^w\parallel {\left(x_n\right)}_n{\parallel }_{p^{*}}^w\le (1+\epsilon ){\parallel T\parallel }_{K_{p,q}}\le {(1+\epsilon )}^2{m}_{K_{p,q}}(K;X).$$

Since $\epsilon >0$ was arbitrary, ${inf\parallel ·\parallel }_{p^{*}}^w{\parallel ·\parallel }_r\le m_{K_{p,q}}(K;X)$.

(b)⇒(c): Let $\epsilon >0$ be given. By (b), there exist ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$ such that $K\subset q-co{\left({\lambda }_n{x}_n\right)}_n$ and $\parallel {\left(x_n\right)}_n{\parallel }_{p^{*}}^w\parallel {\left({\lambda }_n\right)}_n{\parallel }_r\le (1+\epsilon ){inf\parallel ·\parallel }_{p^{*}}^w{\parallel ·\parallel }_r$. By an application of Lemma 1, there exists a sequence ${\left(r_n\right)}_n$ of real numbers with ${lim}_{n\to \infty }{r}_n=\infty$ and $r_n>1$ such that ${\left(r_n{x}_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and $\parallel {\left(r_n{x}_n\right)}_n{\parallel }_{p^{*}}^w\le (1+\epsilon ){\parallel {\left(x_n\right)}_n\parallel }_{p^{*}}^w$. We see that the following set:

$$M:=\left\{{\left(\frac{{\alpha }_n}{r_n}\right)}_n:{\left({\alpha }_n\right)}_n\in B_{{\ell }_{q^{*}}}\right\}$$

is a relatively compact subset of $B_{{\ell }_{q^{*}}}$ and:

$$S:=\sum _{n=1}^{\infty }{\lambda }_n{e}_n\otimes r_n{x}_n\in K_{p,q}({\ell }_{q^{*}},X),$$

where each $e_n$ is the standard unit vector in ${\ell }_q$ ($c_0$ when $q=\infty$).

Now:

$$K\subset q\text{-}co{\left({\lambda }_n{x}_n\right)}_n=\left\{\sum _{n=1}^{\infty }{\lambda }_n{\beta }_n{r}_n{x}_n:{\left({\beta }_n\right)}_n\in M\right\}=S\left(M\right)$$

and:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {inf\parallel ·\parallel }_{K_{p,q}}& \le {\parallel S\parallel }_{K_{p,q}}\hfill \\ & \le \parallel {\left({\lambda }_n\right)}_n{\parallel }_r\parallel {\left(e_n\right)}_n{\parallel }_{q^{*}}^w{\parallel {\left(r_n{x}_n\right)}_n\parallel }_{p^{*}}^w\hfill \\ & \le (1+\epsilon )\parallel {\left({\lambda }_n\right)}_n{\parallel }_r{\parallel {\left(x_n\right)}_n\parallel }_{p^{*}}^w\hfill \\ & \le {(1+\epsilon )}^2{inf\parallel ·\parallel }_{p^{*}}^w{\parallel ·\parallel }_r.\hfill \end{array}\end{array}$$

Since $\epsilon >0$ was arbitrary, ${inf\parallel ·\parallel }_{K_{p,q}}\le {inf\parallel ·\parallel }_{p^{*}}^w{\parallel ·\parallel }_r$.

(c)⇒(a) is trivial and $m_{K_{p,q}}(K;X)\le inf{\parallel ·\parallel }_{K_{p,q}}$. □

Recall that $[K_{p,q}{,\parallel ·\parallel }_{K_{p,q}}{]}^{sur}=[{\mathcal{K}}_{K_{p,q}}{,\parallel ·\parallel }_{{\mathcal{K}}_{K_{p,q}}}]$. As an application of Theorem 1, we have:

Corollary 1 

(cf. [17], Theorem 2.2). Let X and Y be Banach spaces and let $1\le p,q,r\le \infty$ with $1/r=1/p+1/q-1$. Let $T:X\to Y$ be a linear map. Then the following statements are equivalent:

(a) 

$T\in K_{p,q}^{sur}(X,Y)$.

(b) 

There exist ${\left({\lambda }_n\right)}_n\in {\ell }_r$ and ${\left(y_n\right)}_n\in {\ell }_{p^{*}}^u\left(Y\right)$ such that:

$$T\left(B_X\right)\subset q\text{-}co\left({\lambda }_n{y}_n\right).$$

(c) 

There exist a relatively compact $M\subset B_{{\ell }_{q^{*}}}$ ($B_{c_0}$ when q = 1) and an $S\in K_{p,q}({\ell }_{q^{*}},Y)$, such that $T\left(B_X\right)\subset S\left(M\right)$.

In this case:

$${\parallel T\parallel }_{K_{p,q}^{sur}}$$

$$=inf\{\parallel {\left(y_n\right)}_n{\parallel }_{p^{*}}^w\parallel {\left({\lambda }_n\right)}_n{\parallel }_r:{\left(y_n\right)}_n\in {\ell }_{p^{*}}^u\left(Y\right),{\left({\lambda }_n\right)}_n\in {\ell }_r,T\left(B_X\right)\subset q\text{-}co{\left({\lambda }_n{y}_n\right)}_n\}$$

$$=inf\{\parallel S{\parallel }_{K_{p,q}}:relatively compactM\subset B_{{\ell }_{q^{*}}},S\in K_{p,q}({\ell }_{q^{*}},Y),T\left(B_X\right)\subset S\left(M\right)\}.$$

Remark 1. 

One may refer to [6] for a different generalization of p-compactness.

Piñeiro and Delgado [20] introduced an extended concept of null sequences. For $1\le p<\infty$, a sequence ${\left(x_n\right)}_n$ in a Banach space, X is called p-null if for every $\epsilon >0$, there exist a natural number $n_{\epsilon }$ and ${\left(z_k\right)}_k\in \epsilon B_{{\ell }_p\left(X\right)}$ such that:

$$x_n\in p\text{-}co{\left(z_k\right)}_k$$

for all $n\ge n_{\epsilon }$. It was shown in [4] (Proposition 1.5) that p-null sequences and $K_{1,p}$-null sequences coincide.

In [14], the unconditionally p-null sequence was introduced by replacing ${\ell }_p\left(X\right)$ with ${\ell }_p^u\left(X\right)$. Ain and Oja [7] introduced a more general concept of the p-null and unconditionally p-null sequences. They showed that unconditionally p-null sequences and $K_{p^{*},p}$-null sequences coincide, and p-null sequences and $K_{1,p}$-null sequences coincide in a more general setting. We consider the $K_{p,q}$-null sequences.

Theorem 2. 

Let ${\left(x_n\right)}_n$ be a sequence in a Banach space X and let $1\le p,q,r\le \infty$ with $1/r=1/p+1/q-1$. Then, the following statements are equivalent:

(a) 

${\left(x_n\right)}_n$ is a $K_{p,q}$-null sequence.

(b) 

There exist ${\left(z_k\right)}_k\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_k\right)}_k\in {\ell }_r$ such that for every $\epsilon >0$, there is an $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon q\text{-}co{\left({\lambda }_k{z}_k\right)}_k$$

for every $n\ge N$.

(c) 

There exists an $S\in K_{p,q}({\ell }_{q^{*}},X)$ such that for every $\epsilon >0$, there is an $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon S\left(B_{{\ell }_{q^{*}}}\right)$$

for every $n\ge N$.

Proof. 

(c)⇒(a) is trivial.

(a)⇒(b): By (a), there exist a Banach space Z and $T\in K_{p,q}(Z,X)\subset K_{p,q}^{sur}(Z,X)={\mathcal{K}}_{K_{p,q}}(Z,X)$ such that for every $\epsilon >0$, there is an $N\in \mathbb{N}$ such that $x_n\in \epsilon T\left(B_Z\right)$ for every $n\ge N$. Since $T\left(B_Z\right)$ is $K_{p,q}$-compact, by Theorem 1, there exist ${\left(y_k\right)}_k\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_k\right)}_k\in {\ell }_r$ such that:

$$T\left(B_Z\right)\subset q\text{-}co{\left({\lambda }_k{y}_k\right)}_k.$$

Hence, for every $\epsilon >0$, there is an $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon q\text{-}co{\left({\lambda }_k{y}_k\right)}_k$$

for every $n\ge N$.

(b)⇒(c): Let ${\left(z_k\right)}_k\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_k\right)}_k\in {\ell }_r$ be such that for every $\epsilon >0$, there is an $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon q\text{-}co{\left({\lambda }_k{z}_k\right)}_k$$

for every $n\ge N$. Consider the operator:

$$S:=\sum _{n=1}^{\infty }{\lambda }_k{e}_k\otimes z_k\in K_{p,q}({\ell }_{q^{*}},X),$$

where each $e_k$ is the standard unit vector in ${\ell }_q$. Then, for every $\epsilon >0$, there is an $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon q\text{-}co{\left({\lambda }_k{z}_k\right)}_k=\epsilon S\left(B_{{\ell }_{q^{*}}}\right)$$

for every $n\ge N$. □

The prototype of the following corollary is described in [4] (Proposition 1.5).

Corollary 2. 

Let ${\left(x_n\right)}_n$ be a sequence in a Banach space X and let $1\le p,q,r\le \infty$ with $1/r=1/p+1/q-1$. Then, the following statements are equivalent:

(a) 

${\left(x_n\right)}_n$ is a $K_{p,q}$-null sequence.

(b) 

For some $C>0$, for every $\epsilon >0$, there exist a Banach space Z, $T\in K_{p,q}(Z,X)$ with ${\parallel T\parallel }_{K_{p,q}}\le C$ and $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon T\left(B_Z\right)$$

for every $n\ge N$.

(c) 

For some $K>0$, for every $\epsilon >0$, there exist ${\left(z_k\right)}_k\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_k\right)}_k\in {\ell }_r$ with $\parallel {\left(z_k\right)}_k{\parallel }_{p^{*}}^w{\parallel {\left({\lambda }_k\right)}_k\parallel }_r\le K$, and $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon q\text{-}co{\left({\lambda }_k{z}_k\right)}_k$$

for every $n\ge N$.

Proof. 

(a)⇒(b) is trivial.

(b)⇒(c): Suppose that (b) holds. Since $T\left(B_Z\right)$ is $K_{p,q}$-compact, by Theorem 1, there exist ${\left(z_k\right)}_k\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_k\right)}_k\in {\ell }_r$ such that:

$$T\left(B_Z\right)\subset q\text{-}co{\left({\lambda }_k{z}_k\right)}_k$$

and:

$$\parallel {\left(z_k\right)}_k{\parallel }_{p^{*}}^w\parallel {\left({\lambda }_k\right)}_k{\parallel }_r\le m_{K_{p,q}}(T\left(B_Z\right);X)+1={\parallel T\parallel }_{K_{p,q}}+1\le C+1.$$

Hence, for every $\epsilon >0$, there is an $N\in \mathbb{N}$ such that:

$$x_n\in \epsilon q\text{-}co{\left({\lambda }_k{y}_k\right)}_k$$

for every $n\ge N$.

(c)⇒(a): By (c), for every $j\in \mathbb{N}$, there exist ${\left(z_k^j\right)}_k\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_k^j\right)}_k\in {\ell }_r$ with $\parallel {\left(z_k^j\right)}_k{\parallel }_{p^{*}}^w{\parallel {\left({\lambda }_k^j\right)}_k\parallel }_r\le K$, and $N_j\in \mathbb{N}$ with $N_j<N_{j+1}$ such that:

$$x_n\in \frac{1}{8^j}q\text{-}co{\left({\lambda }_k^j{z}_k^j\right)}_k$$

for every $n\ge N_j$. We may assume that $\parallel {\left({\lambda }_k^j\right)}_k{\parallel }_r=1$ and $\parallel {\left(z_k^j\right)}_k{\parallel }_{p^{*}}^w\le K$ for all j. The sequence ${\left(z_m\right)}_m$ is defined as the following array:

$$\begin{array}{ccccccc}\frac{1}{2}{z}_1^1& \to & \frac{1}{2}{z}_2^1& \frac{1}{2}{z}_3^1& \dots & \frac{1}{2}{z}_n^1& \dots \\ & & \downarrow & \downarrow & & \downarrow & \\ \frac{1}{2^2}{z}_1^2& \leftarrow & \frac{1}{2^2}{z}_2^2& \frac{1}{2^2}{z}_3^2& \dots & \frac{1}{2^2}{z}_n^2& \dots \\ & & & \downarrow & & \downarrow & \\ \frac{1}{2^3}{z}_1^3& \leftarrow & \frac{1}{2^3}{z}_2^3& \leftarrow \frac{1}{2^3}{z}_3^3& \dots & \frac{1}{2^3}{z}_n^3& \dots \\ & & & & ⋮& \downarrow & ⋮\\ \frac{1}{2^n}{z}_1^n& \leftarrow & \frac{1}{2^n}{z}_2^n& \dots & \frac{1}{2^n}{z}_{n-1}^n& \leftarrow \frac{1}{2^n}{z}_n^n& \dots \\ & & ⋮& ⋮& & & \end{array}$$

The sequence ${\left({\lambda }_m\right)}_m$ is defined as the following array:

$$\begin{array}{ccccccc}\frac{1}{2}{\lambda }_1^1& \to & \frac{1}{2}{\lambda }_2^1& \frac{1}{2}{\lambda }_3^1& \dots & \frac{1}{2}{\lambda }_n^1& \dots \\ & & \downarrow & \downarrow & & \downarrow & \\ \frac{1}{2^2}{\lambda }_1^2& \leftarrow & \frac{1}{2^2}{\lambda }_2^2& \frac{1}{2^2}{\lambda }_3^2& \dots & \frac{1}{2^2}{\lambda }_n^2& \dots \\ & & & \downarrow & & \downarrow & \\ \frac{1}{2^3}{\lambda }_1^3& \leftarrow & \frac{1}{2^3}{\lambda }_2^3& \leftarrow \frac{1}{2^3}{\lambda }_3^3& \dots & \frac{1}{2^3}{\lambda }_n^3& \dots \\ & & & & ⋮& \downarrow & ⋮\\ \frac{1}{2^n}{\lambda }_1^n& \leftarrow & \frac{1}{2^n}{\lambda }_2^n& \dots & \frac{1}{2^n}{\lambda }_{n-1}^n& \leftarrow \frac{1}{2^n}{\lambda }_n^n& \dots \\ & & ⋮& ⋮& & & \end{array}$$

A standard verification shows that ${\left(z_m\right)}_m\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_m\right)}_m\in {\ell }_r$.

Now, for every j:

$$q\text{-}co{\left(\frac{1}{2^j}{\lambda }_k^j\frac{1}{2^j}{z}_k^j\right)}_k\subset q\text{-}co{\left({\lambda }_m{z}_m\right)}_m.$$

Hence, for every j:

$$x_n\in \frac{1}{2^j}q\text{-}co{\left(\frac{1}{2^j}{\lambda }_k^j\frac{1}{2^j}{z}_k^j\right)}_k\subset \frac{1}{2^j}q\text{-}co{\left({\lambda }_m{z}_m\right)}_m$$

for every $n\ge N_j$. By Theorem 2, we have (a). □

4. The ${\mathcal{K}}_{{\mathit{K}}_{\mathit{p},\mathit{q}}}$-Approximation Property

A Banach space X is said to have the approximation property (AP) if:

$$i{d}_X\in {\overline{\mathcal{F}(X,X)}}^{{\tau }_c},$$

where $\mathcal{F}$ is the ideal of finite rank operators, $i{d}_X$ is the identity map on X and ${\tau }_c$ is the topology of uniform convergence on compact sets. Grothendieck [25] systematically investigated the AP and proved that X has the AP if and only if:

$$\mathcal{K}(Y,X)={\overline{\mathcal{F}(Y,X)}}^{\parallel ·\parallel }$$

for every Banach space Y.

Lassalle and Turco [4] introduced a more general notion of the AP. For a given Banach operator ideal $\mathcal{A}$, a Banach space X is said to have the ${\mathcal{K}}_{\mathcal{A}}$-AP if:

$${\mathcal{K}}_{\mathcal{A}}(Y,X)={\overline{\mathcal{F}(Y,X)}}^{{\parallel ·\parallel }_{{\mathcal{K}}_{\mathcal{A}}}}$$

for every Banach space Y. For $\mathcal{A}=K_{p,q}$, we consider this property, namely, the ${\mathcal{K}}_{K_{p,q}}$-AP. In [17] (Theorem 4.2), for every Banach space X and $1\le p,q\le \infty$ with $1/p+1/q\ge 1$, it was shown that if for every Banach space Y:

$$K_{p,p^{*}}^{sur}(Y,X)={\overline{\mathcal{F}(Y,X)}}^{{\parallel ·\parallel }_{K_{p,p^{*}}^{sur}}},$$

then for every Banach space Y:

$$K_{p,q}^{sur}(Y,X)={\overline{\mathcal{F}(Y,X)}}^{{\parallel ·\parallel }_{K_{p,q}^{sur}}}.$$

Recall that the identity in Section 2 is:

$$[K_{p,q}{,\parallel ·\parallel }_{K_{p,q}}{]}^{sur}=[{\mathcal{K}}_{K_{p,q}}{,\parallel ·\parallel }_{{\mathcal{K}}_{K_{p,q}}}]$$

Then, the result is equivalent to that if a Banach space X has the ${\mathcal{K}}_{K_{p,p^{*}}}$-AP; then X has the ${\mathcal{K}}_{K_{p,q}}$-AP. The proof of the result in [17] uses a factorization theorem for $K_{p,q}^{sur}$. We will give an alternative proof using a characterization of the ${\mathcal{K}}_{K_{p,q}}$-AP.

First, for given a Banach operator ideal $\mathcal{A}$, it was shown in [4] (Proposition 3.1) that X has the ${\mathcal{K}}_{\mathcal{A}}$-AP if and only if:

$$i{d}_X\in {\overline{\mathcal{F}(X,X)}}^{{\tau }_{s\mathcal{A}}}.$$

The topology ${\tau }_{s\mathcal{A}}$ of strong uniform convergence on $\mathcal{A}$-compact sets is given by the seminorms:

$$q_K\left(T\right)=m_{\mathcal{A}}(T\left(K\right);X),$$

where K ranges over all $\mathcal{A}$-compact subsets of X. Namely, for a net $\left(T_{\alpha }\right)$ in the space $\mathcal{L}(X,Y)$ of all operators from X to Y, $T_{\alpha }\stackrel{{\tau }_{s\mathcal{A}}}{⟶}0$ if and only if:

$$\underset{\alpha }{lim}{m}_{\mathcal{A}}(T_{\alpha }\left(K\right);Y)=0$$

for every $\mathcal{A}$-compact subset of X.

Let $1\le p,q,r\le \infty$ with $1/r=1/p+1/q-1$ and let ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$. Then, we can define a $(p,q)$-compact operator:

$$S_{\lambda x}:=\sum _{n=1}^{\infty }{e}_n\otimes {\lambda }_n{x}_n\in K_{p,q}({\ell }_{q^{*}},X),$$

where each $e_n$ is the standard unit vector in ${\ell }_q$. Then, we have:

Proposition 1.

Let X and Y be Banach spaces and let $1\le p,q,r\le \infty$ with $1/r=1/p+1/q-1$. Let $\left(T_{\alpha }\right)$ be a net in $\mathcal{L}(X,Y)$. Then $T_{\alpha }\stackrel{{\tau }_{s{K}_{p,q}}}{⟶}0$ if and only if:

$$\underset{\alpha }{lim}{\parallel S_{\lambda T_{\alpha }x}\parallel }_{{\mathcal{K}}_{K_{p,q}}({\ell }_{q^{*}},Y)}=0$$

for every ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$.

Proof. 

Suppose that $T_{\alpha }\stackrel{{\tau }_{s{K}_{p,q}}}{⟶}0$. Let ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$. By Theorem 1, $S_{\lambda x}\left(B_{{\ell }_{q^{*}}}\right)$ is a $K_{p,q}$-compact subset of X. Then, we have:

$$\underset{\alpha }{lim}{\parallel S_{\lambda T_{\alpha }x}\parallel }_{{\mathcal{K}}_{K_{p,q}}({\ell }_{q^{*}},Y)}=\underset{\alpha }{lim}{m}_{K_{p,q}}(S_{\lambda T_{\alpha }x}\left(B_{{\ell }_{q^{*}}}\right);Y)=\underset{\alpha }{lim}{m}_{K_{p,q}}(T_{\alpha }\left(S_{\lambda x}\left(B_{{\ell }_{q^{*}}}\right)\right);Y)=0.$$

Suppose that the other part holds. Let K be a $K_{p,q}$-compact subset of X. Then, by Theorem 1, there exist ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$ such that:

$$K\subset q\text{-}co{\left({\lambda }_n{x}_n\right)}_n.$$

We have that:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underset{\alpha }{lim}{m}_{K_{p,q}}(T_{\alpha }\left(K\right);Y)& \le \underset{\alpha }{lim}{m}_{K_{p,q}}(T_{\alpha }\left(q\text{-}co{\left({\lambda }_n{x}_n\right)}_n\right);Y)\hfill \\ & =\underset{\alpha }{lim}{m}_{K_{p,q}}(q\text{-}co{\left({\lambda }_n{T}_{\alpha }{x}_n\right)}_n;Y)\hfill \\ & =\underset{\alpha }{lim}{m}_{K_{p,q}}(S_{\lambda T_{\alpha }x}\left(B_{{\ell }_{q^{*}}}\right);Y)\hfill \\ & =\underset{\alpha }{lim}{\parallel S_{\lambda T_{\alpha }x}\parallel }_{{\mathcal{K}}_{K_{p,q}}({\ell }_{q^{*}},Y)}=0.\hfill \end{array}\end{array}$$

Hence, $T_{\alpha }\stackrel{{\tau }_{s{K}_{p,q}}}{⟶}0$. □

Corollary 3.

Let X be a Banach space and let $1\le p,q,r\le \infty$ with $1/r=1/p+1/q-1$. Then, X has the ${\mathcal{K}}_{K_{p,q}}$-AP if and only if:

$$\underset{\alpha }{lim}{\parallel S_{\lambda (T_{\alpha }-i{d}_X)x}\parallel }_{{\mathcal{K}}_{K_{p,q}}({\ell }_{q^{*}},X)}=0$$

for every ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$.

For $1\le p<\infty$, a linear map $T:Y\to X$ is called unconditionally p-compact if $T\left(B_Y\right)$ is an unconditionally p-compact subset of X. This notion was introduced in [14] and the space of all unconditionally p-compact operators from Y to X was denoted by ${\mathcal{K}}_{up}(Y,X)$. Furthermore, the norm on ${\mathcal{K}}_{up}(Y,X)$ was defined by:

$${\parallel T\parallel }_{{\mathcal{K}}_{up}}:=inf\{\parallel {\left(x_n\right)}_n{\parallel }_p^w:{\left(x_n\right)}_n\in {\ell }_p^u\left(X\right),T\left(B_Y\right)\subset p\text{-}co{\left(x_n\right)}_n\}.$$

Consequently, $[{\mathcal{K}}_{K_{p^{*},p}}{,\parallel ·\parallel }_{{\mathcal{K}}_{K_{p^{*},p}}}]=[{\mathcal{K}}_{up}{,\parallel ·\parallel }_{{\mathcal{K}}_{up}}]$.

It was shown in [13] (Theorem 2.3) that a Banach space X has the ${\mathcal{K}}_{K_{p,p^{*}}}$-AP if and only if there exists a net $\left(T_{\alpha }\right)$ in $\mathcal{F}(X,X)$ such that:

$$\underset{\alpha }{lim}{\parallel {\left((T_{\alpha }-i{d}_X)x_n\right)}_n\parallel }_{p^{*}}^w=0$$

for every ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$.

Theorem 3.

Let $1\le p,q\le \infty$ with $1/p+1/q\ge 1$. If a Banach space X has the ${\mathcal{K}}_{K_{p,p^{*}}}$-AP, then X has the ${\mathcal{K}}_{K_{p,q}}$-AP.

Proof. 

We use Corollary 3. Let $1\le r<\infty$ with $1/r=1/p+1/q-1$. Let ${\left(x_n\right)}_n\in {\ell }_{p^{*}}^u\left(X\right)$ and ${\left({\lambda }_n\right)}_n\in {\ell }_r$. Since X has the ${\mathcal{K}}_{K_{p,p^{*}}}$-AP, there exists a net $\left(T_{\alpha }\right)$ in $\mathcal{F}(X,X)$ such that:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underset{\alpha }{lim}{\parallel S_{\lambda (T_{\alpha }-i{d}_X)x}\parallel }_{{\mathcal{K}}_{K_{p,q}}({\ell }_{q^{*}},X)}& =\underset{\alpha }{lim}{m}_{K_{p,q}}(S_{\lambda (T_{\alpha }-i{d}_X)x}\left(B_{{\ell }_{q^{*}}}\right);X)\hfill \\ & =\underset{\alpha }{lim}{m}_{K_{p,q}}(q\text{-}co{\left({\lambda }_n(T_{\alpha }-i{d}_X)x_n\right)}_n;X)\hfill \\ & \le \underset{\alpha }{lim}\parallel {\left({\lambda }_n\right)}_n{\parallel }_r{\parallel {\left((T_{\alpha }-i{d}_X)x_n\right)}_n\parallel }_{p^{*}}^w=0.\hfill \end{array}\end{array}$$

Here, Theorem 1 is used in the last inequality. Thus, $T_{\alpha }\stackrel{{\tau }_{s{K}_{p,q}}}{⟶}i{d}_X$. Hence, X has the ${\mathcal{K}}_{K_{p,q}}$-AP. □

5. Discussion

This work continues the study of compactness, null sequences and the approximation property for operator ideals, and we expect that several more results on those subjects can be developed. We introduce one of the important subjects. For $1\le p<\infty$, it was shown in [5,13,16] that if the dual space $X^{*}$ of a Banach space X has the ${\mathcal{K}}_{K_{p^{*},p}}$-AP, then X has the ${\mathcal{K}}_{K_{1,p}}$-AP, and if $X^{*}$ has the ${\mathcal{K}}_{K_{1,p}}$-AP, then X has the ${\mathcal{K}}_{K_{p^{*},p}}$-AP. For $1\le p,q\le \infty$ with $1/p+1/q\ge 1$, we can consider the ${\mathcal{K}}_{K_{p,q}}$-AP. In view of Theorem 3, if $X^{*}$ has the ${\mathcal{K}}_{K_{1,p^{*}}}$-AP, then X has the ${\mathcal{K}}_{K_{p,q}}$-AP. We have the following question:

Question 1.

Is there any other condition of p and q for the${\mathcal{K}}_{K_{p,q}}$-AP of$X^{*}$to satisfy the${\mathcal{K}}_{K_{p,q}}$-AP of X?