Approximation properties of tensor norms and operator ideals for Banach spaces

Ju Myung Kim

doi:10.1515/math-2020-0116

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Approximation properties of tensor norms and operator ideals for Banach spaces

Ju Myung Kim

Published/Copyright: December 31, 2020

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$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

For a finitely generated tensor norm α , we investigate the α -approximation property ( α -AP) and the bounded α -approximation property (bounded α -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ -bounded α p , q -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) if it has the λ -bounded g p -AP. As a consequence, it follows that if a Banach space X has the λ -bounded g p -AP, then X has the λ -bounded w p -AP.

Keywords: approximation property; tensor norm; Banach operator ideal

MSC 2010: 46B28; 46B45; 47L20

1 Introduction

The main subjects of this paper originate from the classical approximation properties for Banach spaces, which was systematically investigated by Grothendieck [1]. A Banach space X is said to have the approximation property (AP) if

id X ∈ ℱ ( X , X ) ¯ τ c ,

where id X is the identity map on X, ℱ is the ideal of finite rank operators and τ c is the topology of uniform convergence on compact sets.

Let X and Y be Banach spaces. We denote by X ⊗ Y the algebraic tensor product of X and Y. The normed space X ⊗ Y equipped with a norm α is denoted by X ⊗ α Y and its completion is denoted by X ⊗ ^ α Y . The basic two norms on X ⊗ Y are the injective norm ε and the projective norm π which are defined as follows.

ε ( u ; X , Y ) ≔ sup ∑ j = 1 n x ∗ ( x j ) y ∗ ( y j ) : x ∗ ∈ B X ∗ , y ∗ ∈ B Y ∗ ,

where ∑ j = 1 n x j ⊗ y j is any representation of u and we denote by B Z the closed unit ball of a normed space Z.

π ( u ; X , Y ) ≔ inf ∑ j = 1 n ∥ x j ∥ ∥ y j ∥ : u = ∑ j = 1 n x j ⊗ y j , n ∈ ℕ .

It is well known that a Banach space X has the AP if and only if for every Banach space Y, the natural map

J π : Y ⊗ ˆ π X → Y ⊗ ˆ ε X

is injective (cf. [2, Theorem 5.6]). This equivalent statement can be naturally extended to tensor norms. For basic definitions and general background of the theory of tensor norms, we refer to [2,3]. For a finitely generated tensor norm α , a Banach space X is said to have the α -AP if for every Banach space Y, the natural map

J α : Y ⊗ ˆ α X → Y ⊗ ˆ ε X

is injective (cf. [2, Section 21.7]). It is well known that if a Banach space X has the AP, then it has the α -AP for every finitely generated tensor norm α (cf. [2, Proposition 21.7(1)]).

Some of the well-known tensor norms can be obtained from the tensor norm α p , q ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ), which was introduced by Lapresté [4]. For 1 ≤ p < ∞ , ℓ p w ( X ) stands for the Banach space of all X-valued weakly p-summable sequences endowed with the norm ∥ ⋅ ∥ p w . Let 1 ≤ r ≤ ∞ with 1 / r = 1 / p + 1 / q − 1 . For u ∈ X ⊗ Y , let

α p , q ( u ) ≔ inf ( λ j ) j = 1 n r ( x j ) j = 1 n q ∗ w ( y j ) j = 1 n p ∗ w : u = ∑ j = 1 n λ j x j ⊗ y j , n ∈ ℕ ,

where p ∗ is the conjugate index of p. Then α p , q is a finitely generated tensor norm and the transposed tensor norm α p , q t = α q , p (cf. [2, Proposition 12.5]). The special cases g p ≔ α p , 1 and d p ≔ α 1 , p are called the Chevet-Saphar tensor norms [5,6] and α 1 , 1 = π . The tensor norm w p ≔ α p , p ∗ is also well known. Díaz et al. [7] studied the α p , q -AP in terms of certain approximation properties of operator ideals. As a consequence, it was shown that a Banach space X has the α p , q -AP if it has the α p , 1 -AP.

Let λ ≥ 1 . A Banach space X is said to have the λ -bounded AP if

id X ∈ { S ∈ ℱ ( X , X ) : ∥ S ∥ ≤ λ } ¯ τ c .

It is well known that a Banach space X has the λ -bounded AP if and only if for every Banach space Y, the natural map

I π : Y ⊗ π X → ( Y ∗ ⊗ ε X ∗ ) ∗

satisfies π ( u ; Y , X ) ≤ λ I π ( u ) ( Y ∗ ⊗ ε X ∗ ) ∗ for every u ∈ Y ⊗ X (cf. [2, Corollary 16.3.2]). More generally, for a finitely generated tensor norm α , a Banach space X is said to have the λ -bounded α -AP if for every Banach space Y, the natural map

I α : Y ⊗ α X → ( Y ∗ ⊗ α ′ X ∗ ) ∗

satisfies α ( u ; Y , X ) ≤ λ I α ( u ) ( Y ∗ ⊗ α ′ X ∗ ) ∗ for every u ∈ Y ⊗ X (cf. [2, Section 21.7]), where α ′ is the dual tensor norm (cf. [2]) of α . Note that π ' = ε .

The main goal of this paper is to study the α -AP and the λ -bounded α -AP in terms of operator ideals. In Section 2, we extend the result of Díaz et al. [7], and in Section 3, we obtain some bounded versions of the results obtained in Section 2. As an application, it is shown that a Banach space X has the λ -bounded α p , q -AP if it has the λ -bounded α p , 1 -AP. Consequently, if X has the λ -bounded α p , 1 -AP, then X has the λ -bounded α p , p ∗ -AP.

2 The α-approximation property

We denote by [ ℒ , ∥ ⋅ ∥ ] the ideal of all operators and refer to [2,8,9,10] for operator ideals and their some information. A tensor norm α is said to be associated with a Banach operator ideal [ A , ∥ ⋅ ∥ A ] if the canonical map ( A ( M , N ) , ∥ ⋅ ∥ A ) → M ∗ ⊗ α N is an isometry for all finite-dimensional normed spaces M and N. Let X and Y be Banach spaces. For T ∈ ℒ ( X , Y ) , let

T A max ≔ sup q L Y T I M X : dim M , dim Y / L < ∞ ,

where I M X : M → X is the inclusion map and q L Y : Y → Y / L is the quotient map, and

A max ( X , Y ) ≔ { T ∈ ℒ ( X , Y ) : T A max < ∞ } .

We call [ A max , ∥ ⋅ ∥ A max ] the maximal hull of [ A , ∥ ⋅ ∥ A ] . If [ A , ∥ ⋅ ∥ A ] = [ A max , ∥ ⋅ ∥ A max ] , then [ A , ∥ ⋅ ∥ A ] is called maximal. If α is a finitely generated tensor norm, then its associated maximal Banach operator ideal is uniquely determined (cf. [2, Sections 17.1, 17.2 and 17.3]). For a finitely generated tensor norm α , the adjoint ideal [ A adj , ∥ ⋅ ∥ A adj ] is the maximal Banach operator ideal associated with the adjoint tensor norm α ∗ ≔ ( α ' ) t = ( α t ) ' .

Lemma 2.1

[2, Theorem 17.5] Let [ A , ∥ ⋅ ∥ A ] be the maximal Banach operator ideal associated with a finitely generated tensor norm α . Then for all Banach spaces X and Y, A ( X , Y ∗ ) is isometric to ( X ⊗ α ' Y ) ∗ and A ( X , Y ) is isometrically imbedded in ( X ⊗ α ' Y ∗ ) ∗ by the natural dual actions.

Let α be a finitely generated tensor norm. According to [2, Proposition 21.7(4)], a Banach space X has the α -AP if and only if for every Banach space Y, the natural map

J α : Y ∗ ⊗ ˆ α X → Y ∗ ⊗ ˆ ε X ↪ ℒ ( Y , X )

is injective.

Theorem 2.2

Let [ A , ∥ ⋅ ∥ A ] be the maximal Banach operator ideal associated with a finitely generated tensor norm α . Then the following statements are equivalent for a Banach space X.

X has the α -AP.
For every Banach space Y, ℱ ( X , Y ) is dense in A adj ( X , Y ) with the weak^∗ topology on ( X ⊗ ^ α t Y ∗ ) ∗ .
For every Banach space Y, ℱ ( X , Y ∗ ) is dense in A adj ( X , Y ∗ ) with the weak^∗ topology on ( X ⊗ ^ α t Y ) ∗ .

Proof

(a) ⇒ (b): Let Y be a Banach space. Since [ A adj , ∥ ⋅ ∥ A adj ] is associated with α ∗ and ( α ∗ ) ' = ( ( α t ) ' ) ' = α t , by Lemma 2.1, A adj ( X , Y ) is isometrically imbedded in ( X ⊗ ^ α t Y ∗ ) ∗ . Let T ∈ A adj ( X , Y ) . Suppose that T ∉ ℱ ( X , Y ) ¯ weak ∗ . Then by the separation theorem, there exists a u ∈ X ⊗ ^ α t Y ∗ such that for every S ∈ ℱ ( X , Y ) ,

〈 S , u 〉 = 0 but 〈 T , u 〉 ≠ 0 ,

where 〈 ⋅ , ⋅ 〉 is the dual action on ( X ⊗ ^ α t Y ∗ ) ∗ . We will show that u = 0 in X ⊗ ^ α t Y ∗ , which is a contradiction. Let

J α t : X ⊗ ˆ α t Y ∗ → X ⊗ ˆ ε Y ∗ ↪ ℒ ( X ∗ , Y ∗ )

be the natural map. To show that J α t u = 0 in X ⊗ ^ ε Y ∗ , let x ∗ ∈ X ∗ and y ∈ Y . For every v = ∑ k = 1 m x k ⊗ y k ∗ ∈ X ⊗ Y ∗ ,

x ∗ ⊗ y , v = ∑ k = 1 m x ∗ ( x k ) y k ∗ ( y ) = ( ( J α t v ) x ∗ ) ( y ) .

Let ( u n ) be a sequence in X ⊗ Y ∗ such that lim n → ∞ α t ( u n − u ) = 0 . Then

lim n → ∞ x ∗ ⊗ y , u n = x ∗ ⊗ y , u .

Since

| ( ( J α t u n ) x ∗ ) ( y ) − ( ( J α t u ) x ∗ ) ( y ) | ≤ ∥ x ∗ ∥ ∥ y ∥ ε ( J α t ( u n − u ) ; X , Y ∗ ) ≤ ∥ x ∗ ∥ ∥ y ∥ α t ( u n − u ) → 0

as n → ∞ , and for every n, 〈 x ∗ ⊗ y , u n 〉 = ( ( J α t u n ) x ∗ ) ( y ) ,

0 = x ∗ ⊗ y , u = ( ( J α t u ) x ∗ ) ( y ) .

Thus, J α t u = 0 in X ⊗ ^ ε Y ∗ .

The aforementioned argument also shows that

x ∗ ( ( J α u t ) y ) = ( ( J α t u ) x ∗ ) ( y )

for every x ∗ ∈ X ∗ and y ∈ Y , where J α : Y ∗ ⊗ ^ α X → Y ∗ ⊗ ^ ε X ↪ ℒ ( Y , X ) is the natural map. Consequently, J α u t = 0 in Y ∗ ⊗ ^ ε X . Since X has the α -AP, u t = 0 in Y ∗ ⊗ ^ α X and so u = 0 in X ⊗ ^ α t Y ∗ .

(b) ⇒ (c): Let Y be a Banach space. By Lemma 2.1, A adj ( X , Y ∗ ) is isometric to ( X ⊗ ^ α t Y ) ∗ . Let T ∈ A adj ( X , Y ∗ ) . Then by (b),

T ∈ ℱ ( X , Y ∗ ) ¯ weak ∗ on ( X ⊗ ^ α t Y ∗ ∗ ) ∗ .

Since the canonical imbedding from X ⊗ ^ α t Y to X ⊗ ^ α t Y ∗ ∗ is an isometry,

T ∈ ℱ ( X , Y ∗ ) ¯ weak ∗ on ( X ⊗ ^ α t Y ) ∗ .

J α : Y ⊗ ^ α X → Y ⊗ ^ ε X ↪ ℒ ( Y ∗ , X )

is injective. Assume that J α u = 0 in Y ⊗ ^ ε X . To show that u = 0 in Y ⊗ ^ α X , we will show that u t = 0 in X ⊗ ^ α t Y , that is, 〈 T , u t 〉 = 0 for every T ∈ A adj ( X , Y ∗ ) . Let T ∈ A adj ( X , Y ∗ ) be fixed. Since J α u = 0 in Y ⊗ ^ ε X , for every x ∗ ∈ X ∗ and y ∗ ∈ Y ∗ ,

y ∗ ( ( J α t u t ) x ∗ ) = x ∗ ( ( J α u ) y ∗ ) = 0 ,

where J α t : X ⊗ ^ α t Y → X ⊗ ^ ε Y ↪ ℒ ( X ∗ , Y ) is the natural map. As in the proof of (a) ⇒ (b), we see that

〈 x ∗ ⊗ y ∗ , u t 〉 = y ∗ ( ( J α t u t ) x ∗ ) = 0

for every x ∗ ∈ X ∗ and y ∗ ∈ Y ∗ , and so

〈 S , u t 〉 = 0

for every S ∈ ℱ ( X , Y ∗ ) . Since T ∈ ℱ ( X , Y ∗ ) ¯ weak ∗ on ( X ⊗ ^ α t Y ) ∗ , 〈 T , u t 〉 = 0 .□

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≤ 1 and let 1 ≤ r ≤ ∞ with 1 / p + 1 / q + 1 / r ∗ = 1 , where 1 / r + 1 / r ∗ = 1 . A linear map T : X → Y is called ( p , q ) -dominated if there exists a C > 0 such that

( y n ∗ ( T x n ) ) n r ≤ C ( x n ) n p w ( y n ∗ ) n q w

for every ( x n ) n ∈ ℓ p w ( X ) and ( y n ∗ ) n ∈ ℓ q w ( Y ∗ ) . We denote by D p , q ( X , Y ) the collection of all ( p , q ) -dominated operators from X to Y and for T ∈ D p , q ( X , Y ) , let ∥ T ∥ D p , q be the infimum C satisfying all such inequalities. Then [ D p , q , ∥ ⋅ ∥ D p , q ] is a Banach operator ideal (cf. [2, Section 19]). P p ≔ D p , ∞ is well known as the ideal of absolutely p-summing operators (cf. [2,8,9,10]) and D p ≔ D p , p ∗ is the ideal of p-dominated operators. For 1 / p + 1 / q ≥ 1 , let [ ℒ p , q , ∥ ⋅ ∥ ℒ p , q ] be the maximal Banach operator ideal associated with the tensor norm α p , q . ℒ p , q is well known as the ideal of ( p , q ) -factorable operators. Then

ℒ p , q , ⋅ ℒ p , q adj = D p ∗ , q ∗ , ⋅ D p ∗ , q ∗

(see [2, Section 17.12] and [9, Section 17.4]).

Theorem 2.2 applied to the tensor norm α p , q covers [7, Theorem 1].

Corollary 2.3

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 . The following statements are equivalent for a Banach space X.

X has the α p , q -AP.
For every Banach space Y, ℱ ( X , Y ) is dense in D p ∗ , q ∗ ( X , Y ) with the weak ^∗ topology on ( X ⊗ ^ α q , p Y ∗ ) ∗ .
For every Banach space Y, ℱ ( X , Y ∗ ) is dense in D p ∗ , q ∗ ( X , Y ∗ ) with the weak ^∗ topology on ( X ⊗ ^ α q , p Y ) ∗ .

Recall that a Banach space X has the AP if and only if X has the π -AP. Then the most special case of Corollary 2.3 is the following.

Corollary 2.4

The following statements are equivalent for a Banach space X

X has the AP.
For every Banach space Y, ℱ ( X , Y ) is dense in ℒ ( X , Y ) with the weak ^∗ topology on ( X ⊗ ^ π Y ∗ ) ∗ .
For every Banach space Y, ℱ ( X , Y ∗ ) is dense in ℒ ( X , Y ∗ ) with the weak ^∗ topology on ( X ⊗ ^ π Y ) ∗ .

Proof

It is well known that π is associated with the ideal ℐ of integral operators and ℐ adj = ℒ holds isometrically (cf. [2]). Since π t = π , we have the conclusion.□

Theorem 2.5

Let [ A , ∥ ⋅ ∥ A ] be the maximal Banach operator ideal associated with a finitely generated tensor norm α . Then a Banach space X has the α t -AP if and only if for every Banach space Y, ℱ ( Y , X ∗ ) is dense in A adj ( Y , X ∗ ) with the weak ^∗ topology on ( Y ⊗ ^ α t X ) ∗ .

Proof

Assume that X has the α t -AP. Let Y be a Banach space. By Lemma 2.1, A adj ( Y , X ∗ ) is isometric to ( Y ⊗ ^ α t X ) ∗ . Let T ∈ A adj ( Y , X ∗ ) . Suppose that T ∉ ℱ ( Y , X ∗ ) ¯ weak ∗ . Then there exists a u ∈ Y ⊗ ^ α t X such that for every S ∈ ℱ ( Y , X ∗ ) ,

S , u = 0 but T , u ≠ 0 .

Then as in the proof of Theorem 2.2, we can show that the natural map J α t : Y ⊗ ^ α t X → Y ⊗ ^ ε X is not injective. This contradicts the assumption that X has the α t -AP.

To show the converse, let Y be a Banach space. We want to show that the natural map

J α t : Y ⊗ ^ α t X → Y ⊗ ^ ε X ↪ ℒ ( Y ∗ , X )

is injective. Assume that J α t u = 0 in Y ⊗ ^ ε X . Let T ∈ A adj ( Y , X ∗ ) . Since J α t u = 0 in Y ⊗ ^ ε X , we see that for every y ∗ ∈ Y ∗ and x ∗ ∈ X ∗ ,

y ∗ ⊗ x ∗ , u = x ∗ ( ( J α t u ) y ∗ ) = 0 .

Thus, 〈 S , u 〉 = 0 for every S ∈ ℱ ( Y , X ∗ ) . Since T ∈ ℱ ( Y , X ∗ ) ¯ weak ∗ , 〈 T , u 〉 = 0 . Hence, u = 0 in Y ⊗ ^ α t X .□

Corollary 2.6

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 . Then a Banach space X has the α q , p -AP if and only if for every Banach space Y, ℱ ( Y , X ∗ ) is dense in D p ∗ , q ∗ ( Y , X ∗ ) with the weak ^∗ topology on ( Y ⊗ ^ α q , p X ) ∗ .

3 The bounded α -approximation property

Let α be a tensor norm and let X and Y be Banach spaces. Recall from [2, 12.4] that for every u ∈ X ⊗ Y , let

α → ( u ; X , Y ) ≔ inf { α ( u ; M , N ) : u ∈ M ⊗ N , dim M , dim N < ∞ }

and

α ← ( u ; X , Y ) ≔ sup { α ( ( q K X ⊗ q L Y ) ( u ) ; X / K , Y / L ) : dim X / K , dim Y / L < ∞ } .

It follows that α ← ≤ α ≤ α → . A tensor norm α is called totally accessible if α ← = α → .

From [2, Proposition 21.7(2)], a Banach space X has the λ -bounded α -AP if and only if for every Banach space Y,

α ( u ; Y , X ) ≤ λ α ← ( u ; Y , X )

for every u ∈ Y ⊗ X . Since α t ← = ( α ← ) t , it follows that a Banach space X has the λ -bounded α t -AP if and only if for every Banach space Y,

α ( u ; X , Y ) ≤ λ α ← ( u ; X , Y )

for every u ∈ X ⊗ Y .

Lemma 3.1

[2, Theorem 15.5] For all Banach spaces X and Y, and a tensor norm α , the natural maps

I α ← : X ⊗ α ← Y → ( X ∗ ⊗ α ' Y ∗ ) ∗ , I α ← : X ∗ ⊗ α ← Y → ( X ⊗ α ' Y ∗ ) ∗

are isometries.

The following lemma is a reformulation of [2, Lemma 16.2].

Lemma 3.2

Let α be a tensor norm and let X and Y be Banach spaces. Let λ ≥ 1 . Then α ≤ λ α ← on X ⊗ Y if and only if for every ϕ ∈ B ( X ⊗ α Y ) ∗ , there exists a net ( T η ) η in λ B X ∗ ⊗ α ' Y ∗ such that for every x ∈ X and y ∈ Y ,

lim η ( T η x ) ( y ) = 〈 ϕ , x ⊗ y 〉 .

Proof

Suppose that α ≤ λ α ← on X ⊗ Y . Let ϕ ∈ B ( X ⊗ α Y ) ∗ ⊂ ( X ⊗ α ← Y ) ∗ . By Lemma 3.1, we can choose a Hahn-Banach extension ϕ ˆ ∈ ( X ∗ ⊗ α ' Y ∗ ) ∗ ∗ of ϕ . By Goldstine’s theorem, there exists a net ( T η ) η in X ∗ ⊗ Y ∗ with α ′ ( T η ; X ∗ , Y ∗ ) ≤ ϕ ˆ ( X ∗ ⊗ α ′ Y ∗ ) ∗ ∗ such that

lim η f , T η = ϕ ˆ , f

for every f ∈ ( X ∗ ⊗ α ′ Y ∗ ) ∗ . Thus, for every x ∈ X and y ∈ Y ,

lim η ( T η x ) ( y ) = lim η 〈 x ⊗ y , T η 〉 = 〈 ϕ ˆ , x ⊗ y 〉 = 〈 ϕ , x ⊗ y 〉 .

Also, since

ϕ ˆ ( X ∗ ⊗ α ′ Y ∗ ) ∗ ∗ = ϕ ( X ⊗ α ← Y ) ∗ ≤ λ ϕ ( X ⊗ α Y ) ∗ ≤ λ ,

the net ( T η ) is in λ B X ∗ ⊗ α ′ Y ∗ .

To show the converse, let u = ∑ k = 1 m x k ⊗ y k ∈ X ⊗ Y . Then there exists ϕ ∈ B ( X ⊗ α Y ) ∗ such that α ( u ; X , Y ) = 〈 ϕ , u 〉 . By assumption, there exists a net ( T η ) in λ B X ∗ ⊗ α ′ Y ∗ such that

lim η 〈 u , T η 〉 = lim η ∑ k = 1 m ( T η x k ) ( y k ) = 〈 ϕ , u 〉 .

Hence,

α ( u ; X , Y ) = 〈 ϕ , u 〉 ≤ λ sup u , v : v ∈ B X ∗ ⊗ α ′ Y ∗ = λ u ( X ∗ ⊗ α ′ Y ∗ ) ∗ = λ α ← ( u ; X , Y ) . □

Lemma 3.3

[2, Proposition 21.8] Let [ A , ∥ ⋅ ∥ A ] be the maximal Banach operator ideal associated with a finitely generated tensor norm α and let λ ≥ 1 . Let X and Y be Banach spaces. Then α ≤ λ α ← on Y ∗ ⊗ X if and only if for every T ∈ B A adj ( X , Y ∗ ∗ ) , there exists a net ( T η ) η in λ B X ∗ ⊗ α ∗ Y such that for every x ∈ X and y ∗ ∈ Y ∗ ,

lim η y ∗ ( T η x ) = ( T x ) ( y ∗ ) .

We denote the strong operator topology and the weak operator topology on ℒ , respectively, by τ so and τ wo . For a net ( T α ) α in ℒ ( X , Y ∗ ) , we say that T α → 0 in the weak^∗ operator topology if

lim η ( T α x ) ( y ) → 0

for every x ∈ X and y ∈ Y . We denote the weak ∗ operator topology by τ w ∗ o .

Theorem 3.4

Let [ A , ∥ ⋅ ∥ A ] be the maximal Banach operator ideal associated with a finitely generated tensor norm α and let λ ≥ 1 . Then the following statements are equivalent for a Banach space X.

X has the λ -bounded α -AP.
For every Banach space Y and every T ∈ A adj ( X , Y ) ,
T ∈ { S ∈ ℱ ( X , Y ) : α ∗ ( S ; X ∗ , Y ) ≤ λ ∥ T ∥ A adj } ¯ τ so .
For every Banach space Y and every T ∈ A adj ( X , Y ∗ ) ,

T ∈ { S ∈ ℱ ( X , Y ∗ ) : α ∗ ( S ; X ∗ , Y ∗ ) ≤ λ ∥ T ∥ A adj } ¯ τ w ∗ o .

Proof

(b) ⇒ (c) is trivial.

(a) ⇒ (b): This proof is essentially due to [11, Theorem 4.1]. Let Y be a Banach space and let T ∈ A adj ( X , Y ) . Consider i Y T ∈ A adj ( X , Y ∗ ∗ ) , where i Y : Y → Y ∗ ∗ is the canonical isometry. Since X has the λ -bounded α -AP, by Lemma 3.3, there exists a net ( T η ) η in ℱ ( X , Y ) with α ∗ ( T η ; X ∗ , Y ) ≤ λ such that for every x ∈ X and y ∗ ∈ Y ∗ ,

lim η y ∗ ( i Y T A adj T η x ) = ( i Y T x ) ( y ∗ ) = y ∗ ( T x ) .

Since α ∗ ( i Y T A adj T η ; X ∗ , Y ) ≤ λ T A adj ,

T ∈ { S ∈ ℱ ( X , Y ) : α ∗ ( S ; X ∗ , Y ) ≤ λ ∥ T ∥ A adj } ¯ τ wo = { S ∈ ℱ ( X , Y ) : α ∗ ( S ; X ∗ , Y ) ≤ λ ∥ T ∥ A adj } ¯ τ so .

(c) ⇒ (a): Let Y be a Banach space. Since α t ← = ( α ← ) t , α ≤ λ α ← on Y ⊗ X if and only if α t ≤ λ α t ← on X ⊗ Y . So, in order to show that X has the λ -bounded α -AP, we will show that α t ≤ λ α t ← on X ⊗ Y using Lemma 3.2.

Now, let ϕ ∈ B ( X ⊗ α t Y ) ∗ . By Lemma 2.1, we can choose the representation T ϕ ∈ A adj ( X , Y ∗ ) of ϕ with ∥ T ϕ ∥ A adj ≤ 1 . Then by (c), there exists a net ( S η ) η in λ B X ∗ ⊗ α ∗ Y ∗ such that for every x ∈ X and y ∈ Y ,

lim η ( S η x ) ( y ) = ( T ϕ x ) ( y ) = ϕ , x ⊗ y . □

Hence by Lemma 3.2, we complete the proof.

Corollary 3.5

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 and let λ ≥ 1 . The following statements are equivalent for a Banach space X.

X has the λ -bounded α p , q -AP.
For every Banach space Y and every T ∈ D p ∗ , q ∗ ( X , Y ) ,
T ∈ S ∈ ℱ ( X , Y ) : S D p ∗ , q ∗ ≤ λ T D p ∗ , q ∗ ¯ τ so .
For every Banach space Y and every T ∈ D p ∗ , q ∗ ( X , Y ∗ ) ,

T ∈ S ∈ ℱ ( X , Y ∗ ) : ∥ S ∥ D p ∗ , q ∗ ≤ λ ∥ T ∥ D p ∗ , q ∗ ¯ τ w ∗ o .

Proof

If [ A , ∥ ⋅ ∥ A ] is the maximal Banach operator ideal associated with a totally accessible finitely generated tensor norm α , then by Lemmas 2.1 and 3.1, α = ∥ ⋅ ∥ A on ℱ . Since α p , q ∗ is totally accessible (cf. [2, Theorem 21.5]), by Theorem 3.4, we have the conclusion. The equivalence (a) ⇔ (b) is also a consequence of [11, Theorem 4.1].□

For the following result, we will need [11, Corollary 2.14], which can be reformulated as follows.

Lemma 3.6

Let 1 ≤ p ≤ ∞ and let λ ≥ 1 . The following statements are equivalent for a Banach space X.

For every Banach space Y and every T ∈ P p ( X , Y ) ,
T ∈ S ∈ ℱ ( X , Y ) : S P p ≤ λ T P p ¯ τ so .
For every Banach space Y and every T ∈ P p ( X , Y ) ,

id X ∈ S ∈ ℱ ( X , X ) : T S P p ≤ λ T P p ¯ τ so .

According to [11, Definition 1.2], for a Banach operator ideal A , a Banach space X is said to have the weak λ -BAP for A if for every Banach space Y and every T ∈ A ( X , Y ) ,

id X ∈ S ∈ ℱ ( X , X ) : T S A ≤ λ T A ¯ τ so .

Theorem 3.7

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 and let λ ≥ 1 . If a Banach space X has the λ -bounded g p -AP, then X has the weak λ -BAP for D p ∗ , q ∗ .

Proof

By Corollary 3.5 and Lemma 3.6, if X has the λ -bounded g p -AP, then for every Banach space Z and every T ∈ P p ∗ ( X , Z ) ,

id X ∈ S ∈ ℱ ( X , X ) : T S P p ∗ ≤ λ T P p ∗ ¯ τ so .

Now, let Y be a Banach space and let T ∈ D p ∗ , q ∗ ( X , Y ) . Let δ > 0 . Then by Kwapień’s factorization theorem (cf. [2, Theorem 19.3]), there exist a Banach space Z, R ∈ P p ∗ ( X , Z ) and U ∗ ∈ P q ∗ ( Y ∗ , Z ∗ ) with ∥ U ∗ ∥ P q ∗ ∥ R ∥ P p ∗ ≤ ( 1 + δ ) ∥ T ∥ D p ∗ , q ∗ such that the following diagram is commutative.

By the aforementioned statement, for every finite x 1 , . . . , x m ∈ X and every ε > 0 , there exists an S ∈ ℱ ( X , X ) with ∥ R S ∥ P p ∗ ≤ λ ∥ R ∥ P p ∗ such that

∥ S x i − x i ∥ ≤ ε

for every i = 1 , . . . , m . Since

T S D p ∗ , q ∗ ≤ U ∗ P q ∗ R S P p ∗ ≤ ( 1 + δ ) λ T D p ∗ , q ∗ ,

we have shown that for every δ > 0 ,

id X ∈ S ∈ ℱ ( X , X ) : T S D p ∗ , q ∗ ≤ ( 1 + δ ) λ T D p ∗ , q ∗ ¯ τ so .

Let x 1 , . . . , x m ∈ X and let ε > 0 . Choose a δ > 0 so that

( δ λ / ( 1 + δ ) λ ) max 1 ≤ k ≤ m x k ≤ ε / 2 .

Then, there exists an S ∈ { S ∈ ℱ ( X , X ) : ∥ T S ∥ D p ∗ , q ∗ ≤ ( 1 + δ ) λ ∥ T ∥ D p ∗ , q ∗ } such that for every i = 1 , . . . , m , ∥ S x i − x i ∥ ≤ ε / 2 . Consider

( λ / ( 1 + δ ) λ ) S ∈ S ∈ ℱ ( X , X ) : T S D p ∗ , q ∗ ≤ λ T D p ∗ , q ∗ .

Then for every i = 1 , . . . , m ,

λ ( 1 + δ ) λ S x i − x i ≤ λ (1 + δ ) λ S x i − x i + δ λ (1 + δ ) λ max 1 ≤ k ≤ m x k ≤ ε .

Hence, id X ∈ { S ∈ ℱ ( X , X ) : ∥ T S ∥ D p ∗ , q ∗ ≤ λ ∥ T ∥ D p ∗ , q ∗ } ¯ τ so .□

In [7, Proposition 2], it was shown that if a Banach space X has the g p -AP, then X has the α p , q -AP. From Theorem 3.7 and Corollary 3.5, we have:

Corollary 3.8

Let 1 ≤ p , q ≤ ∞ with 1/ p + 1/ q ≥ 1 and let λ ≥ 1 . If a Banach space X has the λ -bounded g p -AP, then X has the λ -bounded α p , q -AP.

Theorem 3.9

Let [ A , ∥ ⋅ ∥ A ] be the maximal Banach operator ideal associated with a finitely generated tensor norm α and let λ ≥ 1 . Then a Banach space X has the λ -bounded α t -AP if and only if for every Banach space Y and every T ∈ A adj ( Y , X ∗ ) ,

T ∈ S ∈ ℱ ( Y , X ∗ ) : α ∗ ( S ; Y ∗ , X ∗ ) ≤ λ ∥ T ∥ A adj ¯ τ w ∗ o .

Proof

Let

i : X ⊗ α Y → Y ⊗ α t X

be the isometry defined by i ( u ) = u t .

Suppose that X has the λ -bounded α t -AP. Let Y be a Banach space and let T ∈ A adj ( Y , X ∗ ) . By Lemma 2.1, we can choose the representation ϕ T ∈ ( Y ⊗ α t X ) ∗ of T. Consider ϕ T i ∈ ( X ⊗ α Y ) ∗ . Since X has the λ -bounded α t -AP, α ≤ λ α ← on X ⊗ Y . Then by Lemma 3.2, there exists a net ( u η ) η in λ B X ∗ ⊗ α ′ Y ∗ such that for every x ∈ X and y ∈ Y ,

lim η x ⊗ y , u η = ϕ T i ϕ T i ( X ⊗ α Y ) ∗ , x ⊗ y .

Let us consider the net ( ∥ ϕ T i ∥ ( X ⊗ α Y ) ∗ u η t ) η in Y ∗ ⊗ X ∗ = ℱ ( Y , X ∗ ) . Then

α ∗ ( ϕ T i ( X ⊗ α Y ) ∗ u η t ; Y ∗ , X ∗ ) = ϕ T i ( X ⊗ α Y ) ∗ α ' ( u η ; X ∗ , Y ∗ ) ≤ λ T A adj

and for every y ∈ Y and x ∈ X ,

lim η ( ϕ T i ( X ⊗ α Y ) ∗ u η t y ) ( x ) = lim η ϕ T i ( X ⊗ α Y ) ∗ x ⊗ y , u η = ϕ T i , x ⊗ y = ( T y ) ( x ) .

Hence, T ∈ S ∈ ℱ ( Y , X ∗ ) : α ∗ ( S ; Y ∗ , X ∗ ) ≤ λ T A adj ¯ τ w ∗ o .

To show the converse, we also use Lemma 3.2. Let Y be a Banach space and let ϕ ∈ B ( X ⊗ α Y ) ∗ . Consider ϕ i − 1 ∈ B ( Y ⊗ α t X ) ∗ . By Lemma 2.1, we can choose the representation T ϕ i − 1 ∈ A adj ( Y , X ∗ ) of ϕ i − 1 with ∥ T ϕ i − 1 ∥ A adj ≤ 1 . By assumption, there exists a net ( S η ) η in λ B Y ∗ ⊗ α ∗ X ∗ such that for every y ∈ Y and x ∈ X ,

lim η ( S η y ) ( x ) = ( T ϕ i − 1 y ) ( x ) .

Consider the net ( S η t ) η in X ∗ ⊗ Y ∗ . Then α ' ( S η t ; X ∗ , Y ∗ ) = α ∗ ( S η ; Y ∗ , X ∗ ) ≤ λ and for every x ∈ X and y ∈ Y ,

lim η ( S η t x ) ( y ) = lim η ( S η y ) ( x ) = ( T ϕ i − 1 y ) ( x ) = ϕ i − 1 , y ⊗ x = ϕ , x ⊗ y .

Thus by Lemma 3.2, α ≤ λ α ← on X ⊗ Y . Hence, X has the λ -bounded α t -AP.□

From Theorem 3.9, we have:

Corollary 3.10

Let 1 ≤ p , q ≤ ∞ with 1 / p + 1 / q ≥ 1 and let λ ≥ 1 . A Banach space X has the λ -bounded α q , p -AP if and only if for every Banach space Y and every T ∈ D p ∗ , q ∗ ( Y , X ∗ ) ,

T ∈ S ∈ ℱ ( Y , X ∗ ) : S D p ∗ , q ∗ ≤ λ T D p ∗ , q ∗ ¯ τ w ∗ o .

4 Open problems

The following question is a well-known problem (cf. [2, Section 21.12]).

Problem 1

Is the tensor norm w p ( 1 < p < ∞ , p ≠ 2 ) totally accessible?

Since a finitely generated tensor norm α is totally accessible if and only if every Banach space has the 1-bounded α -AP, the problem can be reformulated as follows.

Problem 1

Does every Banach space have the 1-bounded w p -AP ( 1 < p < ∞ , p ≠ 2 ) ?

According to Corollaries 3.5 and 3.10, a Banach space X has the 1-bounded w p -AP if and only if for every Banach space Y and every T ∈ D p ∗ ( X , Y ) ,

T ∈ S ∈ ℱ ( X , Y ) : S D p ∗ ≤ T D p ∗ ¯ τ so

if and only if for every Banach space Y and every T ∈ D p ( Y , X ∗ ) ,

T ∈ S ∈ ℱ ( Y , X ∗ ) : S D p ≤ T D p ¯ τ w ∗ o .

Therefore, the problem can be reformulated as follows.

Problem 1

Let 1 < p < ∞ , p ≠ 2 . For all Banach spaces X, Y and every T ∈ D p ∗ ( X , Y ) ,

T ∈ S ∈ ℱ ( X , Y ) : S D p ∗ ≤ T D p ∗ ¯ τ so ?

Or, for every T ∈ D p ( Y , X ∗ ) ,

T ∈ S ∈ ℱ ( Y , X ∗ ) : S D p ≤ T D p ¯ τ w ∗ o ?

Reinov [12] constructed Banach spaces failing to have the g p -AP and the d p -AP ( 1 ≤ p ≤ ∞ , p ≠ 2 ) . From [7, Proposition 2], if a Banach space has the g p -AP, then it has the w p -AP. It is not known whether every Banach space has the w p -AP ( 1 < p < ∞ , p ≠ 2 ) . According to Corollaries 2.3 and 2.6, a Banach space X has the w p -AP if and only if ℱ ( X , Y ) is dense in D p ∗ ( X , Y ) with the weak ^∗ topology on ( X ⊗ ^ w p ∗ Y ∗ ) ∗ for every Banach space Y if and only if ℱ ( Y , X ∗ ) is dense in D p ( Y , X ∗ ) with the w e a k ∗ topology on ( Y ⊗ ^ w p X ) ∗ for every Banach space Y. We ask:

Problem 2

Let 1 < p < ∞ , p ≠ 2 . For all Banach spaces X and Y, is the space ℱ ( X , Y ) dense in D p ∗ ( X , Y ) with the weak ∗ topology on ( X ⊗ ^ w p ∗ Y ∗ ) ∗ ?

Or, is the space ℱ ( Y , X ∗ ) dense in D p ( Y , X ∗ ) with the w e a k ∗ topology on ( Y ⊗ ^ w p X ) ∗ ?

Acknowledgments

The author would like to thank the referees for valuable comments. This work was supported by the National Research Foundation of Korea (NRF-2018R1D1A1B07043566).

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Received: 2019-06-12

Revised: 2020-11-17

Accepted: 2020-11-17

Published Online: 2020-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2020-0116

Keywords for this article

approximation property; tensor norm; Banach operator ideal

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