Home (p, q)-Compactness in spaces of holomorphic mappings
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(p, q)-Compactness in spaces of holomorphic mappings

  • Antonio Jiménez-Vargas EMAIL logo and David Ruiz-Casternado
Published/Copyright: May 18, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

Based on the concept of ( p , q ) -compact operator for p ∈ [ 1 , ∞ ] and q ∈ [ 1 , p * ] , we introduce and study the notion of ( p , q ) -compact holomorphic mapping between Banach spaces. We prove that the space formed by such mappings is a surjective p q ∕ ( p + q ) -Banach bounded-holomorphic ideal that can be generated by composition with the ideal of ( p , q ) -compact operators. In addition, we study Mujica’s linearization of such mappings, its relation with the ( u * v * + t v * + t u * ) ∕ t u * v * -Banach bounded-holomorphic composition ideal of the ( t , u , v ) -nuclear holomorphic mappings for t , u , v ∈ [ 1 , ∞ ] , its holomorphic transposition via the injective hull of the ideal of ( p , q * , 1 ) -nuclear operators, the Möbius invariance of ( p , q ) -compact holomorphic mappings on D , and its full compact factorization through a compact holomorphic mapping, a ( p , q ) -compact operator, and a compact operator.

Keywords: vector-valued holomorphic function; linearization; factorization theorems; (p, q)-compact operator; (p, q)-compact holomorphic mapping; (t, u, v)-nuclear holomorphic mapping
MSC 2010: 47B07; 46E15; 46E40; 47L20

1 Introduction

Let E and F be Banach spaces over K . As usual, B E stands for the closed unit ball of E , E * for the dual space of E , and ℒ ( E , F ) for the Banach space of all bounded linear operators from E into F , with the operator canonical norm. For a set A ⊆ E , lin ( A ) and abco ¯ ( A ) represent the linear hull and the norm-closed absolutely convex hull of A in E , respectively.

Given p ∈ [ 1 , ∞ ) , ℓ p ( E ) denotes the Banach space of all absolutely p -summable sequences ( x n ) in E endowed with the norm:

‖ ( x n ) ‖ p = ∑ n = 1 ∞ ‖ x n ‖ p 1 ∕ p ,

and c 0 ( E ) is the Banach space of all sequences in E converging to zero equipped with the norm:

‖ ( x n ) ‖ ∞ = sup { ‖ x n ‖ : n ∈ N } .

In the case of sequences in K , we will just write ℓ p and c 0 .

Let p ∈ [ 1 , ∞ ] and let p * denote the conjugate index of p defined by p * = p ∕ ( p − 1 ) if p ≠ 1 , p * = ∞ if p = 1 , and p * = 1 if p = ∞ .

For each p ∈ ( 1 , ∞ ) , the p-convex hull of a sequence ( x n ) ∈ ℓ p ( E ) is defined by

p - conv ( x n ) = ∑ n = 1 ∞ a n x n : ( a n ) ∈ B ℓ p * .

Moreover, the 1-convex hull of a sequence ( x n ) ∈ ℓ 1 ( E ) is given by

1 - conv ( x n ) = ∑ n = 1 ∞ a n x n : ( a n ) ∈ B c 0 ,

and the ∞ - convex hull of a sequence ( x n ) ∈ c 0 ( E ) by

∞ - conv ( x n ) = ∑ n = 1 ∞ a n x n : ( a n ) ∈ B ℓ 1 .

According to Grothendieck’s criterion of compactness [1], a set K ⊆ E is relatively compact if and only if there exists a sequence ( x n ) ∈ c 0 ( E ) such that K ⊆ abco ¯ ( { x n : n ∈ N } ) . Note that abco ¯ ( { x n : n ∈ N } ) is compact and coincides with ∞ - conv ( x n ) .

Motivated by this result, a stronger property of compactness in Banach spaces was introduced by Sinha and Karn [2] in 2002. Namely, given p ∈ [ 1 , ∞ ] , a set K ⊆ E is said to be relatively p-compact if there is a sequence ( x n ) ∈ ℓ p ( E ) (or ( x n ) ∈ c 0 ( E ) if p = ∞ ) such that K ⊆ p - conv ( x n ) . Lassalle and Turco [3] provided a measure of the p-compactness of K by defining

m p ( K ) = inf { ‖ ( x n ) ‖ p : ( x n ) ∈ ℓ p ( E ) , K ⊆ p - conv ( x n ) } , if 1 ≤ p < ∞ , inf { ‖ ( x n ) ‖ ∞ : ( x n ) ∈ c 0 ( E ) , K ⊆ ∞ - conv ( x n ) } , if p = ∞ .

An operator T ∈ ℒ ( E , F ) is said to be p-compact if T ( B E ) is a relatively p -compact subset of F . Theorem 4.2 in [2] states that the class of p -compact operators between Banach spaces K p is a Banach operator ideal equipped with the norm k p , where k p ( T ) = m p ( T ( B E ) ) for all T ∈ K p ( E , F ) .

The study of holomorphic mappings between Banach spaces with relatively compact range was initiated by Mujica [4] and continued by Sepulcre and the two authors of this work in [5]. The investigation on the p -compactness in the polynomial and holomorphic settings was addressed by Aron et al. [6], Aron et al. [7], and Jiménez-Vargas [8].

Influenced by these concepts, Ain et al. [9] introduced a more general property of compactness in 2012. Precisely, given p ∈ [ 1 , ∞ ] and q ∈ [ 1 , p * ] , a set K ⊆ E is said to be relatively (p,q)-compact if there exists a sequence ( x n ) ∈ ℓ p ( E ) ( ( x n ) ∈ c 0 ( E ) if p = ∞ ) such that K ⊆ q * - conv ( x n ) . As in the p -compact case, a measure of the (p,q)-compactness of K could be defined as

m ( p , q ) ( K ) = inf { ‖ ( x n ) ‖ p : ( x n ) ∈ ℓ p ( E ) , K ⊆ q * - conv ( x n ) } , if 1 ≤ p < ∞ , 1 ≤ q ≤ p * , inf { ‖ ( x n ) ‖ ∞ : ( x n ) ∈ c 0 ( E ) , K ⊆ ∞ - conv ( x n ) } , if p = ∞ , q = 1 .

Note that the ( ∞ , 1 ) -compactness coincides with the compactness in view of Grothendieck’s criterion, while the ( p , p * ) -compactness is precisely the p -compactness of Sinha and Karn [2].

In an analogous form, an operator T ∈ ℒ ( E , F ) is said to be ( p , q ) -compact if T ( B E ) is a relatively ( p , q ) -compact subset of F . Proposition 2.1 in [9] shows that the class of ( p , q ) -compact operators between Banach spaces, denoted K ( p , q ) , is an operator ideal. Furthermore, given s = p q ∕ ( p + q ) , [ K ( p , q ) , k ( p , q ) ] is an s -Banach operator ideal endowed with the s -norm k ( p , q ) ( T ) = m ( p , q ) ( T ( B E ) ) for all T ∈ K ( p , q ) ( E , F ) , which becomes a Banach operator ideal if and only if q = p * .

Since then, ( p , q ) -compact sets and ( p , q ) -compact operators between Banach spaces have been studied by various authors. For example, Ain and Oja [10,11] characterized relatively ( p , q ) -compact sets and studied ( p , q ) -null sequences, while Kim [12,13] investigated the injective and surjective hulls and an approximation property of the ideal of ( p , q ) -compact operators.

Our aim in this study is now to extend the property of ( p , q ) -compactness to the holomorphic setting as follows.

Let U be an open subset of a complex Banach space E and let F be a complex Banach space, p ∈ [ 1 , ∞ ] and q ∈ [ 1 , p * ] . Let ℋ ( U , F ) denote the linear space of all holomorphic mappings from U to F . A mapping f ∈ ℋ ( U , F ) is said to be ( p , q ) -compact if f ( U ) is a relatively ( p , q ) -compact subset of F . If ℋ K ( p , q ) ∞ ( U , F ) denotes the space of all ( p , q ) -compact holomorphic mappings from U into F , we define k ( p , q ) ℋ ∞ ( f ) = m ( p , q ) ( f ( U ) ) for every f ∈ ℋ K ( p , q ) ∞ ( U , F ) .

This article contains a complete study on ( p , q ) -compact holomorphic mappings. We now describe the content of this article. Let us recall that the space of all bounded holomorphic mappings from U into F , denoted ℋ ∞ ( U , F ) , is a Banach space endowed with the supremum norm. If G ∞ ( U ) is the canonical predual space of ℋ ∞ ( U ) ≔ ℋ ∞ ( U , C ) obtained by Mujica in [4], we prove that a mapping f ∈ ℋ ∞ ( U , F ) is ( p , q ) -compact if and only if its linearization T f : G ∞ ( U ) → F is a ( p , q ) -compact operator. This fact allows us to extend to the holomorphic setting some known results on ( p , q ) -compact operators. For instance, we prove that every mapping f ∈ ℋ K ( p , q ) ∞ ( U , F ) admits a factorization in the form f = T ∘ g , where G is a complex Banach space, g ∈ ℋ ∞ ( U , G ) , and T is a ( p , q ) -compact operator from G into F . Furthermore, k ( p , q ) ℋ ∞ ( f ) = inf { k ( p , q ) ( T ) ‖ g ‖ ∞ } , where the infimum is taken over all factorization of f as above.

We also prove that [ ℋ K ( p , q ) ∞ , k ( p , q ) ℋ ∞ ] is a surjective s -Banach bounded-holomorphic ideal, where s = p q ∕ ( p + q ) . This fact extends some results stated in [5,8,14] on spaces of holomorphic mappings with relatively compact or relatively p -compact range.

The notion of ( t , u , v ) -nuclear holomorphic mapping for t , u , v ∈ [ 1 , ∞ ] is introduced, and it is proved that the space formed by such mappings is an s -Banach bounded-holomorphic ideal, where s = 1 ∕ t + 1 ∕ u * + 1 ∕ v * . This allows us to ensure that every ( p , 1 , q * ) -nuclear holomorphic mapping is ( p , q ) -compact holomorphic. We provide three characterizations for ( p , q ) -compact holomorphic mappings: (1) as those bounded holomorphic mappings whose Mujica’s linearization is a ( p , q ) -compact operator; (2) as those that can be generated by composition with the ideal of ( p , q ) -compact operators; and (3) as those for which its holomorphic transpose belongs to the injective hull of the ideal of ( p , q * , 1 ) -nuclear operators.

The Möbius invariance of the class of ( p , q ) -holomorphic mappings defined on D (the complex open unit disc) is also addressed. Finally, we establish a general result of factorization for ( p , q ) -compact operators between Banach spaces, which extends some known results and permits us to characterize the members of ℋ K ( p , q ) ∞ ( U , F ) as those bounded holomorphic mappings that admit a full compact factorization through a compact holomorphic mapping, a ( p , q ) -compact operator, and a compact operator.

2 Results

We will start by collecting some results on holomorphic mappings due to Mujica [4] that will be applied throughout this article.

Theorem 2.1

[4] Let U be an open subset of a complex Banach space E. Let G ∞ ( U ) denote the norm-closed linear subspace of ℋ ∞ ( U ) * generated by the functionals δ ( x ) ∈ ℋ ∞ ( U ) * with x ∈ U , defined by δ ( x ) ( f ) = f ( x ) for all f ∈ ℋ ∞ ( U ) .

  1. The mapping g U : U → G ∞ ( U ) , defined by g U ( x ) = δ ( x ) for all x ∈ U , is holomorphic with ‖ δ ( x ) ‖ = 1 for all x ∈ U .

  2. For every complex Banach space F and every mapping f ∈ ℋ ∞ ( U , F ) , there exists a unique operator T f ∈ ℒ ( G ∞ ( U ) , F ) such that T f ∘ g U = f . Furthermore, ‖ T f ‖ = ‖ f ‖ ∞ .

  3. For every complex Banach space F, the mapping f ↦ T f is an isometric isomorphism from ℋ ∞ ( U , F ) onto ℒ ( G ∞ ( U ) , F ) .

  4. ℋ ∞ ( U ) is isometrically isomorphic to G ∞ ( U ) * , via J U : ℋ ∞ ( U ) → G ∞ ( U ) * given by J U ( f ) ( g U ( x ) ) = f ( x ) for all f ∈ ℋ ∞ ( U ) and x ∈ U .

  5. B G ∞ ( U ) coincides with abco ¯ ( g U ( U ) ) .

From now on, unless otherwise, E will denote a complex Banach space, U an open subset of E , and F a complex Banach space. The subspaces of ℒ ( E , F ) formed by compact operators, finite-rank bounded operators, and approximable operators from E into F will be denoted by K ( E , F ) , ℱ ( E , F ) , and ℱ ¯ ( E , F ) , respectively.

In view of the following result, we will only focus on the case ( p , q ) ∈ [ 1 , ∞ ) × [ 1 , p * ) . We must point out that the Banach space of all holomorphic mappings with relatively compact range from U into F , denoted ℋ K ∞ ( U , F ) , equipped with the supremum norm ∥ ⋅ ∥ ∞ was studied in [4,5,14], and that the Banach space of all holomorphic mappings with relatively p -compact range from U into F for p ∈ [ 1 , ∞ ] , denoted ℋ K p ∞ ( U , F ) , endowed with the norm k p ℋ ∞ was dealt in [8].

Proposition 2.2

The following statements are satisfied:

  1. ℋ K ( ∞ , 1 ) ∞ ( U , F ) = ℋ K ∞ ( U , F ) and k ( ∞ , 1 ) ℋ ∞ ( f ) = ‖ f ‖ ∞ , for all f ∈ ℋ K ( ∞ , 1 ) ∞ ( U , F ) .

  2. For p ∈ [ 1 , ∞ ) , ℋ K ( p , p * ) ∞ ( U , F ) = ℋ K p ∞ ( U , F ) , and k ( p , p * ) ℋ ∞ ( f ) = k p ℋ ∞ ( f ) , for all f ∈ ℋ K ( p , p * ) ∞ ( U , F ) .

Proof

( i ) Let f ∈ ℋ K ( ∞ , 1 ) ∞ ( U , F ) and let ( x n ) ∈ c 0 ( F ) be such that f ( U ) ⊆ ∞ - conv ( x n ) . Note that ∞ - conv ( x n ) is compact in F , and then, f ( U ) is relatively compact in F . Thus, f ∈ ℋ K ∞ ( U , F ) with ‖ f ‖ ∞ ≤ ‖ ( x n ) ‖ ∞ , and taking the infimum over all such sequences ( x n ) , we have ‖ f ‖ ∞ ≤ m ( ∞ , 1 ) ( f ( U ) ) = k ( ∞ , 1 ) ℋ ∞ ( f ) .

Conversely, let f ∈ ℋ K ∞ ( U , F ) . Then, f ( U ) is a relatively compact subset of F and, by the Grothendieck’s criterion of compactness, for every ε > 0 , there is a sequence ( x n ) ∈ c 0 ( F ) such that f ( U ) ⊆ ∞ - conv ( x n ) with ‖ ( x n ) ‖ ∞ ≤ ‖ f ‖ ∞ + ε . Hence, f ∈ ℋ K ( ∞ , 1 ) ∞ ( U , F ) with k ( ∞ , 1 ) ℋ ∞ ( f ) ≤ ‖ f ‖ ∞ .

( i i ) Let f ∈ ℋ K ( p , p * ) ∞ ( U , F ) and let ( x n ) ∈ ℓ p ( F ) be such that f ( U ) ⊆ p - conv ( x n ) . Hence, f ( U ) is a relatively p -compact subset of F , and thus, f ∈ ℋ K p ∞ ( U , F ) .

Conversely, let f ∈ ℋ K p ∞ ( U , F ) , i.e., f ( U ) is a relatively p -compact subset of F . Hence, there exists a sequence ( x n ) ∈ ℓ p ( F ) such that f ( U ) ⊆ p - conv ( x n ) . This allows us to assure that f ∈ ℋ K ( p , p * ) ∞ ( U , F ) .

Finally, note that k p ℋ ∞ ( f ) = m p ( f ( U ) ) = m ( p , p * ) ( f ( U ) ) = k ( p , p * ) ℋ ∞ ( f ) .□

2.1 Linearization and factorization

We next characterize ( p , q ) -compact holomorphic mappings f : U → F in terms of the ( p , q ) -compactness of its linearization T f ∈ ℒ ( G ∞ ( U ) , F ) .

Theorem 2.3

Let p ∈ [ 1 , ∞ ) , q ∈ [ 1 , p * ) , and f ∈ ℋ ∞ ( U , F ) . The following conditions are equivalent:

  1. f : U → F is a ( p , q ) -compact holomorphic mapping.

  2. T f : G ∞ ( U ) → F is a ( p , q ) -compact operator.

In this case, k ( p , q ) ℋ ∞ ( f ) = k ( p , q ) ( T f ) . Furthermore, the mapping f ↦ T f is an isometric isomorphism from ( ℋ K ( p , q ) ∞ ( U , F ) , k ( p , q ) ℋ ∞ ) onto ( K ( p , q ) ( G ∞ ( U ) , F ) , k ( p , q ) ) .

Proof

Applying Theorem 2.1, we obtain the following inclusions:

f ( U ) = T f ( g U ( U ) ) ⊆ T f ( abco ¯ ( g U ( U ) ) ) = T f ( B G ∞ ( U ) ) ⊆ abco ¯ ( T f ( g U ( U ) ) ) = abco ¯ ( f ( U ) ) .

( i ) ⇒ ( i i ) : Suppose that f ∈ ℋ K ( p , q ) ∞ ( U , F ) , and thus, f ( U ) is a relatively ( p , q ) -compact set in F . In light of [15, p. 1094], abco ¯ ( f ( U ) ) is also relatively ( p , q ) -compact and m ( p , q ) ( f ( U ) ) = m ( p , q ) ( abco ¯ ( f ( U ) ) ) . Hence, there exists a sequence ( x n ) ∈ ℓ p ( E ) so that abco ¯ ( f ( U ) ) ⊆ q * - conv ( x n ) . It follows that T f ( B G ∞ ( U ) ) ⊆ q * - conv ( x n ) by the second inclusion above. Hence, T f ∈ K ( p , q ) ( G ∞ ( U ) , F ) with

k ( p , q ) ( T f ) = m ( p , q ) ( T f ( B G ∞ ( U ) ) ) ≤ m ( p , q ) ( abco ¯ ( f ( U ) ) ) = m ( p , q ) ( f ( U ) ) = k ( p , q ) ℋ ∞ ( f ) .

( i i ) ⇒ ( i ) : Assume that T f ∈ K ( p , q ) ( G ∞ ( U ) , F ) . Then, T f ( B G ∞ ( U ) ) is relatively ( p , q ) -compact in F . By using the first inclusion above, f ( U ) is also relatively ( p , q ) -compact, i.e., f ∈ ℋ K ( p , q ) ∞ ( U , F ) , with

k ( p , q ) ℋ ∞ ( f ) = m ( p , q ) ( f ( U ) ) ≤ m ( p , q ) ( T f ( B G ∞ ( U ) ) ) = k ( p , q ) ( T f ) .

Combining assertion (iii) in Theorem 2.1 and what was proved above, we can ensure the last assertion of the result.□

Let 1 ≤ p ≤ r ≤ ∞ , 1 ≤ q ≤ p * , and 1 ≤ s ≤ r * . Assume that s ≤ q with

1 r + 1 s ≤ 1 p + 1 q .

Then, Corollary 3.3 in [9] states that K ( p , q ) ( G ∞ ( U ) , F ) ⊆ K ( r , s ) ( G ∞ ( U ) , F ) with k ( r , s ) ( f ) ≤ k ( p , q ) ( f ) for all f ∈ K ( p , q ) ( G ∞ ( U ) , F ) .

Taking into account this fact, Theorem 2.3 yields the following result on inclusions between ℋ K ( p , q ) ∞ -spaces.

Corollary 2.4

Let p , q , r , s be satisfying the same conditions as above. Then, ℋ K ( p , q ) ∞ ( U , F ) ⊆ ℋ K ( r , s ) ∞ ( U , F ) with k ( r , s ) ℋ ∞ ( f ) ≤ k ( p , q ) ℋ ∞ ( f ) for all f ∈ ℋ K ( p , q ) ∞ ( U , F ) .

As a consequence, we deduce [8, Corollary 1.10]: if 1 ≤ p ≤ q ≤ ∞ , then ℋ K p ∞ ( U , F ) ⊆ ℋ K q ∞ ( U , F ) with k q ℋ ∞ ( f ) ≤ k p ℋ ∞ ( f ) for all f ∈ ℋ K p ∞ ( U , F ) .

The combination of Theorem 2.3 and [14, Theorem 2.4] gives the following result about the factorization of ( p , q ) -compact holomorphic mappings.

Corollary 2.5

Let p ∈ [ 1 , ∞ ) , q ∈ [ 1 , p * ) , and f ∈ ℋ ∞ ( U , F ) . The following are equivalent:

  1. f : U → F is ( p , q ) -compact holomorphic.

  2. f = T ∘ g , for some complex Banach space G, g ∈ ℋ ∞ ( U , G ) and T ∈ K ( p , q ) ( G , F ) .

In this case, we have

k ( p , q ) ℋ ∞ ( f ) = ‖ f ‖ K ( p , q ) ∘ ℋ ∞ ≔ inf { k ( p , q ) ( T ) ‖ g ‖ ∞ } ,

where this infimum (in fact attained at T f ∘ g U ) is taken over all factorizations of f as above. Furthermore, the correspondence f ↦ T f is an isometric isomorphism from ( ℋ K ( p , q ) ∞ ( U , F ) , ‖ ⋅ ‖ K ( p , q ) ∘ ℋ ∞ ) onto ( K ( p , q ) ( G ∞ ( U ) , F ) , k ( p , q ) ) .

Let us recall (see [4, p. 72]) that a mapping f ∈ ℋ ∞ ( U , F ) has finite-rank if lin ( f ( U ) ) is a finite dimensional subspace of F . The set of all finite-rank bounded holomorphic mappings from U into F is denoted by ℋ ℱ ∞ ( U , F ) . In view of [5, Theorem 2.1] and Theorem 2.3, we can assure that ℋ ℱ ∞ ( U , F ) is a linear subspace of ℋ K ( p , q ) ∞ ( U , F ) . We next introduce a greater class of holomorphic mappings.

Definition 2.6

Let p ∈ [ 1 , ∞ ) and q ∈ [ 1 , p * ) . A mapping f ∈ ℋ ∞ ( U , F ) is said to be ( p , q ) -approximable if there exists a sequence ( f n ) in ℋ ℱ ∞ ( U , F ) such that k ( p , q ) ℋ ∞ ( f n − f ) → n → ∞ 0 . We denote by ℋ ℱ ¯ ( p , q ) ∞ ( U , F ) the space of all ( p , q ) -approximable bounded holomorphic maps from U to F .

Corollary 2.7

Let p ∈ [ 1 , ∞ ) and q ∈ [ 1 , p * ) . Every ( p , q ) -approximable holomorphic mapping is ( p , q ) -compact holomorphic.

Proof

Let f ∈ ℋ ℱ ¯ ( p , q ) ∞ ( U , F ) . Then, there is a sequence ( f n ) in ℋ ℱ ∞ ( U , F ) such that k ( p , q ) ℋ ∞ ( f n − f ) → 0 as n → ∞ . Since T f n ∈ ℱ ( G ∞ ( U ) , F ) by [5, Theorem 2.1], ℱ ( G ∞ ( U ) , F ) ⊆ K ( p , q ) ( G ∞ ( U ) , F ) by [9, Proposition 2.1] and k ( p , q ) ( T f n − T f ) = k ( p , q ) ( T f n − f ) = k ( p , q ) ℋ ∞ ( f n − f ) for all n ∈ N by Theorems 2.1 and 2.3, we obtain that T f ∈ K ( p , q ) ( G ∞ ( U ) , F ) because [ K ( p , q ) , k ( p , q ) ] is an s -Banach operator ideal [9, Proposition 2.1 and p. 151]. We conclude that f ∈ ℋ K ( p , q ) ∞ ( U , F ) by Theorem 2.3.□

2.2 s -Banach ideal property

By [14, Definition 2.1], a bounded-holomorphic ideal is a subclass ℐ ℋ ∞ of the class ℋ ∞ of all bounded holomorphic mappings such that for any complex Banach space E , any open subset U of E and any complex Banach space F , the components

ℐ ℋ ∞ ( U , F ) ≔ ℐ ℋ ∞ ∩ ℋ ∞ ( U , F )

satisfy the following three properties:

  1. ℐ ℋ ∞ ( U , F ) is a linear subspace of ℋ ∞ ( U , F ) .

  2. For any g ∈ ℋ ∞ ( U ) and y ∈ F , the map g ⋅ y : x ↦ g ( x ) y from U into F is in ℐ ℋ ∞ ( U , F ) .

  3. The ideal property: If H , G are complex Banach spaces, V is an open subset of H , h ∈ ℋ ( V , U ) , f ∈ ℐ ℋ ∞ ( U , F ) , and S ∈ ℒ ( F , G ) , then S ∘ f ∘ h ∈ ℐ ℋ ∞ ( V , G ) .

Given s ∈ ( 0 , 1 ] , let us recall that an s-norm on a linear space X over K is a function f : X → R satisfying that x = 0 whenever f ( x ) = 0 , f ( λ x ) = ∣ λ ∣ f ( x ) for all λ ∈ K and x ∈ X , and f ( x + y ) s ≤ f ( x ) s + f ( y ) s for all x , y ∈ X . We say that ( X , f ) is an s-normed space, and it is said that ( X , f ) is an s-Banach space if every Cauchy sequence in ( X , f ) converges in ( X , f ) .

Inspired by the notion of s -Banach operator ideal introduced by Pietsch [16, 6.2.2], we present the holomorphic analogue that extends the concept of (Banach) normed bounded-holomorphic ideal stated in [14, Definition 2.1]. A 1-Banach bounded-holomorphic ideal is simply a Banach bounded-holomorphic ideal.

Definition 2.8

Let s ∈ ( 0 , 1 ] . A bounded-holomorphic ideal ℐ ℋ ∞ is said to be s -normed ( s -Banach) if there exists a function ∥ ⋅ ∥ ℐ ℋ ∞ : ℐ ℋ ∞ → R 0 + such that for every complex Banach space E , every open subset U of E , and every complex Banach space F , the following three conditions are satisfied:

  1. ( ℐ ℋ ∞ ( U , F ) , ∥ ⋅ ∥ ℐ ℋ ∞ ) is an s -normed ( s -Banach) space with ∥ f ∥ ∞ ≤ ∥ f ∥ ℐ ℋ ∞ for all f ∈ ℐ ℋ ∞ ( U , F ) .

  2. ∥ g ⋅ y ∥ ℐ ℋ ∞ = ∥ g ∥ ∞ ∥ y ∥ for all g ∈ ℋ ∞ ( U ) and y ∈ F .

  3. If H , G are complex Banach spaces, V is an open subset of H , h ∈ ℋ ( V , U ) , f ∈ ℐ ℋ ∞ ( U , F ) , and S ∈ ℒ ( F , G ) , then ∥ S ∘ f ∘ h ∥ ℐ ℋ ∞ ≤ ∥ S ∥ ∥ f ∥ ℐ ℋ ∞ .

An s -normed bounded-holomorphic ideal [ ℐ ℋ ∞ , ∥ ⋅ ∥ ℐ ℋ ∞ ] is said to be:

  1. regular if for any f ∈ ℋ ∞ ( U , F ) , we have that f ∈ ℐ ℋ ∞ ( U , F ) with ‖ f ‖ ℐ ℋ ∞ = ‖ κ F ∘ f ‖ ℐ ℋ ∞ whenever κ F ∘ f ∈ ℐ ℋ ∞ ( U , F * * ) , where κ F denotes the canonical isometric linear embedding of F into F * * .

  2. surjective if for any mapping f ∈ ℋ ∞ ( U , F ) , any open subset V of a complex Banach space G , and any surjective mapping π ∈ ℋ ( V , U ) , we have that f ∈ ℐ ℋ ∞ ( U , F ) with ‖ f ‖ ℐ ℋ ∞ = ‖ f ∘ π ‖ ℐ ℋ ∞ whenever f ∘ π ∈ ℐ ℋ ∞ ( V , F ) .

We are now in a position to study the structure of ℋ K ( p , q ) ∞ as an s -Banach bounded holomorphic ideal.

Theorem 2.9

Let p ∈ [ 1 , ∞ ) , q ∈ [ 1 , p * ) , and s = p q ∕ ( p + q ) . Then, [ ℋ K ( p , q ) ∞ , k ( p , q ) ℋ ∞ ] is a surjective s-Banach bounded-holomorphic ideal. Furthermore, [ ℋ K ( p , q ) ∞ ( U , F ) , k ( p , q ) ℋ ∞ ] is regular for the class of reflexive Banach spaces F.

Proof

Note that [ ℋ K ( p , q ) ∞ , k ( p , q ) ℋ ∞ ] is a bounded-holomorphic ideal by Corollary 2.5 and [14, Corollary 2.5]. By [9, p. 151], [ K ( p , q ) , k ( p , q ) ] is an s -Banach space, and therefore, we deduce that [ ℋ K ( p , q ) ∞ , k ( p , q ) ℋ ∞ ] is also so by using the fact that both spaces are isometrically isomorphic by Corollary 2.5.

(S) Let f ∈ ℋ ∞ ( U , F ) and assume that f ∘ π ∈ ℋ K ( p , q ) ∞ ( V , F ) , where V is an open subset of a complex Banach space G and π ∈ ℋ ( V , U ) is surjective. Since f ( U ) = ( f ∘ π ) ( V ) , it follows immediately that f ∈ ℋ K ( p , q ) ∞ ( U , F ) with k ( p , q ) ℋ ∞ ( f ) = k ( p , q ) ℋ ∞ ( f ∘ π ) . Hence, [ ℋ K ( p , q ) ∞ , k ( p , q ) ℋ ∞ ] is surjective.

(R) Assume that F is reflexive, and thus, ℓ p ( F * * ) = κ F ( ℓ p ( F ) ) . Let f ∈ ℋ ∞ ( U , F ) and suppose that κ F ∘ f ∈ ℋ K ( p , q ) ∞ ( U , F * * ) . Consider a sequence ( x n ) ∈ ℓ p ( F ) such that ( κ F ∘ f ) ( U ) ⊆ q * - conv ( κ F ( x n ) ) . Thus, κ F ( f ( U ) ) ⊆ κ F ( q * - conv ( x n ) ) , and due to the injectivity of κ F , we have f ( U ) ⊆ q * - conv ( x n ) . Hence, f ∈ ℋ K ( p , q ) ∞ ( U , F ) with k ( p , q ) ℋ ∞ ( f ) ≤ ‖ ( x n ) ‖ p = ‖ ( κ F ( x n ) ) ‖ p and taking the infimum over all such sequences ( κ F ( x n ) ) , we obtain k ( p , q ) ℋ ∞ ( f ) ≤ k ( p , q ) ℋ ∞ ( κ F ∘ f ) . The converse inequality follows since [ ℋ K ( p , q ) ∞ , k ( p , q ) ℋ ∞ ] satisfies the condition (N3) in Definition 2.8.□

In view of Proposition 2.2, Theorem 2.9 extends Proposition 3.2 in [14] and Theorem 2.5 in [8] on the structure as a Banach bounded-holomorphic ideal of [ ℋ K ∞ , ∥ ⋅ ∥ ∞ ] and [ ℋ K p ∞ , k p ℋ ∞ ] , respectively.

2.3 Relation with ( t , u , v ) -nuclear holomorphic mappings

For p ∈ [ 1 , ∞ ] , ℓ p weak ( E ) denotes the Banach space of all weakly p -summable sequences in E , endowed with the norm:

‖ ( x n ) ‖ p weak = sup x * ∈ B E * ‖ ( x * ( x n ) ) ‖ p .

By [16, Definition 18.1.1], an operator T ∈ ℒ ( E , F ) is said to be ( t , u , v ) -nuclear (with t , u , v ∈ [ 1 , ∞ ] and 1 + 1 ∕ t ≥ 1 ∕ u + 1 ∕ v ) if

T = ∑ n = 1 ∞ λ n x n * ⋅ y n

in the norm topology of ℒ ( E , F ) , where ( λ n ) ∈ ℓ t , ( x n * ) ∈ ℓ v * weak ( E * ) and ( y n ) ∈ ℓ u * weak ( F ) . In the case t = ∞ , we put ( λ n ) ∈ c 0 . We will say that ∑ n ≥ 1 λ n x n * ⋅ y n is a ( t , u , v ) -nuclear representation of T. We denote by N ( t , u , v ) the space of all ( t , u , v ) -nuclear operators, and if we set 1 ∕ s = 1 ∕ t + 1 ∕ u * + 1 ∕ v * , then it becomes an s -Banach operator ideal under the norm:

ν ( t , u , v ) ( T ) ≔ inf { ‖ ( λ n ) ‖ t ‖ ( x n * ) ‖ v * weak ‖ ( y n ) ‖ u * weak } ,

where the infimum is taken over all ( t , u , v ) -nuclear representations of T (see [16, Theorem 18.1.2]).

In [16, Theorem 18.1.3], Pietsch proved that an operator T ∈ ℒ ( E , F ) is ( t , u , v ) -nuclear if and only if there exists a commutative diagram:

where S ∈ ℒ ( E , ℓ v * ) , R ∈ ℒ ( ℓ u , F ) , and M λ is the diagonal operator from ℓ v * into ℓ u given by M λ ( ( x n ) ) = ( λ n x n ) for all ( x n ) ∈ ℓ v * , where λ = ( λ n ) ∈ ℓ t if 1 ≤ t < ∞ and λ = ( λ n ) ∈ c 0 if t = ∞ . In this case,

ν ( t , u , v ) ( T ) = inf { ‖ R ‖ ‖ ( λ n ) ‖ t ‖ S ‖ } ,

and the infimum is taken over all possible factorizations of T .

This motivates the introduction of a new class of holomorphic mappings.

Definition 2.10

Let t , u , v ∈ [ 1 , ∞ ] such that 1 + 1 ∕ t ≥ 1 ∕ u + 1 ∕ v . A mapping f : U → F is said to be ( t , u , v ) -nuclear holomorphic if there exist an operator T ∈ ℒ ( ℓ u , F ) , a diagonal operator M λ ∈ ℒ ( ℓ v * , ℓ u ) induced by a sequence λ = ( λ n ) ∈ ℓ t ( λ ∈ c 0 if t = ∞ ), and a mapping g ∈ ℋ ∞ ( U , ℓ v * ) such that f = T ∘ M λ ∘ g , i.e., the following diagram commutes

The triple ( T , M λ , g ) is called a ( t , u , v ) -nuclear holomorphic factorization of f . We set

ν ( t , u , v ) ℋ ∞ ( f ) = inf { ‖ T ‖ ‖ M λ ‖ ‖ g ‖ ∞ } ,

where the infimum is taken over all such factorizations of f . We will denote by ℋ N ( t , u , v ) ∞ ( U , F ) the set of all ( t , u , v ) -nuclear holomorphic mappings from U into F .

Our next goal is to show that [ ℋ N ( t , u , v ) ∞ , ν ( t , u , v ) ℋ ∞ ] is an s -Banach bounded-holomorphic ideal. For it, we first study its linearization.

Theorem 2.11

Let t , u , v ∈ [ 1 , ∞ ] such that 1 + 1 ∕ t ≥ 1 ∕ u + 1 ∕ v , and let f ∈ ℋ ∞ ( U , F ) . The following conditions are equivalent:

  1. f : U → F is a ( t , u , v ) -nuclear holomorphic mapping.

  2. T f : G ∞ ( U ) → F is a ( t , u , v ) -nuclear operator.

In this case, ν ( t , u , v ) ℋ ∞ ( f ) = ν ( t , u , v ) ( T f ) . Furthermore, the mapping f ↦ T f is an isometric isomorphism from ( ℋ N ( t , u , v ) ∞ ( U , F ) , ν ( t , u , v ) ℋ ∞ ) onto ( N ( t , u , v ) ( G ∞ ( U ) , F ) , ν ( t , u , v ) ) .

Proof

( i ) ⇒ ( i i ) : Assume that f ∈ ℋ N ( t , u , v ) ∞ ( U , F ) . Then, f = T ∘ M λ ∘ g , where T ∈ ℒ ( ℓ u , F ) , ( λ n ) ∈ ℓ t (or c 0 if t = ∞ ) and g ∈ ℋ ∞ ( U , ℓ v * ) . Theorem 2.1 gives

T f ∘ g U = f = T ∘ M λ ∘ g = T ∘ M λ ∘ T g ∘ g U ,

where T g ∈ ℒ ( G ∞ ( U ) , ℓ v * ) . Hence, T f = T ∘ M λ ∘ T g . By [16, 18.1.3], it follows that T f ∈ N ( t , u , v ) ( G ∞ ( U ) , F ) with

ν ( t , u , v ) ( T f ) ≤ ‖ T ‖ ‖ M λ ‖ ‖ T g ‖ = ‖ T ‖ ‖ M λ ‖ ‖ g ‖ ∞ .

Taking the infimum over all such factorizations of f , we have ν ( t , u , v ) ( T f ) ≤ ν ( t , u , v ) ℋ ∞ ( f ) .

( i i ) ⇒ ( i ) : Suppose that T f ∈ N ( t , u , v ) ( G ∞ ( U ) , F ) and let ε > 0 . Again, by [16, 18.1.3], there exist ( λ n ) ∈ ℓ t (or c 0 if t = ∞ ), R ∈ ℒ ( G ∞ ( U ) , ℓ v * ) , and S ∈ ℒ ( ℓ u , F ) such that T f = S ∘ M λ ∘ R and

‖ S ‖ ‖ M λ ‖ ‖ R ‖ ≤ ( 1 + ε ) ν ( t , u , v ) ( T f ) .

Then, f = T f ∘ g U = S ∘ M λ ∘ R ∘ g U . Consider the mapping g : U → ℓ v * defined by

g ( x ) = ( R ∘ g U ) ( x ) ( x ∈ U ) .

It is clear that g ∈ ℋ ( U , ℓ v * ) and

‖ g ( x ) ‖ = ‖ ( R ∘ g U ) ( x ) ‖ ≤ ‖ R ‖ ‖ δ ( x ) ‖ = ‖ R ‖ ,

for all x ∈ U . Thus, g ∈ ℋ ∞ ( U , ℓ v * ) and f = S ∘ M λ ∘ g , i.e., f ∈ ℋ N ( t , u , v ) ∞ ( U , F ) with

ν ( t , u , v ) ℋ ∞ ( f ) ≤ ‖ S ‖ ‖ M λ ‖ ‖ g ‖ ∞ ≤ ‖ S ‖ ‖ M λ ‖ ‖ R ‖ ≤ ( 1 + ε ) ν ( t , u , v ) ( T f ) .

Just letting ε → 0 , we obtain ν ( t , u , v ) ℋ ∞ ( f ) ≤ ν ( t , u , v ) ( T f ) .

The last assertion can be shown using assertion (iii) in Theorem 2.1.□

Applying Theorem 2.11 and [14, Theorem 2.4], we obtain the following result on factorization of ( t , u , v ) -nuclear holomorphic mappings.

Corollary 2.12

Let t , u , v ∈ [ 1 , ∞ ] with 1 + 1 ∕ t ≥ 1 ∕ u + 1 ∕ v and f ∈ ℋ ∞ ( U , F ) . The following conditions are equivalent:

  1. f : U → F is ( t , u , v ) -nuclear holomorphic.

  2. f = T ∘ g , for some complex Banach space G , g ∈ ℋ ∞ ( U , G ) and T ∈ N ( t , u , v ) ( G , F ) .

In this case, we have

ν ( t , u , v ) ℋ ∞ ( f ) = ‖ f ‖ N ( t , u , v ) ∘ ℋ ∞ ≔ inf { ν ( t , u , v ) ( T ) ‖ g ‖ ∞ } ,

where the infimum is taken over all factorizations of f as in (ii), and this infimum is attained in T f ∘ g U . Furthermore, the mapping f ↦ T f is an isometric isomorphism from ( ℋ N ( t , u , v ) ∞ ( U , F ) , ‖ ⋅ ‖ N ( t , u , v ) ∘ ℋ ∞ ) onto ( N ( t , u , v ) ( G ∞ ( U ) , F ) , ν ( t , u , v ) ) .

Now, using [14, Corollary 2.5], the fact that [ N ( t , u , v ) , ν ( t , u , v ) ] is an s -Banach space (see [16, Theorem 18.1.2]) and Corollary 2.12, we arrive at the announced fact.

Corollary 2.13

Let t , u , v ∈ [ 1 , ∞ ] and 1 ∕ s = 1 ∕ t + 1 ∕ u * + 1 ∕ v * . Then, [ ℋ N ( t , u , v ) ∞ , ν ( t , u , v ) ℋ ∞ ] is an s-Banach bounded-holomorphic ideal.

The following relationship is easily obtained.

Corollary 2.14

Let p ∈ [ 1 , ∞ ) and q ∈ [ 1 , p * ) . Then, ℋ N ( p , 1 , q * ) ∞ ( U , F ) ⊆ ℋ K ( p , q ) ∞ ( U , F ) and k ( p , q ) ℋ ∞ ( f ) ≤ ν ( p , 1 , q * ) ℋ ∞ ( f ) for all f ∈ ℋ N ( p , 1 , q * ) ∞ ( U , F ) .

Proof

By Theorem 2.11, T f ∈ N ( p , 1 , q * ) ( G ∞ ( U ) , F ) with ν ( p , 1 , q * ) ( T f ) = ν ( p , 1 , q * ) ℋ ∞ ( f ) . By [9, Theorems 3.2 and 3.4], T f ∈ K ( p , q ) ( G ∞ ( U ) , F ) and k ( p , q ) ( T f ) ≤ ν ( p , 1 , q * ) ( T f ) . Hence, f ∈ ℋ K ( p , q ) ∞ ( U , F ) with

k ( p , q ) ℋ ∞ ( f ) = k ( p , q ) ( T f ) ≤ ν ( p , 1 , q * ) ( T f ) = ν ( p , 1 , q * ) ℋ ∞ ( f ) ,

by Theorem 2.3.□

Let us recall (see [8, Definition 1.18]) that the surjective hull of a bounded-holomorphic ideal ℐ ℋ ∞ is the smallest surjective ideal, which contains ℐ ℋ ∞ , and it is denoted by ( ℐ ℋ ∞ ) sur .

We have proven that ℋ K ( p , q ) ∞ is a surjective s -Banach bounded-holomorphic ideal, which contains ℋ N ( p , 1 , q * ) ∞ , and therefore, ( ℋ N ( p , 1 , q * ) ∞ ) sur ⊆ ℋ K ( p , q ) ∞ , but we do not know whether both sets are equal as it occurs in [9, Theorem 3.2], in the linear context.

2.4 Transposition

Let us recall that the transpose of a mapping f ∈ ℋ ∞ ( U , F ) is the operator f t ∈ ℒ ( F * , ℋ ∞ ( U ) ) defined by

f t ( y * ) = y * ∘ f ( y * ∈ F * ) .

Moreover, ‖ f t ‖ = ‖ f ‖ ∞ and f t = J U − 1 ∘ ( T f ) * (see [5, Proposition 1.6]).

It is well known by [9, Theorem 4.2] that the ideal K ( p , q ) is related by duality with the injective hull of the ideal of ( p , q * , 1 ) -nuclear operators. Let us recall that if ( A , ‖ ⋅ ‖ A ) is an s -Banach operator ideal, the components of the s -Banach operator ideal A inj , the injective hull of A , are defined as

A inj ( E , F ) = { T ∈ ℒ ( E , F ) : ι F ∘ T ∈ A ( E , ℓ ∞ ( B F * ) ) } ,

with ‖ T ‖ A inj = ‖ ι F ∘ T ‖ A for T ∈ A inj ( E , F ) , where ι F : F → ℓ ∞ ( B F * ) is the map defined by y ↦ ( y * ( y ) ) y * ∈ B F * .

Thus, we can characterize ( p , q ) -compact holomorphic mappings via their transposes.

Theorem 2.15

Let p ∈ [ 1 , ∞ ) and q ∈ [ 1 , p * ) . If f ∈ ℋ ∞ ( U , F ) , then f ∈ ℋ K ( p , q ) ∞ ( U , F ) if and only if f t ∈ N ( p , q * , 1 ) inj ( G ∞ ( U ) , F ) . In this case, k ( p , q ) ℋ ∞ ( f ) = ‖ f t ‖ N ( p , q * , 1 ) inj .

Proof

Applying Theorem 2.3, [9, Theorem 4.2], and [16, 8.4.2], we have

f ∈ ℋ K ( p , q ) ∞ ( U , F ) ⇔ T f ∈ K ( p , q ) ( G ∞ ( U ) , F ) ⇔ ( T f ) * ∈ N ( p , q * , 1 ) inj ( F * , G ∞ ( U ) * ) ⇔ f t = J U − 1 ∘ ( T f ) * ∈ N ( p , q * , 1 ) inj ( F * , ℋ ∞ ( U ) ) .

Furthermore, k ( p , q ) ℋ ∞ ( f ) = k ( p , q ) ( T f ) = ‖ ( T f ) * ‖ N ( p , q * , 1 ) inj = ‖ f t ‖ N ( p , q * , 1 ) inj .□

2.5 Möbius invariance

The Möbius group of D , denoted by Aut ( D ) , consists of all one-to-one holomorphic functions ϕ that map D onto itself. Each ϕ ∈ Aut ( D ) has the form ϕ = λ ϕ a , with λ ∈ T and a ∈ D , where

ϕ a ( z ) = a − z 1 − a ¯ z ( z ∈ D ) .

Given a complex Banach space F , let us recall (see [17]) that a linear space A ( D , F ) of holomorphic mappings from D into F , endowed with a seminorm p A , is said to be Möbius-invariant if for all ϕ ∈ Aut ( D ) and f ∈ A ( D , F ) , we have that f ∘ ϕ ∈ A ( D , F ) with p A ( f ∘ ϕ ) = p A ( f ) .

We obtain the following result closely related to the invariance by Möbius transformations of ℋ K ( p , q ) ∞ ( D , F ) .

Proposition 2.16

Let p ∈ [ 1 , ∞ ) and q ∈ [ 1 , p * ) . If f ∈ ℋ K ( p , q ) ∞ ( D , F ) and ϕ ∈ Aut ( D ) , then f ∘ ϕ ∈ ℋ K ( p , q ) ∞ ( D , F ) with k ( p , q ) ℋ ∞ ( f ∘ ϕ ) = k ( p , q ) ℋ ∞ ( f ) .

Proof

Let f ∈ ℋ K ( p , q ) ∞ ( D , F ) and ϕ ∈ Aut ( D ) . Note that there exists a sequence ( x n ) ∈ ℓ p ( F ) such that f ( D ) ⊆ q * - conv ( x n ) . Thus, due to ϕ maps D onto itself, we have that ( f ∘ ϕ ) ( D ) ⊆ q * - conv ( x n ) . Hence, f ∘ ϕ ∈ ℋ K ( p , q ) ∞ ( D , F ) with k ( p , q ) ℋ ∞ ( f ∘ ϕ ) ≤ k ( p , q ) ℋ ∞ ( f ) . Since ϕ − 1 ∈ Aut ( D ) , the previous proof yields the converse inequality k ( p , q ) ℋ ∞ ( f ) ≤ k ( p , q ) ℋ ∞ ( f ∘ ϕ ) .□

2.6 Full compact factorization

With the aim to present such a factorization theorem for ( p , q ) -compact holomorphic mappings, we will prove previously that an operator T ∈ ℒ ( E , F ) is ( p , q ) -compact if and only if it factors through two compact operators and a ( p , q ) -compact operator. This fact improves [18, Theorem 3.1] and [19, Proposition 2.9] but its proof is based on them. For it, we will first prove that a ( p , q ) -compact operator factors through a quotient space of ℓ 1 .

Theorem 2.17

Let E and F be Banach spaces, T ∈ ℒ ( E , F ) , p ∈ [ 1 , ∞ ) , and q ∈ [ 1 , p * ) . The following assertions are equivalent:

  1. T is ( p , q ) -compact.

  2. There exist a sequence y ∈ ℓ p ( F ) , operators T y ∈ K ( p , q ) ( ℓ q , F ) and R 0 ∈ ℒ ( E , ℓ q ∕ ker ( T y ) ) , a closed subspace M ⊆ ℓ 1 , and operators T 0 ∈ K ( p , q ) ( ℓ q ∕ ker ( T y ) , ℓ 1 ∕ M ) and S ∈ K ( ℓ 1 ∕ M , F ) such that T = S ∘ T 0 ∘ R 0 .

In this case, k ( p , q ) ( T ) = inf { ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R 0 ‖ } , where the infimum is taken over all factorizations of T as in ( i i ) .

Proof

( i ) ⇒ ( i i ) : Let T ∈ K ( p , q ) ( E , F ) . Then, given ε > 0 , there is a sequence y = ( y n ) ∈ ℓ p ( F ) with ∥ ( y n ) ∥ p ≤ k ( p , q ) ( T ) + ε such that

T ( B E ) ⊆ q * - conv ( y n ) ≔ ∑ n = 1 ∞ α n y n : ( α n ) ∈ B ℓ q .

Define the bounded linear operators T y : ℓ q → F and T y ^ : ℓ q ∕ ker ( T y ) → F by

T y ( α ) = ∑ n = 1 ∞ α n y n and T y ^ ( [ α ] ) = T y ( α ) ,

for all α = ( α n ) ∈ ℓ q , respectively. Clearly, T y ∈ K ( p , q ) ( ℓ q , F ) , and thus, T y ^ ∈ K ( p , q ) ( ℓ q ∕ ker ( T y ) , F ) with k ( p , q ) ( T y ^ ) ≤ ∥ ( y n ) ∥ p .

For each x ∈ E , there exists a sequence β = ( β n ) ∈ ℓ q such that T ( x ) = ∑ n = 1 ∞ β n y n . Hence, R 0 ( x ) = [ β ] defines a bounded linear operator from E into ℓ q ∕ ker ( T y ) with ∥ R 0 ∥ ≤ 1 . Clearly, T = T y ^ ∘ R 0 .

We can assume that y n ≠ 0 for all n ∈ N . Let ( β n ) be a sequence in B c 0 with β n > 0 for all n ∈ N such that ( y n ∕ β n ) ∈ ℓ p ( F ) with ∥ ( y n ∕ β n ) ∥ p ≤ ∥ ( y n ) ∥ p + ε . Taking λ = ( λ n ) = ( ∥ y n ∥ ∕ β n ) and z = ( z n ) = ( y n ∕ λ n ) , then λ ∈ ℓ p , z ∈ B c 0 ( F ) , and

T ( B E ) ⊆ ∑ n = 1 ∞ α n λ n z n : ( α n ) ∈ B ℓ q .

Define the bounded linear operators S 0 : ℓ 1 → F and S : ℓ 1 ∕ ker ( S 0 ) → F by

S 0 ( γ ) = ∑ n = 1 ∞ γ n z n and S ( [ γ ] ) = S 0 ( γ ) ,

for all γ = ( γ n ) ∈ ℓ 1 , respectively. Clearly, S ∈ K ( ℓ 1 ∕ ker ( S 0 ) , F ) with ∣ ∣ S ∣ ∣ ≤ ∥ ( z n ) ∥ ∞ ≤ 1 . Now, we can define the bounded linear operator T 0 : ℓ q ∕ ker ( T y ) → ℓ 1 ∕ ker ( S 0 ) by

T 0 ( [ α ] ) = [ ( λ n α n ) ] ,

for all α = ( α n ) ∈ ℓ q . Since ( λ n e n ) ∈ ℓ p ( ℓ 1 ) , where { e n : n ∈ N } is the canonical basis of ℓ 1 ; note that T 0 ∈ K ( p , q ) ( ℓ q ∕ ker ( T y ) , ℓ 1 ∕ ker ( S 0 ) ) with k ( p , q ) ( T 0 ) ≤ ∥ ( λ n e n ) ∥ p = ∥ ( λ n ) ∥ p . Plainly, T y ^ = S ∘ T 0 , and thus, T = S ∘ T 0 ∘ R 0 with ∣ ∣ S ∣ ∣ k ( p , q ) ( T 0 ) ∥ R 0 ∥ ≤ k ( p , q ) ( T ) + 2 ε . Since ε is arbitrary, it follows that inf { ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R 0 ‖ } ≤ k ( p , q ) ( T ) .

( i i ) ⇒ ( i ) : Assume that T = S ∘ T 0 ∘ R 0 with S , T 0 , R 0 being as in the statement. By the ideal property of K ( p , q ) , we have that T ∈ K ( p , q ) ( E , F ) with k ( p , q ) ( T ) ≤ ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R 0 ‖ , and taking the infimum over all such representations of T , we obtain that k ( p , q ) ( T ) ≤ inf { ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R 0 ‖ } .□

Although a great part of the demonstration of the following result is similar to that of Theorem 2.17, we provide its complete proof for the sake of completeness.

Theorem 2.18

Let E and F be the Banach spaces, T ∈ ℒ ( E , F ) , p ∈ [ 1 , ∞ ) , and q ∈ [ 1 , p * ) . The following conditions are equivalent:

  1. T is ( p , q ) -compact.

  2. There exist Banach spaces H and G, an operator T 0 ∈ K ( p , q ) ( H , G ) , and operators S ∈ K ( G , F ) and R ∈ K ( E , H ) such that T = S ∘ T 0 ∘ R .

In this case, k ( p , q ) ( T ) = inf { ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R ‖ } , where the infimum is taken over all factorizations of T as in ( i i ) .

Proof

( i ) ⇒ ( i i ) : Let T ∈ K ( p , q ) ( E , F ) . Given ε > 0 , we can take y = ( y n ) ∈ ℓ p ( F ) with ∥ ( y n ) ∥ p ≤ k ( p , q ) ( T ) + ε so that

T ( B E ) ⊆ q * - conv ( y n ) ≔ ∑ n = 1 ∞ α n y n : ( α n ) ∈ B ℓ q .

Assuming y n ≠ 0 for all n ∈ N , let ( β n ) be a sequence in B c 0 with β n > 0 for all n ∈ N such that ( y n ∕ β n ) ∈ ℓ p ( F ) with ∥ ( y n ∕ β n ) ∥ p ≤ ∥ ( y n ) ∥ p + ε . If λ = ( λ n ) = ( ∥ y n ∥ ∕ β n ) and z = ( z n ) = ( y n ∕ λ n ) , then λ ∈ ℓ p , z ∈ B c 0 ( F ) , and

T ( B E ) ⊆ ∑ n = 1 ∞ α n λ n z n : ( α n ) ∈ B ℓ q ⊆ ∑ n = 1 ∞ a n z n : ( a n ) ∈ L ,

where L ≔ ( a n ) ∈ B ℓ q : ∑ n = 1 ∞ ( ∣ a n ∣ q ∕ λ n q ) ≤ 1 is a compact set in B ℓ q .

Define the bounded linear operators T z : ℓ q → F and T z ^ : ℓ q ∕ ker ( T z ) → F by

T z ( a ) = ∑ n = 1 ∞ a n z n and T z ^ ( [ a ] ) = T z ( a ) ,

for all a = ( a n ) ∈ ℓ q , respectively. Clearly, T z ^ ∈ K ( p , q ) ( ℓ q ∕ ker ( T z ) , F ) with k ( p , q ) ( T z ^ ) ≤ ∥ ( z n ) ∥ p .

For each x ∈ B E , there exists a sequence a = ( a n ) ∈ L such that T ( x ) = ∑ n = 1 ∞ a n z n . Hence, Q z ( x ) = [ a ] defines a bounded linear operator from E into ℓ q ∕ ker ( T z ) with ∥ Q z ∥ ≤ 1 . Since Q z ( B E ) ⊆ π ( L ) , where π : ℓ q → ℓ q ∕ ker ( T z ) is the canonical projection, it follows that Q z ∈ K ( E , ℓ q ∕ ker ( T z ) ) . Clearly, T = T z ^ ∘ Q z , and the ideal property of K ( p , q ) yields

k ( p , q ) ( T ) ≤ k ( p , q ) ( T z ^ ) ∥ Q z ∥ ≤ k ( p , q ) ( T z ^ ) ≤ ∥ ( z n ) ∥ p ≤ ∥ ( y n ) ∥ p + ε ≤ k ( p , q ) ( T ) + 2 ε .

Now, applying Theorem 2.17 to T z ^ , we can find a y ∈ ℓ p ( F ) , T z , y ∈ K ( p , q ) ( ℓ q , F ) , and R 0 ∈ ℒ ( ℓ q ∕ ker ( T z ) , ℓ q ∕ ker ( T z , y ) ) , a closed subspace M ⊆ ℓ 1 , and T 0 ∈ K ( p , q ) ( ℓ q ∕ ker ( T z , y ) , ℓ 1 ∕ M ) and S ∈ K ( ℓ 1 ∕ M , F ) such that T z ^ = S ∘ T 0 ∘ R 0 with ∣ ∣ S ∣ ∣ k ( p , q ) ( T 0 ) ∥ R 0 ∥ ≤ k ( p , q ) ( T z ^ ) + 2 ε . Hence, T = S ∘ T 0 ∘ R , where R ≔ R 0 ∘ Q z ∈ K ( E , ℓ q ∕ ker ( T z , y ) ) , with ∣ ∣ S ∣ ∣ k ( p , q ) ( T 0 ) ∥ R ∥ ≤ k ( p , q ) ( T ) + 4 ε . Therefore, inf { ‖ S z ^ ‖ k ( p , q ) ( T 0 ) ‖ R ‖ } ≤ k ( p , q ) ( T ) .

( i i ) ⇒ ( i ) : Assume that T = S ∘ T 0 ∘ R with S , T 0 , R being as in the statement. By the ideal property of K ( p , q ) , we have that T ∈ K ( p , q ) ( E , F ) with k ( p , q ) ( T ) ≤ ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R ‖ , and taking the infimum over all such representations of T , we obtain k ( p , q ) ( T ) ≤ inf { ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R ‖ } .□

We are now able to provide a full compact factorization for ( p , q ) -compact holomorphic mappings.

Corollary 2.19

Let p ∈ [ 1 , ∞ ) , q ∈ [ 1 , p * ) , and let f ∈ ℋ ∞ ( U , F ) . The following conditions are equivalent:

  1. f : U → F is ( p , q ) -compact holomorphic.

  2. There exist complex Banach spaces H and G, an operator T 0 ∈ K ( p , q ) ( H , G ) , a mapping g ∈ ℋ K ∞ ( U , H ) , and an operator S ∈ K ( G , F ) such that f = S ∘ T 0 ∘ g .

In this case, k ( p , q ) ℋ ∞ ( f ) = inf { ‖ S ‖ k ( p , q ) ( T 0 ) ‖ g ‖ ∞ } , where the infimum is extended over all factorizations of f as in (ii).

Proof

( i ) ⇒ ( i i ) : Suppose that f ∈ ℋ K ( p , q ) ∞ ( U , F ) . By Theorem 2.3, T f ∈ K ( p , q ) ( G ∞ ( U ) , F ) with k ( p , q ) ( T f ) = k ( p , q ) ℋ ∞ ( f ) . Applying Theorem 2.18, for each ε > 0 , there exist complex Banach spaces H and G , an operator T 0 ∈ K ( p , q ) ( H , G ) , and operators S ∈ K ( G , F ) and R ∈ K ( G ∞ ( U ) , H ) such that T f = S ∘ T 0 ∘ R with ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R ‖ ≤ k ( p , q ) ( T f ) + ε . Note that R = T g with ‖ g ‖ ∞ = ‖ R ‖ for some g ∈ ℋ K ∞ ( U , H ) by [5, Corollary 2.11]. Hence, f = T f ∘ g U = S ∘ T 0 ∘ T g ∘ g U = S ∘ T 0 ∘ g , with

‖ S ‖ k ( p , q ) ( T 0 ) ‖ g ‖ ∞ = ‖ S ‖ k ( p , q ) ( T 0 ) ‖ R ‖ ≤ k ( p , q ) ( T f ) + ε = k ( p , q ) ℋ ∞ ( f ) + ε .

Just letting ε → 0 , we have ‖ S ‖ k ( p , q ) ( T 0 ) ‖ g ‖ ∞ ≤ k ( p , q ) ℋ ∞ ( f ) .

( i i ) ⇒ ( i ) : Suppose that f = S ∘ T 0 ∘ g is a factorization as in (ii). By the ideal property of K ( p , q ) , we have that S ∘ T 0 ∈ K ( p , q ) ( H , F ) and then, by Corollary 2.5, f ∈ ℋ K ( p , q ) ∞ ( U , F ) with

k ( p , q ) ℋ ∞ ( f ) ≤ k ( p , q ) ( S ∘ T 0 ) ‖ g ‖ ∞ ≤ ‖ S ‖ k ( p , q ) ( T 0 ) ‖ g ‖ ∞ .

Taking the infimum over all representations of f , we have k ( p , q ) ℋ ∞ ( f ) ≤ inf { ‖ S ‖ k ( p , q ) ( T 0 ) ‖ g ‖ ∞ } .□

  1. Funding information: The authors acknowledge the support of Junta de Andalucía grant FQM194, and of Ministerio de Ciencia e Innovación grant PID2021-122126NB-C31 funded by “ERDF A way of making Europe” and by MCIN/AEI/10.13039/501100011033.

  2. Author contributions: All the authors contributed equally to this work.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-12-02
Revised: 2024-02-07
Accepted: 2024-02-08
Published Online: 2024-05-18

© 2024 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/math-2023-0183
Keywords for this article
vector-valued holomorphic function; linearization; factorization theorems; (p, q)-compact operator; (p, q)-compact holomorphic mapping; (t, u, v)-nuclear holomorphic mapping
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  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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