On holomorphic mappings with compact type range

Using Mujica's linearization theorem, we extend to the holomorphic setting some classical characterizations of compact (weakly compact, Rosenthal, Asplund) linear operators between Banach spaces such as the Schauder, Gantmacher and Gantmacher-Nakamura theorems and the Davis-Figiel-Johnson-Pelczynski, Rosenthal and Asplund factorization theorems.


Introduction
From a local point of view, the properties of compactness, weak compactness, Rosenthal and Asplund for holomorphic mappings from U into F were addressed by R. M. Aron and M. Schottenloher [2], R. Ryan [24], M. Lindström [13] and N. Robertson [21], respectively. Let us recall that a mapping f : U → F is said to be locally compact (respectively, locally weakly compact, locally Rosenthal, locally Asplund) if every point x ∈ U has a neighborhood V x ⊆ U such that f (V x ) is relatively compact (respectively, relatively weakly compact, Rosenthal, Asplund) in F . Clearly, every mapping f : U → F having relatively compact range (respectively, relatively weakly compact range, Rosenthal range, Asplund range) is locally compact (respectively, locally weakly compact, locally Rosenthal, locally Asplund), however, the converse is not true in general for mappings f ∈ H ∞ (U, F ) (see Example 3.2 in [16] for the first two types of mappings).
We have organized this note as follows. In Section 1, we recall Mujica's linearization theorem and some of its consequences that will be needed to establish our results. With the aid of the notion of transpose mapping of a bounded holomorphic mapping, Section 2 is devoted to the analogues for bounded holomorphic mappings of the results due to Schauder, Gantmacher and Nakamura on the compactness and weak compactness of the adjoint of a bounded linear operator between Banach spaces. We suggest the reader to compare our results with Schauder and Gantmacher type theorems for holomorphic mappings of bounded type established by M. González and J. M. Gutiérrez [8,9,10] and R. Ryan [23,24].
As a main result, we show that some factorization theorems for bounded linear operators between Banach spaces can be extended to the holomorphic setting. This is the case of the Davis-Figiel-Johnson-Pe lczynski factorization theorem [5] which states that every weakly compact operator factors through a reflexive Banach space, the Rosenthal factorization theorem (see [1]) which asserts that every Rosenthal operator factors through a Banach space not containing ℓ 1 , and the Asplund factorization theorem which assures that every Asplund operator factors through an Asplund space (see [4,Theorem 5.3.5]).
We refer to the book of R. E. Megginson [14] for a complete study on weak topologies and linear operators on Banach spaces; to the monograph of J. Mujica [15] for the theory of holomorphic mappings on Banach spaces; and to the book of A. Pietsch [20] for the theory of operator ideals.
Notation. Through the paper, given a complex Banach space E, we denote by B E , • BE, S E and E * the closed unit ball, the open unit ball, the unit sphere and the dual space of E, respectively. For a set A ⊆ E, lin(A), lin(A), co(A), co(A), aco(A) and aco(A) stand for the linear hull, the norm-closed linear hull, the convex hull, the norm-closed convex hull, the absolutely convex hull, the norm-closed absolutely convex hull of A in E, respectively. If E and F are locally convex Hausdorff spaces, L(E; F ) denotes the vector space of all continuous linear operators from E into F . Unless stated otherwise, if E and F are Banach spaces, we will understand that they are endowed with the norm topology. Given T ∈ L(E; F ), T * : F * → E * denotes the adjoint operator of T .
We will sometimes use the following notation: for each x ∈ E and x * ∈ E * , x * , x is defined to be x * (x).

Preliminaries
Let U be an open subset of a complex Banach space E. We know that H ∞ (U ) is a Banach space under the supremum norm and it is actually a dual Banach space. In fact, there are different ways to construct a predual of H ∞ (U ). The most straightforward one is as the norm-closed linear subspace of H ∞ (U ) * generated by the functionals δ(x) ∈ H ∞ (U ) * with x ∈ U , defined by Theorem 2.1 in [16] justifies the following notation. For consistency with the preceding notation, we may consider the mapping δ U : In [16], Mujica established the following properties of G ∞ (U ) and δ U .
(2) For every complex Banach space F and every mapping f ∈ H ∞ (U, F ), there exists a unique operator As a consequence, we have (5) The closed unit ball of G ∞ (U ) coincides with the norm-closed absolutely convex hull of δ U (U ).
From Theorem 1.2 (4), we immediately deduce that, on bounded subsets of H ∞ (U ), the weak* topology agrees with the topology of pointwise convergence. ( Using Corollary 1.3, we obtain the following result. Proof. Let C f : H ∞ (V ) → H ∞ (U ) be the composition operator defined by .
From above we infer that and this implies that We finish this section with some results related to the transpose of a bounded holomorphic mapping.
Let U be an open subset of a complex Banach space E, let F be a complex Banach space and This justifies the following. Definition 1.5. Let U be an open subset of a complex Banach space E, let F be a complex Banach space and f ∈ H ∞ (U, F ). We will call the transpose mapping f t : Letting ε → 0, one obtains ||f t || ≥ f ∞ , as desired. Moreover, note that for all ϕ ∈ F * and x ∈ U , and since We have proved the following. Proposition 1.6. Let U be an open subset of a complex Banach space E, let F be a complex Banach space and let f ∈ H ∞ (U, F ).
We next see that the mapping f → f t identifies H ∞ (U, F ) with the subspace of L(F * ; H ∞ (U )) formed by all weak*-to-weak* continuous linear operators from F * into H ∞ (U ) (see [ , w * )) by Theorem 1.2 (2) and [14,Theorem 3.1.11]. Moreover, we have ||f t || = f ∞ by Proposition 1.6. It remains to show the surjectivity of the mapping in the statement. Take T ∈ L((F * , w * ); (H ∞ (U ), w * )). Then

Linearization of holomorphic mappings with compact type range
Let us recall that a bounded linear operator between Banach spaces T : A(E, F ) the linear spaces of bounded finite-rank linear operators, approximable linear operators (i.e., operators which are the norm limits of bounded finite-rank operators), compact linear operators, bounded separable linear operators, weakly compact linear operators, Rosenthal linear operators and Asplund linear operators from E into F , respectively. The following inclusions are known: Our aim is to study the following holomorphic variants of these concepts. If U is an open subset of a complex Banach space E and F is a complex Banach space, we will consider bounded holomorphic mappings f : U → F that have a range f (U ) ⊆ F satisfying an algebraic or topological property as, for instance, finite-dimensional range, relatively compact range, separable range, relatively weakly compact range, Rosenthal range or Asplund range.
Note that if T ∈ L(E, F ), then T is a compact (respectively, separable, weakly compact, Rosenthal, Asplund) linear operator if and only if the holomorphic mapping T | • BE has relatively compact (respectively, separable, relatively weakly compact, Rosenthal, Asplund) range.
The study of the connections between these compactness properties of a mapping f ∈ H ∞ (U, F ) and its corresponding associated operator T f ∈ L(G ∞ (U ); F ) was initiated by Mujica in Propositions 3.1 and 3.4 of [16]. Apparently, these results are the only known on this question and they have been included here with their proofs for the convenience of the reader. We have divided our study for the different types of holomorphic mappings considered.
2.1. Bounded finite-rank holomorphic mappings. Let us recall (see [16, p. 72]) that a mapping f : U → F has finite rank if lin(f (U )) is a finite dimensional subspace of F in which case this dimension is called the rank of f and denoted by rank(f ). Let H ∞ F (U, F ) denote the linear space of all bounded finite-rank holomorphic mappings from U to F . (1) f : U → F has finite rank.
(1) ⇔ (2) ([16, Proposition 3.1 (a)]): If f has finite rank, then lin(f (U )) is finite dimensional and therefore closed in F . We have = lin(T f (δ U (U ))) = lin(f (U )) = lin(f (U )) and hence T f has finite rank. Conversely, if T f has finite rank, then f has finite rank since Furthermore, in this case we have rank(f ) = rank(T f ).

2.2.
Holomorphic mappings with relatively compact range. We denote by H ∞ K (U, F ) the linear space of all holomorphic mappings from U to F that have relatively compact range. The equivalence (1) ⇔ (3) of the next result is a version for holomorphic mappings with relatively compact range of the classical Schauder's theorem on the relationship of the compactness of a bounded linear operator between Banach spaces and its adjoint. (1) f : U → F has relatively compact range. ( (4) f t : F * → H ∞ (U ) is bounded-weak*-to-norm continuous. , if T f ∈ K(G ∞ (U ), F ), then T f (δ U (U )) must be relatively compact in F , and therefore f ∈ H ∞ K (U, F ) since f (U ) = T f (δ U (U )) by Theorem 1.2 (2). Conversely, since B G ∞ (U ) = aco(δ U (U )) by Theorem 1.2 (5), one has T f (B G ∞ (U ) ) = T f (aco(δ U (U ))) ⊆ aco(T f (δ U (U ))) ⊆ aco(T f (δ U (U ))).
So, if f ∈ H ∞ K (U, F ), then T f (δ U (U )) is relatively compact in F , hence aco(T f (δ U (U ))) is compact in F by Mazur's compactness theorem ([14, Theorem 2.8.15]) and the fact that aco(A) = co(DA) for any subset A of a normed space E, where D denotes the closed unit disc of C. Therefore ; H ∞ (U )), by [14,Theorem 3.4.16], where bw * denotes the bounded weak* topology.

Proposition 2.3. Let U be an open subset of a complex
Banach space E and let F be a complex Banach space. The mapping f → f t is an isometric isomorphism from H ∞ K (U, F ) onto L((F * , bw * ); H ∞ (U )).

2.4.
Approximable bounded holomorphic mappings. We now enlarge the set of bounded finite-rank holomorphic mappings as follows. We say that a mapping f ∈ H ∞ (U, F ) is approximable if it is the limit in the uniform norm of a sequence of mappings of H ∞ F (U, F ). The set of such mappings will be denoted by H ∞ F (U, F ). Next, we see that every approximable mapping f ∈ H ∞ (U, F ) has relatively compact range.  [14,Corollary 3.4.9], and so f ∈ H ∞ K (U, F ) by Theorem 2.
2. An application of the principle of local reflexivity obtained by C. V. Hutton [11] shows that an operator T ∈ L(E; F ) can be approximated by bounded finite-rank linear operators from E into F if and only if T * can be approximated by bounded finite-rank linear operators from F * into E * . We now invoke Hutton's theorem to obtain the following. Theorem 2.6. Let U be an open subset of a complex Banach space E, let F be a complex Banach space and f ∈ H ∞ (U, F ). The following are equivalent: (1) f : U → F can be approximated by bounded finite-rank holomorphic mappings.
(2) T f : G ∞ (U ) → F can be approximated by bounded finite-rank linear operators.
Proof. We have  [5] asserts that any weakly compact linear operator between Banach spaces factors through a reflexive Banach space. We now extend this result to holomorphic mappings with relatively weakly compact range and give also the analogs of Gantmacher and Gantmacher-Nakamura theorems for such mappings.
We will denote by H ∞ W (U, F ) the linear space of all holomorphic mappings from U to F that have relatively weakly compact range. Clearly, (1) f : U → F has relatively weakly compact range.
(3) There exist a reflexive complex Banach space G, an operator T ∈ L(G; F ) and a mapping g ∈ H ∞ (U, G) such that f = T • g. (2) ⇒ (3): Applying the Davis-Figiel-Johnson-Pe lczynski theorem, there exist a reflexive complex Banach space G and operators T ∈ L(G; F ) and S ∈ L(G ∞ (U ); G) such that T f = T • S.
) is relatively weakly compact because T is weak-to-weak continuous by [14,Theorem 2.5.11] and g(U ) is relatively weakly compact in G since it is a bounded subset of the reflexive Banach space G (see [14,Theorem 2.8.2]).
We next identify H ∞ W (U, F ) with the subspace of L((F * , w * ); (G ∞ (U ), w * )) formed by all weak*to-weak continuous linear operators from F * into H ∞ (U ). Proposition 2.8. Let U be an open subset of a complex Banach space E and let F be a complex Banach space. The mapping f → f t is an isometric isomorphism from H ∞ W (U, F ) onto L((F * , w * ); (H ∞ (U ), w)).
Proof. In view of Theorem 2.7 and Proposition 1.6, we only need to show that the mapping in the statement is surjective. Let T ∈ L((F * , w * ); (H ∞ (U ), w)). Then J U •T ∈ L((F * , w * ); (G ∞ (U ) * , w)) by [14,Theorem 2.5.11], and this last set is contained in L((F * , w * ); (G ∞ (U ) * , w * )) since the weak topology is stronger than the weak* topology on the dual of a normed space. It follows that J U •T = S * for some S ∈ L(G ∞ (U ); F ) by [14,Theorem 3.1.11]. Hence S * ∈ L((F * , w * ); (G ∞ (U ) * , w)) and, by Gantmacher-Nakamura's theorem [14, Theorem 3.5.14], S ∈ W(G ∞ (U ), F ). Now, S = T f for some f ∈ H ∞ W (U, F ) by Theorem 1.2 (3) and Theorem 2.7. Finally, We now prove a similar result for holomorphic mappings with Rosenthal range, and the Rosenthal factorization theorem then allows us to factor these mappings through a Banach space not containing an isomorphic copy of ℓ 1 . We denote by H ∞ R (U, F ) the linear space of all bounded holomorphic mappings from U to F that have Rosenthal range. Clearly, H ∞ W (U, F ) is contained in H ∞ R (U, F ). Theorem 2.9. Let U be an open subset of a complex Banach space E, let F be a complex Banach space and let f ∈ H ∞ (U, F ). The following are equivalent: (1) f : U → F has Rosenthal range.
(3) There exist a complex Banach space G which does not contain an isomorphic copy of ℓ 1 , an operator T ∈ L(G; F ) and a mapping g ∈ H ∞ (U, G) such that f = T • g. Proof.
(1) ⇔ (2): It can be proved similarly as the same equivalence in Theorem 2.2 taking into account that the norm-closed absolutely convex hull of a Rosenthal subset of a Banach space is itself Rosenthal (see [13, p. 357]).

2.7.
Holomorphic mappings with Asplund range. By [4, Definition 5.1.2], a bounded set D ⊆ E is said to be Asplund or to have the Asplund property if every convex continuous function f : E → R is D-differentiable on a residual subset of E. A Banach space E is called an Asplund space if every convex continuous function f : E → R is Fréchet differentiable on a dense G δ set in E. This definition is due to E. Asplund [3] under the name strong differentiability space. We refer to R. D. Bourgin [4, Theorem 5.2.11] and R. R. Phelps [19,Theorem 2.34] for some equivalent formulations of the concept of Asplund set in a Banach space. In particular, we remark that the Asplund spaces are the Banach spaces for which each separable subspace has a separable dual. We will also use the paper [25] by C. Stegall for the properties of Asplund sets and Asplund operators in Banach spaces (see also Theorem 5.5.4 in [4]).