p -SUMMING BLOCH MAPPINGS ON THE COMPLEX UNIT DISC

A bstract . The notion of p -summing Bloch mapping from the complex unit open disc D into a complex Banach space X is introduced for any 1 ≤ p ≤ ∞ . It is shown that the linear space of such mappings, equipped with a natural seminorm π B p , is M¨obius-invariant. Moreover, its subspace consisting of all those mappings which preserve the zero is an injective Banach ideal of normalized Bloch mappings. Bloch versions of the Pietsch’s domination / factorization Theorem and the Maurey’s extrapolation Theorem are presented. We also introduce the spaces of X -valued Bloch molecules on D and identify the spaces of normalized p -summing Bloch mappings from D into X ∗ under the norm π B p with the duals of such spaces of molecules under the Bloch version of the p ∗ -Chevet–Saphar tensor norms d p ∗ .


Introduction
The known concept of absolutely p-summing operator between Banach spaces, introduced by Grothendieck [10] for p = 1 and by Pietsch [18] for any p > 0, can be adapted to address the property of summability in the setting of Bloch mappings from the complex unit open disc D into a complex Banach space X as follows.
The study of summability has been addressed for different classes of mappings by some authors.For example, for multilinear operators by Achour and Mezrag [1] and Dimant [8], for Lipschitz mappings by Farmer and Johnson [9] and Saadi [20], and for holomorphic mappings by Matos [12] and Pellegrino [15], among other settings.See also the survey by Pellegrino, Rueda and Sánchez-Pérez [16] for the summability on multilinear operators and homogeneous polynomials.
If H(D, X) denotes the space of all holomorphic mappings from D into X, let us recall that a mapping f ∈ H(D, X) is called Bloch if there exists a constant c ≥ 0 such that (1 − |z| 2 ) f ′ (z) ≤ c for all z ∈ D.
The Bloch space B(D, X) is the linear space of all those mappings f ∈ H(D, X) such that equipped with the Bloch seminorm p B .The normalized Bloch space B(D, X) is the Banach space of all Bloch mappings from D into X such that f (0) = 0, equipped with the Bloch norm p B .In particular, we will write B(D) instead of B(D, C).We refer the reader to [2,21] for the scalarvalued theory, and to [4,5] for the vector-valued theory on these spaces.
For any 1 ≤ p ≤ ∞, we say that a mapping f ∈ H(D, X) is p-summing Bloch if there is a constant c ≥ 0 such that for any n ∈ N, λ 1 , . . ., λ n ∈ C and z 1 , . . ., z n ∈ D, we have The infimum of all the constants c for which such an inequality holds, denoted π B p ( f ), defines a seminorm on the linear space, denoted Π B p (D, X), of all p-summing Bloch mappings f : D → X.Furthermore, this seminorm becomes a norm on the subspace Π B p (D, X) consisting of all those mappings f ∈ Π B p (D, X) so that f (0) = 0.These spaces enjoy nice properties in both complex and functional analytical frameworks.In the former setting, we show that the space (Π B p (D, X), π B p ) is invariant by Möbius transformations of D. In the latter context and in a clear parallelism with the theory of absolutely p-summing linear operators (see [7,Chapter 2]), we prove that [Π B p , π B p ] is an injective Banach ideal of normalized Bloch mappings whose elements can be characterized by means of Pietsch domination/factorization.Applying this Pietsch domination, we present a Bloch version of Maurey's extrapolation Theorem [13].
On the other hand, the known duality of the Bloch spaces (see [2,4,21]) is extended to the spaces (Π B p (D, X * ), π B p ) by identifying them with the duals of the spaces of the so-called X-valued Bloch molecules on D, equipped with the Bloch versions of the p * -Chevet-Saphar tensor norms d p * .We conclude the paper with some open problems.
The proofs of some of our results are similar to those of their corresponding linear versions, but a detailed reading of them shows that the adaptation of the linear techniques to the Bloch setting is far from being simple.Our approach depends mainly on the application of some concepts and results concerning the theory on a strongly unique predual of the space B(D), called Bloch-free Banach space over D that was introduced in [11].
Notation.For two normed spaces X and Y, L(X, Y) denotes the normed space of all bounded linear operators from X to Y, equipped with the operator canonical norm.In particular, the topological dual space L(X, C) is denoted by X * .For x ∈ X and x * ∈ X * , we will sometimes write x * , x = x * (x).As usual, B X and S X stand for the closed unit ball of X and the unit sphere of X, respectively.Let T and D denote the unit sphere and the unit open disc of C, respectively.
Given 1 ≤ p ≤ ∞, let p * denote the conjugate index of p defined by

p-Summing Bloch mappings on the unit disc
This section gathers the most important properties of p-summing Bloch mappings on D. From now on, unless otherwise stated, X will denote a complex Banach space.
The following class of Bloch functions will be used throughout the paper.For each z ∈ D, the function f z : D → C defined by .
Since q/p > 1 and (q/p) * = q/(q − p), Hölder Inequality yields , and thus we obtain for all z ∈ D, and thus 1.2.Injective Banach ideal property.Let us recall (see [11,Definition 5.11]) that a normalized Bloch ideal is a subclass I B of the class of all normalized Bloch mappings B such that for every complex Banach space X, the components satisfy the following properties: (I1) I B (D, X) is a linear subspace of B(D, X), (I2) For every g ∈ B(D) and x ∈ X, the mapping g A normalized Bloch ideal I B is said to be normed (Banach) if there is a function • I B : I B → R + 0 such that for every complex Banach space X, the following three conditions are satisfied: where Y is a complex Banach space, then A normed normalized Bloch ideal [I B , • I B ] is said to be: (I) Injective if for any mapping f ∈ B(D, X), any complex Banach space Y and any isometric linear embedding ι : X → Y, we have that We are now ready to establish the following result which can be compared to [18,.
, and therefore , Thus we have proved that Π B p (D, X), π B p is a normed space.To show that it is a Banach space, it is enough to see that every absolutely convergent series is convergent.So let ( f n ) n≥1 be a sequence in for all n ∈ N, and by taking limits with n → ∞ yields for all n ∈ N, and thus f is the π B p -limit of the series [11,Proposition 5.13].If g = 0, there is nothing to prove.Assume g 0. We have [11,Proposition 5.13].We have , where we have used that p B (g The reverse inequality follows from (N3).1.3.Möbius invariance.The Möbius group of D, denoted Aut(D), is formed by all biholomorphic bijections φ : Given a complex Banach space X, let us recall (see [3]) that a linear space A(D, X) of holomorphic mappings from D into X, endowed with a seminorm p A , is Möbius-invariant if it holds: , and using this fact we also deduce that π In this way we have proved the following.Let us recall that B(D) is a dual Banach space (see [2]) and therefore we can consider this space equipped with its weak* topology.Let P(B B(D) ) denote the set of all Borel regular probability measures µ on (B B(D) , w * ).
(ii) (Pietsch domination).There is a constant c ≥ 0 and a Borel regular probability measure for all z ∈ D. In this case, π B p ( f ) is the infimum of all constants c ≥ 0 satisfying the preceding inequality, and this infimum is attained.
Proof.(i) ⇒ (ii): We will apply an unified abstract version of Piestch domination Theorem (see [6,17]).For it, consider the functions , and therefore f is R − S -abstract p-summing.Hence, by applying [17, Theorem 3.1], there is a constant c ≥ 0 and a measure µ ∈ P(B B(D) ) such that for all z ∈ D and λ ∈ C, and therefore for all z ∈ D. Furthermore, we have for every z ∈ D by taking, for example, n ∈ N, Pietsch factorization.We now present the analogue for p-summing Bloch mappings of Pietsch factorization theorem for p-summing operators (see [18,Theorem 3], also [7,Theorem 2.13]). Given ) → L ∞ (µ) denote the formal inclusion operators.We will also use the mapping and for a complex Banach space X, the isometric linear embedding ι X : The following easy fact will be applied below.
(ii) (Pietsch factorization).There exist a regular Borel probability measure µ on (B B(D) , w * ), an operator T ∈ L(L p (µ), ℓ ∞ (B X * )) and a mapping h ∈ B(D, L ∞ (µ)) such that the following diagram commutes: , where the infimum is taken over all such factorizations of ι X • f ′ as above, and this infimum is attained.
for all z ∈ D. By Lemma 1.5, there is a mapping for any n ∈ N, α 1 , . . ., α n ∈ C * and z 1 , . . ., z n ∈ D. By the injectivity of the Banach space ℓ ∞ (B X * ) (see [7, p. 45]), there exists The concept of holomorphic mapping with a relatively (weakly) compact Bloch range was introduced in [11].The Bloch range of a function f ∈ H(D, X) is the set A mapping f ∈ H(D, X) is said to be (weakly) compact Bloch if rang B ( f ) is a relatively (weakly) compact subset of X.
we conclude that f is weakly compact Bloch.For p = 1, the result follows from Proposition 1.1 and from what was proved above.
(ii) It follows from (i) that if f ∈ Π B p (D, X) and X is reflexive, then rang B ( f ) is relatively compact in X, hence f is compact Bloch.1.6.Maurey extrapolation.We now use Pietsch domination of p-summing Bloch mappings to give a Bloch version of Maurey's extrapolation Theorem [13].
Proof.Lemma 1.5 and Proposition 1.2 assures that for each µ ∈ P(B B(D) ), there is a mapping We now follow the proof of [7,Theorem 3.17].Since Π B q (D, ℓ q ) = Π B p (D, ℓ q ) and π B q ≤ π B p on Π B p (D, ℓ q ) by Proposition 1.1, the Closed Graph Theorem yields a constant c > 0 such that π B p ( f ) ≤ cπ B q ( f ) for all f ∈ Π B q (D, ℓ q ).Since L q (µ) is an L q,λ -space for each λ > 1, we can assure that given n ∈ N and z 1 , . . ., z n ∈ D, the subspace E = lin I ∞,q (h µ (z 1 )), . . ., I ∞,q (h µ (z n )) ⊆ L q (µ) embeds λ-isomorphically into ℓ q , that is, E is contained in a subspace F ⊆ L q (µ) for which there exists an isomorphism T : ≤ cλ for all λ > 1, and thus π B p (I ∞,q • h µ ) ≤ c.Now, by Theorem 1.4, there exists a measure µ ∈ P(B B(D) ) such that for all z ∈ D. In the last equality, we have used that for all z ∈ D and g ∈ B B(D) .
Take a complex Banach space X and let f ∈ Π B q (D, X).In view of Proposition 1.1, we only must show that f ∈ Π B 1 (D, X).Theorem 1.4 provides again a measure µ 0 ∈ P(B B(D) ) such that for all z ∈ D. We claim that there is a constant C > 0 and a measure λ ∈ P(B B(D) ) such that for all z ∈ D. Indeed, define λ = ∞ n=0 (1/2 n+1 )µ n ∈ P(B B(D) ), where (µ n ) n≥1 is the sequence in P(B B(D) ) given by µ n+1 = µ n for all n ∈ N 0 , where the measure µ n is defined using Theorem 1.4.Since 1 < p < q, there exists θ ∈ (0, 1) such that p = θ • 1 + (1 − θ)q, and applying Hölder's Inequality with 1/θ (note that (1/θ) * = 1/(1 − θ)), we have for each n ∈ N 0 and all z ∈ D. Using Hölder's Inequality and the inequality for all z ∈ D. From above, we deduce that 1 2 for all z ∈ D, and this proves our claim taking C = 2(2c) 1 θ .Therefore we can write

Banach-valued Bloch molecules on the unit disc
Our aim in this section is to study the duality of the spaces of p-summing Bloch mappings from D into X * .We begin by recalling some concepts and results stated in [11] on the Bloch-free Banach space over D.
For each z ∈ D, a Bloch atom of D is the bounded linear functional γ z : B(D) → C given by Let X be a complex Banach space.Given z ∈ D and x ∈ X, it is immediate that the functional

The elements of lin({γ
is linear and continuous with We now present a tensor product space whose elements, according to [11,Definition 2.6], could be referred to as X-valued Bloch molecules on D. Definition 2.1.Let X be a complex Banach space.Define the linear space where n ∈ N, λ i ∈ C, z i ∈ D and x i ∈ X for i = 1, . . ., n, but such a representation of γ is not unique. The action of the functional γ = n i=1 λ i γ z i ⊗ x i ∈ lin(Γ(D)) ⊗ X on a mapping f ∈ B(D, X * ) can be described as 2.1.Pairing.The space lin(Γ(D)) ⊗ X is a linear subspace of B(D, X * ) * and, in fact, we have: ) is a dual pair, via the bilinear form given by Proof.Note that •, • is a well-defined bilinear map on (lin(Γ(D)) ⊗ X) × B(D, X * ) since γ, f = γ( f ).On one hand, if γ ∈ lin(Γ(D)) ⊗ X and γ, f = 0 for all f ∈ B(D, X * ), then γ = 0, and thus B(D, X * ) separates points of lin(Γ(D)) ⊗ X.On the other hand, if f ∈ B(D, X * ) and γ, f = 0 for all γ ∈ lin(Γ(D)) ⊗ X, then f ′ (z), x = γ z ⊗ x, f = 0 for all z ∈ D and x ∈ X, hence f ′ (z) = 0 for all z ∈ D, therefore f is a constant function on D, then f = 0 since f (0) = 0 and thus lin(Γ(D)) ⊗ X separates points of B(D, X * ).
Since lin(Γ(D)) ⊗ X, B(D, X * ) is a dual pair, we can identify B(D, X * ) with a linear subspace of (lin(Γ(D)) ⊗ X) ′ (the algebraic dual of lin(Γ(D)) ⊗ X) by means of the following easy result.

Projective norm.
As usual (see [19]), given two linear spaces E and F, the tensor product space E ⊗ F equipped with a norm α will be denoted by E ⊗ α F, and the completion of E ⊗ α F by E ⊗ α F. An important example of tensor norm is the projective norm π on u ∈ E ⊗ F defined by where the infimum is taken over all the representations of u as above.

p-Chevet-Saphar
Bloch norms.The p-Chevet-Saphar norms d p on the tensor product of two Banach spaces E ⊗ F are well known (see, for example, [19,Section 6.2]).
Our study of the duality of the spaces Π B p (D, X * ) requires the introduction of the following Bloch versions of such norms defined now on lin(Γ(D)) ⊗ X.
The p-Chevet-Saphar Bloch norms d B p for 1 ≤ p ≤ ∞ are defined on a X-valued Bloch molecule γ ∈ lin(Γ(D)) ⊗ X as where the infimum is taken over all such representations of γ as n i=1 λ i γ z i ⊗ x i .Motivated by the analogue concept on the tensor product space (see [19, p. 127]), we introduce the following.Definition 2.5.Let X be a complex Banach space.A norm α on lin(Γ(D)) ⊗ X is said to be a Bloch reasonable crossnorm if it has the following properties: (i) α(γ z ⊗ x) ≤ γ z x for all z ∈ D and x ∈ X, (ii) For every g ∈ B(D) and x * ∈ X * , the linear functional g Theorem 2.6.d B p is a Bloch reasonable crossnorm on lin(Γ(D)) ⊗ X for any 1 ≤ p ≤ ∞.Proof.We will only prove it for 1 < p < ∞.The other cases follow similarly.

An easy verification gives
Using Young's Inequality, it follows that x 2,i p .
Since r, s were arbitrary in R + , taking above , for any h ∈ B B(D) and x * ∈ B X * , by applying Hölder's Inequality.Since the quantity n i=1 λ i h ′ (z i )x * (x i ) does not depend on the representation of γ because n i=1 taking the infimum over all representations of γ we deduce that for any h ∈ B B(D) and x * ∈ B X * .Now, if d B p (γ) = 0, the preceding inequality yields for all h ∈ B B(D) and x * ∈ B X * .For each x * ∈ B X * , this implies that n i=1 λ i x * (x i )γ z i = 0, and since Γ(D) is a linearly independent subset of G(D) by [11,Remark 2.8], it follows that x * (x i )λ i = 0 for all i ∈ {1, . . ., n}, hence λ i = 0 for all i ∈ {1, . . ., n} since B X * separates the points of X, and thus γ = n i=1 λ i γ z i ⊗ x i = 0. Finally, we will show that d B p is a Bloch reasonable crossnorm on lin(Γ(D)) ⊗ X.Firstly, given z ∈ D and x ∈ X, we have Secondly, given g ∈ B(D) and x * ∈ X * , we have .
Taking infimum over all the representations of γ, we deduce that The next result shows that d B p can be computed using a simpler formula in the cases p = 1 and p = ∞.In fact, the 1-Chevet-Saphar Bloch norm is justly the projective norm.
Proposition 2.7.For γ ∈ lin(Γ(D)) ⊗ X, we have where the infimum is taken over all such representations of γ as n i=1 λ i γ z i ⊗ x i .Proof.Let γ ∈ lin(Γ(D)) ⊗ X and let n i=1 λ i γ z i ⊗ x i be a representation of γ.We have On the other hand, we have and taking the infimum over all representations of γ gives inf Conversely, we can assume without loss of generality that x i 0 for all i ∈ {1, . . ., n} and since and taking the infimum over all representations of γ, we conclude that Furthermore, its inverse comes given by Moreover, on the unit ball of Π B p (D, X * ) the weak* topology coincides with the topology of pointwise σ(X * , X)-convergence.
Let f ∈ Π B p (D, X * ) and let Λ 0 ( f ) : lin(Γ(D)) ⊗ X → C be its associate linear functional given by and taking infimum over all the representations of γ, we deduce that |Λ 0 ( f Hence there is a unique continuous mapping Λ As in the proof of Proposition 2.4, it is similarly proved that F ϕ ∈ H(D, X * ) and there exists a mapping f ϕ ∈ B(D, X * ) with p B ( f ϕ ) ≤ ϕ such that f ′ ϕ = F ϕ .We now prove that f ϕ ∈ Π B p (D, X * ).Fix n ∈ N, λ 1 , . . ., λ n ∈ C and z 1 , . . ., z n ∈ D. Let ε > 0. For each i ∈ {1, . . ., n}, there exists x i ∈ X with x i ≤ 1 + ε such that f ′ ϕ (z i ), x i = f ′ ϕ (z i ) .It is clear that the map T : C n → C, defined by  , and we conclude that f ϕ ∈ Π B p (D, X * ) with π B p ( f ϕ ) ≤ ϕ .Finally, for any γ = n i=1 λ i γ z i ⊗ x i ∈ lin(Γ(D)) ⊗ X, we get i (w z ) − f ′ (w z ), x for all i ∈ I and some w z ∈ [0, z], and thus ( f i (z), x ) i∈I → f (z), x .This tells us that ( f i ) i∈I converges to f in the topology of pointwise σ(X * , X)-convergence.Hence the identity on Π B p (D, X * ) is a continuous bijection from the weak* topology to the topology of pointwise σ(X * , X)-convergence.On the unit ball, the first topology is compact and the second one is Hausdorff, and so they must coincide.
In particular, in view of Theorem 2. We conclude this paper with some open questions we hope researchers will take up.In Theorem 1.6, note that if f ∈ Π B 2 (D, X), then Hence ι X • f ′ factors in this way through the Hilbert space L 2 (µ).It would be interesting to introduce and study the class of Bloch mappings whose derivatives factor through a Hilbert space.Motivated by the seminal paper of Farmer and Johnson [9] that raised a similar question in the setting of Lipschitz p-summing mappings, what results about p-summing linear operators have analogues for p-summing Bloch mappings?

1. 4 .
Pietsch domination.We establish a version for p-summing Bloch mappings on D of the known Pietsch domination Theorem for p-summing linear operators between Banach spaces [18, Theorem 2].

Corollary 1 . 7 .
Let 1 ≤ p < ∞. (i) Every p-summing Bloch mapping from D to X is weakly compact Bloch.(ii) If X is reflexive, then every p-summing Bloch mapping from D to X is compact Bloch.Proof.(i) Assume first p > 1.If f ∈ Π B p (D, X), then Theorem 1.6 gives a regular Borel probability measure µ on (B B(D) , w * ), an operator T z : z ∈ D}) in B(D) * are called Bloch molecules of D. The Bloch-free Banach space over D, denoted G(D), is the norm-closed linear hull of {γ z : z ∈ D} in B(D) * .The mapping Γ : D → G(D), defined by Γ(z) = γ z for all z ∈ D, is holomorphic with γ z = 1/(1 − |z| 2 ) for all z ∈ D (see [11, Proposition 2.7]).