# Factorization of derivatives of Bloch mappings
through a Hilbert space

M. G. Cabrera-Padilla  Departamento de Matemáticas, Universidad de Almería,
04120 Almería, Spain<EMAIL_ADDRESS>,  A. Jiménez-Vargas  Departamento
de Matemáticas, Universidad de Almería, 04120, Almería, Spain<EMAIL_ADDRESS>and  D. Ruiz-Casternado  Departamento de Matemáticas, Universidad de Almería,
04120, Almería, Spain<EMAIL_ADDRESS>
###### Abstract.

We introduce the concept of vector-valued holomorphic mapping on the complex
unit disc whose derivative factors through a Hilbert space and state the main
properties of the space formed by such Bloch mappings equipped with a natural
norm: linearization, Bloch transposition, surjective and injective Banach
ideal property, invariance by Möbius transformations, Kwapień-type
characterization by Bloch domination, and duality.

###### Key words and phrases:

Vector-valued Bloch mapping, compact Bloch mapping, Banach-valued Bloch
molecule, Bloch-free Banach space.

###### 2020 Mathematics Subject Classification:

30H30, 46E15, 46E40, 47B38

Research partially supported by Junta de Andalucía grant FQM194, and by grant
PID2021-122126NB-C31 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A
way of making Europe”.

## Introduction

Bounded linear operators between Banach spaces that factor through a Hilbert
space appeared in Grothendieck’s Résume [11], with the name of _$h$ -integral
operators_. Their study was encouraged by Lindenstrauss and Pełczyński [15],
and their duality theory addressed by Kwapień [13]. Basic results on this
class of operators can be consulted in the monographs by Diestel, Jarchow and
Tonge [8, Chapter 7], Li and Queffélec [14, Chapter 5] and Pisier [17, Chapter
2].

The extension of this theory to different nonlinear settings has been promoted
by some authors. This is the case of completely bounded operators between
operator spaces by Pisier [18], Lipschitz mappings between metric spaces by
Chávez-Domínguez [6], multilinear operators between Banach spaces by
Fernández-Unzueta and García-Hernández [10], and $2$-dominated polynomials by
Rueda and Sánchez Pérez [19].

We will study in this note the factorization through a Hilbert space of the
derivatives of vector-valued Bloch mappings on the complex unit open disc
$\mathbb{D}$. Under a different approach, the problem of factoring derivatives
of holomorphic functions has been addressed by Cohn [7] in the Hardy space
$H^{p}$ and by Dyakonov [9] in the Nevanline and Smirnov classes.

Let $X$ be a complex Banach space and let $\mathcal{H}(\mathbb{D},X)$ be the
space of all holomorphic mappings from $\mathbb{D}$ into $X$. A mapping
$f\in\mathcal{H}(\mathbb{D},X)$ is said to be _Bloch_ if there is a constant
$c\geq 0$ such that $(1-|z|^{2})\left\|f^{\prime}(z)\right\|\leq c$ for all
$z\in\mathbb{D}$. The Bloch space $\mathcal{B}(\mathbb{D},X)$ is the linear
space of all mappings $f\in\mathcal{H}(\mathbb{D},X)$ so that

$p_{\mathcal{B}}(f):=\sup\left\\{(1-|z|^{2})\left\|f^{\prime}(z)\right\|\colon
z\in\mathbb{D}\right\\}<\infty,$

equipped with the Bloch seminorm $p_{\mathcal{B}}$. The normalized Bloch space
$\widehat{\mathcal{B}}(\mathbb{D},X)$ is the Banach space of all mappings
$f\in\mathcal{B}(\mathbb{D},X)$ such that $f(0)=0$, under the the Bloch norm
$p_{\mathcal{B}}$. To simplify, we will write
$\widehat{\mathcal{B}}(\mathbb{D})$ instead of
$\widehat{\mathcal{B}}(\mathbb{D},\mathbb{C})$. Spaces of Bloch functions have
been studied, among others, by Anderson, Clunie and Pommerenke [1] and Zhu
[21] in the complex-valued case, and by Arregui and Blasco [3, 4] in the
vector-valued case.

Given two Banach spaces $X$ and $Y$, we denote by $\mathcal{L}(X,Y)$ the
Banach space of all bounded linear operators from $X$ into $Y$, equipped with
the operator canonical norm. As usual, $X^{*}$ stands for
$\mathcal{L}(X,\mathbb{C})$.

A mapping $f\in\mathcal{H}(\mathbb{D},X)$ is said to _factor its derivative
through a Hilbert space_ if there exists a Hilbert space $H$, a mapping
$g\in\widehat{\mathcal{B}}(\mathbb{D},H)$ and an operator
$T\in\mathcal{L}(H,X)$ such that $f^{\prime}=T\circ g^{\prime}$, that is, the
following diagram commutes:

$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\textstyle{\mathbb{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{X}$

For such a $f$, we set
$\gamma^{\mathcal{B}}_{2}(f)=\inf\left\\{\left\|T\right\|p_{\mathcal{B}}(g)\right\\}$,
where the infimum runs over all possible factorizations of $f^{\prime}$ as
above. We denote by $\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$ the linear space
of all mappings $f\in\mathcal{H}(\mathbb{D},X)$ for which $f^{\prime}$ admits
such a factorization. Moreover,
$\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ is the linear subspace
consisting of all mappings in $\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$ that
preserve the zero.

The key tool to obtain our results is a linearization process of Bloch
mappings on $\mathbb{D}$ by using a predual space of
$\widehat{\mathcal{B}}(\mathbb{D})$, called _Bloch-free space over
$\mathbb{D}$_ and denoted $\mathcal{G}(\mathbb{D})$. We will devote the first
lines of the following section to recall this process that was introduced in
[12] and applied to study $p$-summing Bloch mappings on $\mathbb{D}$ in [5].
Once this is done, we will present the most important properties of
$\Gamma^{\mathcal{B}}_{2}$-spaces in five subsections.

First, we characterize the factorization through a Hilbert space of the
derivative of a zero-preserving Bloch mapping from $\mathbb{D}$ into $X$ by
means of the linear factorization through a Hilbert space of its linearization
from $\mathcal{G}(\mathbb{D})$ into $X$ as well as of its Bloch transposition
from $X^{*}$ into $\widehat{\mathcal{B}}(\mathbb{D})$. Next, using this
characterization, we prove that
$[\Gamma^{\widehat{\mathcal{B}}}_{2},\gamma^{\mathcal{B}}_{2}]$ is a
surjective and injective Banach ideal of normalized Bloch mappings. In [5],
the concept of $p$-summing Bloch mapping from $\mathbb{D}$ into $X$ for $1\leq
p\leq\infty$ was introduced and studied. We show that every $2$-summing Bloch
mapping $f\colon\mathbb{D}\to X$ with $f(0)=0$ factors through a Hilbert
space. We also devote a brief subsection to show that the space
$(\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X),\gamma^{\mathcal{B}}_{2})$ enjoys the
property of invariance by Möbius transformations of $\mathbb{D}$. In
Subsection 1.4, the main result of this paper provides a Bloch version of a
known theorem of Kwapień [13] by characterizing holomorphic mappings whose
derivatives factor through a Hilbert space in terms of a Bloch
subordination/domination criterion of finite sequences of elements of
$\mathbb{C}\times\mathbb{D}$. Lastly, we introduce a convenient norm on the
space of the so-called _$X$ -valued Bloch molecules on $\mathbb{D}$_ so that
the dual of the completion of this last space is identified with
$(\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*}),\gamma^{\mathcal{B}}_{2})$.

##  1\. Preliminaries and results

We first collect some useful results on the linearization of Bloch mappings
borrowed from [12]. For each $z\in\mathbb{D}$, a Bloch atom of $\mathbb{D}$ is
the bounded linear functional
$\gamma_{z}\colon\widehat{\mathcal{B}}(\mathbb{D})\to\mathbb{C}$ given by

$\gamma_{z}(f)=f^{\prime}(z)\qquad(f\in\widehat{\mathcal{B}}(\mathbb{D})).$

The elements of $\mathrm{lin}(\\{\gamma_{z}\colon z\in\mathbb{D}\\})$ in
$\widehat{\mathcal{B}}(\mathbb{D})^{*}$ are called Bloch molecules of
$\mathbb{D}$. The Bloch-free Banach space over $\mathbb{D}$, denoted
$\mathcal{G}(\mathbb{D})$, is the norm-closed linear hull of
$\left\\{\gamma_{z}\colon z\in\mathbb{D}\right\\}$ in
$\widehat{\mathcal{B}}(\mathbb{D})^{*}$. As usual, for any
$T\in\mathcal{L}(X,Y)$, $T^{*}\colon Y^{*}\to X^{*}$ denotes the adjoint
operator of $T$.

######  Theorem 1.1.

[12]

  1. (i)

For every $z\in\mathbb{D}$, the function $f_{z}\colon\mathbb{D}\to\mathbb{C}$
defined by

$f_{z}(w)=\frac{(1-|z|^{2})w}{1-\overline{z}w}\qquad(w\in\mathbb{D}),$

belongs to $\widehat{\mathcal{B}}(\mathbb{D})$ with
$p_{\mathcal{B}}(f_{z})=1=(1-|z|^{2})f_{z}^{\prime}(z)$.

  2. (ii)

The mapping $\Gamma\colon\mathbb{D}\to\mathcal{G}(\mathbb{D})$, defined by
$\Gamma(z)=\gamma_{z}$ for all $z\in\mathbb{D}$, is holomorphic with
$\left\|\gamma_{z}\right\|=1/(1-|z|^{2})$ for all $z\in\mathbb{D}$.

  3. (iii)

$\widehat{\mathcal{B}}(\mathbb{D})$ is isometrically isomorphic to
$\mathcal{G}(\mathbb{D})^{*}$, via the evaluation mapping
$\Lambda\colon\widehat{\mathcal{B}}(\mathbb{D})\to\mathcal{G}(\mathbb{D})^{*}$
given by

$\Lambda(f)(\gamma)=\sum_{k=1}^{n}\lambda_{k}f^{\prime}(z_{k})\qquad(f\in\widehat{\mathcal{B}}(\mathbb{D}),\;\gamma=\sum_{k=1}^{n}\lambda_{k}\gamma_{z_{k}}\in\mathrm{lin}(\Gamma(\mathbb{D}))).$

  4. (iv)

For every holomorphic function $h\colon\mathbb{D}\to\mathbb{D}$ with $h(0)=0$,
the composition operator
$C_{h}\colon\widehat{\mathcal{B}}(\mathbb{D})\to\widehat{\mathcal{B}}(\mathbb{D})$,
defined by $C_{h}(f)=f\circ h$, is in
$\mathcal{L}(\widehat{\mathcal{B}}(\mathbb{D}),\widehat{\mathcal{B}}(\mathbb{D}))$
with $p_{\mathcal{B}}(h)\leq||C_{h}||\leq 1$.

  5. (v)

For every holomorphic function $h\colon\mathbb{D}\to\mathbb{D}$ such that
$h(0)=0$, there exists a unique operator
$\widehat{h}\in\mathcal{L}(\mathcal{G}(\mathbb{D}),\mathcal{G}(\mathbb{D}))$
such that $\widehat{h}\circ\Gamma=h^{\prime}\cdot(\Gamma\circ h)$. In fact,
$\left(\widehat{h}\right)^{*}\equiv C_{h}$.

  6. (vi)

For every complex Banach space $X$ and every mapping
$f\in\widehat{\mathcal{B}}(\mathbb{D},X)$, there exists a unique operator
$S_{f}\in\mathcal{L}(\mathcal{G}(\mathbb{D}),X)$ such that
$S_{f}\circ\Gamma=f^{\prime}$. Furthermore, $||S_{f}||=p_{\mathcal{B}}(f)$.

  7. (vii)

The correspondence $f\mapsto S_{f}$ is an isometric isomorphism from
$\widehat{\mathcal{B}}(\mathbb{D},X)$ onto
$\mathcal{L}(\mathcal{G}(\mathbb{D}),X)$.

  8. (viii)

Given $f\in\widehat{\mathcal{B}}(\mathbb{D},X)$, the map $f^{t}\colon
X^{*}\to\widehat{\mathcal{B}}(\mathbb{D})$, defined by
$f^{t}(x^{*})=x^{*}\circ f$ and called Bloch transpose of $f$, is in
$\mathcal{L}(X^{*},\widehat{\mathcal{B}}(\mathbb{D}))$ with
$\|f^{t}\|=p_{\mathcal{B}}(f)$. Furthermore,
$f^{t}=\Lambda^{-1}\circ(S_{f})^{*}$. $\hfill\qed$

We now present the most important properties of vector-valued Bloch mappings
on $\mathbb{D}$ whose derivatives factor through a Hilbert space.

###  1.1. Linearization / Bloch transposition

We study the relationships of a mapping
$f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with both its
linearization $S_{f}\in\mathcal{L}(\mathcal{G}(\mathbb{D}),X)$ and its Bloch
transposition $f^{t}\in\mathcal{L}(X^{*},\widehat{\mathcal{B}}(\mathbb{D}))$.
Compare to [8, Proposition 7.2].

Let us recall that given Banach spaces $X$ and $Y$, an operator
$T\in\mathcal{L}(X,Y)$ is said to _factor through a Hilbert space_ if there
exists a Hilbert space $H$ and operators $B\in\mathcal{L}(X,H)$ and
$A\in\mathcal{L}(H,Y)$ such that $T=A\circ B$. The set of all operators
$T\in\mathcal{L}(X,Y)$ admitting such a factorization is denoted by
$\Gamma_{2}(X,Y)$, and it becomes a Banach space equipped with the norm
$\gamma_{2}(T)=\inf\left\\{\left\|A\right\|\left\|B\right\|\right\\}$, where
the infimum extends over all possible factorizations of $T$ as above.

We begin with the following easy observation.

######  Lemma 1.2.

Let $X$ be a complex Banach space. Then
$\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)\subseteq\mathcal{B}(\mathbb{D},X)$
with $p_{\mathcal{B}}(f)\leq\gamma_{2}^{\mathcal{B}}(f)$ for all
$f\in\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$.

###### Proof.

If $f\in\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$, then $f^{\prime}=T\circ
g^{\prime}$, where $g\in\mathcal{B}(\mathbb{D},H)$ and $T\in\mathcal{L}(H,X)$
for some Hilbert space $H$. We have

$(1-|z|^{2})\left\|f^{\prime}(z)\right\|=(1-|z|^{2})\left\|T(g^{\prime}(z))\right\|\leq\left\|T\right\|(1-|z|^{2})\left\|g^{\prime}(z)\right\|\leq\left\|T\right\|p_{\mathcal{B}}(g)$

for all $z\in\mathbb{D}$, and therefore $f\in\mathcal{B}(\mathbb{D},X)$ with
$p_{\mathcal{B}}(f)\leq\left\|T\right\|p_{\mathcal{B}}(g)$. Taking infimum
over all such factorizations of $f^{\prime}$, we obtain
$p_{\mathcal{B}}(f)\leq\gamma_{2}^{\mathcal{B}}(f)$. ∎

######  Theorem 1.3.

Let $X$ be a complex Banach space and
$f\in\widehat{\mathcal{B}}(\mathbb{D},X)$. The following affirmations are
equivalent:

  1. (i)

$f\colon\mathbb{D}\to X$ factors its derivative through a Hilbert space.

  2. (ii)

$S_{f}\colon\mathcal{G}(\mathbb{D})\to X$ factors through a Hilbert space.

  3. (iii)

$f^{t}\colon X^{*}\to\widehat{\mathcal{B}}(\mathbb{D})$ factors through a
Hilbert space.

In this case,
$\gamma^{\mathcal{B}}_{2}(f)=\gamma_{2}(S_{f})=\gamma_{2}(f^{t})$.
Furthermore, the mapping $f\mapsto S_{f}$ is an isometric isomorphism from
$\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ onto
$\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)$.

###### Proof.

$(i)\Rightarrow(ii)$: If
$f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$, then
$f^{\prime}=T\circ g^{\prime}$, where
$g\in\widehat{\mathcal{B}}(\mathbb{D},H)$ and $T\in\mathcal{L}(H,X)$ for some
Hilbert space $H$. By Theorem 1.1, we have the factorizations
$f^{\prime}=S_{f}\circ\Gamma$ and $g^{\prime}=S_{g}\circ\Gamma$, where
$S_{f}\in\mathcal{L}(\mathcal{G}(\mathbb{D}),X)$ and
$S_{g}\in\mathcal{L}(\mathcal{G}(\mathbb{D}),H)$. Hence
$S_{f}\circ\Gamma=T\circ S_{g}\circ\Gamma$ and this implies that $S_{f}=T\circ
S_{g}$ since $\Gamma(\mathbb{D})$ is a dense linear subspace of
$\mathcal{G}(\mathbb{D})$. Therefore
$S_{f}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)$ with
$\gamma_{2}(S_{f})\leq\left\|T\right\|\left\|S_{g}\right\|=\left\|T\right\|p_{\mathcal{B}}(g)$.
Taking infimum over all factorizations of $f^{\prime}$ as above, we have
$\gamma_{2}(S_{f})\leq\gamma^{\mathcal{B}}_{2}(f)$.

$(ii)\Rightarrow(i)$: If $S_{f}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)$, then
$S_{f}=T\circ S$, where $S\in\mathcal{L}(\mathcal{G}(\mathbb{D}),H)$ and
$T\in\mathcal{L}(H,X)$ for some Hilbert space $H$. Hence
$f^{\prime}=S_{f}\circ\Gamma=T\circ S\circ\Gamma$ by Theorem 1.1. Since
$S\circ\Gamma\in\mathcal{H}(\mathbb{D},H)$, Lemma 2.9 in [12] gives us a
mapping $h\in\mathcal{H}(\mathbb{D},H)$ with $h(0)=0$ such that
$h^{\prime}=S\circ\Gamma$. Since

$(1-|z|^{2})\left\|h^{\prime}(z)\right\|=(1-|z|^{2})\left\|S(\Gamma(z))\right\|\leq\left\|S\right\|$

for all $z\in\mathbb{D}$, it follows that
$h\in\widehat{\mathcal{B}}(\mathbb{D},H)$ with
$p_{\mathcal{B}}(h)\leq\left\|S\right\|$. Hence $f^{\prime}=T\circ h^{\prime}$
and therefore $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f)\leq\left\|T\right\|\left\|S\right\|$. Taking the
infimum over all factorizations of $S_{f}$, we obtain
$\gamma^{\mathcal{B}}_{2}(f)\leq\gamma_{2}(S_{f})$, as desired.

$(ii)\Leftrightarrow(iii)$: Applying first [8, Proposition 7.2], and then
Theorem 1.1 with [8, Theorem 7.1], we get

$S_{f}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)\Leftrightarrow(S_{f})^{*}\in\Gamma_{2}(X^{*},\mathcal{G}(\mathbb{D})^{*})\Leftrightarrow
f^{t}=\Lambda^{-1}\circ(S_{f})^{*}\in\Gamma_{2}(X^{*},\widehat{\mathcal{B}}(\mathbb{D})),$

and, in this case,
$\gamma_{2}(S_{f})=\gamma_{2}((S_{f})^{*})=\gamma_{2}(f^{t})$.

The last assertion of the statement follows immediately by applying that the
correspondence $f\mapsto S_{f}$ from $\widehat{\mathcal{B}}(\mathbb{D},X)$
into $\mathcal{L}(\mathcal{G}(\mathbb{D}),X)$ is surjective and from what was
proved above. ∎

###  1.2. Ideal property

The concept of a Banach ideal of normalized Bloch mappings was introduced in
[12, Definition 5.11], inspired by the notion of Banach operator ideal due to
Pietsch [16].

######  Proposition 1.4.

$[\Gamma^{\widehat{\mathcal{B}}}_{2},\gamma^{\mathcal{B}}_{2}]$ is a
surjective and injective Banach ideal of normalized Bloch mappings.

###### Proof.

We are going to show that
$[\Gamma^{\widehat{\mathcal{B}}}_{2},\gamma^{\mathcal{B}}_{2}]$ satisfies the
properties of the aforementioned definition. Let $X$ be a complex Banach
space.

(N1):
$(\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X),\gamma^{\mathcal{B}}_{2})$
is a Banach space with $p_{\mathcal{B}}(f)\leq\gamma^{\mathcal{B}}_{2}(f)$ for
all $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$.

_Proof:_ It follows directly from Lemma 1.2 and Theorem 1.3.

(N2): For every $g\in\widehat{\mathcal{B}}(\mathbb{D})$ and $x\in X$, the
mapping $g\cdot x\colon z\mapsto g(z)x$ from $\mathbb{D}$ into $X$ is in
$\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(g\cdot x)=p_{\mathcal{B}}(g)\left\|x\right\|$.

_Proof:_ It is easy to show that $g\cdot x\in\hat{\mathcal{B}}(\mathbb{D},X)$
with $(g\cdot x)^{\prime}=g^{\prime}\cdot x$ and $p_{\mathcal{B}}(g\cdot
x)=p_{\mathcal{B}}(g)\left\|x\right\|$. Since $(g\cdot x)^{\prime}=(S_{g}\cdot
x)\circ\Gamma$, it follows that $S_{g\cdot x}=S_{g}\cdot x$ with
$p_{\mathcal{B}}(g\cdot x)=\left\|S_{g\cdot x}\right\|=\left\|S_{g}\cdot
x\right\|=p_{\mathcal{B}}(g)\left\|x\right\|$ by Theorem 1.1, and therefore
$S_{g\cdot x}\in\mathscr{F}(\mathcal{G}(\mathbb{D}),X)$, where
$\mathscr{F}(\mathcal{G}(\mathbb{D}),X)$ denotes the subspace of
$\mathcal{L}(\mathcal{G}(\mathbb{D}),X)$ formed by all finite-rank operators.
Since $\mathscr{F}\subseteq\Gamma_{2}$, we have $S_{g\cdot
x}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)$ and thus $g\cdot
x\in\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(g\cdot x)=\left\|S_{g\cdot
x}\right\|=p_{\mathcal{B}}(g)\left\|x\right\|$ by Theorem 1.3. To obtain the
reverse inequality, apply (N1).

(N3) _Ideal property_ : If $h\colon\mathbb{D}\to\mathbb{D}$ is a holomorphic
function with $h(0)=0$, $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$
and $T\in\mathcal{L}(X,Y)$ where $Y$ is a complex Banach space, then $T\circ
f\circ h\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},Y)$ with
$\gamma^{\mathcal{B}}_{2}(T\circ f\circ
h)\leq\|T\|\gamma^{\mathcal{B}}_{2}(f)$.

_Proof:_ Note that $S_{T\circ f\circ h}=T\circ
S_{f}\circ\widehat{h}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),Y)$ by [12,
Proposition 5.13], Theorem 1.3 and the ideal property of
$[\Gamma_{2},\gamma_{2}]$ (see [8, Proposition 7.1]). Then, by Theorem 1.3,
$T\circ f\circ h\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},Y)$ with

$\gamma^{\mathcal{B}}_{2}(T\circ f\circ h)=\gamma_{2}(T\circ
S_{f}\circ\widehat{h})\leq\left\|T\right\|\gamma_{2}(S_{f})||\widehat{h}||\leq\left\|T\right\|\gamma^{\mathcal{B}}_{2}(f).$

(I) _Injectivity_ : For any $f\in\widehat{\mathcal{B}}(\mathbb{D},X)$, any
complex Banach space $Y$ and any metric injection $\iota\in\mathcal{L}(X,Y)$,
we have $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f)=\gamma^{\mathcal{B}}_{2}(\iota\circ f)$ whenever
$\iota\circ f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},Y)$.

_Proof:_ Assume $\iota\circ
f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},Y)$. Then $\iota\circ
S_{f}=S_{\iota\circ f}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),Y)$ by [12,
Proposition 5.13] and Theorem 1.3. Since the Banach operator ideal
$[\Gamma_{2},\gamma_{2}]$ is injective by [8, Proposition 7.3], it follows
that $S_{f}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)$ with
$\gamma_{2}(S_{f})=\gamma_{2}(\iota\circ S_{f})$ or, equivalently,
$f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f)=\gamma^{\mathcal{B}}_{2}(\iota\circ f)$ again by
Theorem 1.3.

(S) _Surjectivity_ : For any $f\in\widehat{\mathcal{B}}(\mathbb{D},X)$ and any
holomorphic function $\pi\colon\mathbb{D}\to\mathbb{D}$ with $\pi(0)=0$ such
that
$\hat{\pi}\in\mathcal{L}(\mathcal{G}(\mathbb{D}),\mathcal{G}(\mathbb{D}))$ is
a metric surjection, we have
$f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f)=\gamma^{\mathcal{B}}_{2}(f\circ\pi)$ whenever
$f\circ\pi\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$.

_Proof:_ Assume
$f\circ\pi\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$. Since
$S_{f}\circ\hat{\pi}=S_{f\circ\pi}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)$ by
[12, Proposition 5.13] and Theorem 1.3, and the Banach operator ideal
$[\Gamma_{2},\gamma_{2}]$ is surjective by [8, Proposition 7.3], we have
$S_{f}\in\Gamma_{2}(\mathcal{G}(\mathbb{D}),X)$ with
$\gamma_{2}(S_{f})=\gamma_{2}(S_{f}\circ\hat{\pi})$, hence
$f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f)=\gamma_{2}(S_{f})$ and thus
$\gamma^{\mathcal{B}}_{2}(f)=\gamma_{2}(S_{f\circ\pi})=\gamma^{\mathcal{B}}_{2}(f\circ\pi)$
by Theorem 1.3. ∎

Let us recall (see [5]) that a mapping $f\in\mathcal{H}(\mathbb{D},X)$ is said
to be $2$-summing Bloch if there is a constant $c\geq 0$ such that for any
$n\in\mathbb{N}$, $\lambda_{1},\ldots,\lambda_{n}\in\mathbb{C}$ and
$z_{1},\ldots,z_{n}\in\mathbb{D}$, we have

$\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}(z_{i})\right\|^{2}\right)^{\frac{1}{2}}\leq
c\sup_{g\in
B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|g^{\prime}(z_{i})\right|^{2}\right)^{\frac{1}{2}}.$

The infimum of such constants $c$, denoted $\pi^{\mathcal{B}}_{2}(f)$, defines
a complete norm on the linear space, denoted
$\Pi^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$, consisting of all $2$-summing
Bloch mappings $f\colon\mathbb{D}\to X$ for which $f(0)=0$.

If $f\in\Pi^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$, Theorem 1.6 in [5]
provides us a regular Borel probability measure $\mu$ on
$(B_{\widehat{\mathcal{B}}(\mathbb{D})},w^{*})$, an operator
$T\in\mathcal{L}(L_{2}(\mu),\ell_{\infty}(B_{X^{*}}))$ and a mapping
$h\in\widehat{\mathcal{B}}(\mathbb{D},L_{\infty}(\mu))$ such that

$\iota_{X}\circ f^{\prime}=T\circ I_{\infty,2}\circ
h^{\prime}\colon\mathbb{D}\stackrel{{\scriptstyle
h^{\prime}}}{{\rightarrow}}L_{\infty}(\mu)\stackrel{{\scriptstyle
I_{\infty,2}}}{{\rightarrow}}L_{2}(\mu)\stackrel{{\scriptstyle
T}}{{\rightarrow}}\ell_{\infty}(B_{X^{*}}),$

where $I_{\infty,2}\colon L_{\infty}(\mu)\to L_{2}(\mu)$ is the formal
inclusion operator and $\iota_{X}\colon X\to\ell_{\infty}(B_{X^{*}})$ is the
isometric linear embedding defined by
$\left\langle\iota_{X}(x),x^{*}\right\rangle=x^{*}(x)$. Moreover,
$\pi^{\mathcal{B}}_{2}(f)=\inf\left\\{\left\|T\right\|p_{\mathcal{B}}(h)\right\\}$,
where the infimum is taken over all such factorizations of $\iota_{X}\circ
f^{\prime}$ as above.

Hence $\iota_{X}\circ
f\in\Gamma_{2}^{\widehat{\mathcal{B}}}(\mathbb{D},\ell_{\infty}(B_{X^{*}}))$.
By the injectivity of the Banach ideal
$[\Gamma^{\widehat{\mathcal{B}}}_{2},\gamma^{\mathcal{B}}_{2}]$, we have
$f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f)=\gamma^{\mathcal{B}}_{2}(\iota_{X}\circ f)$.
Since $(\iota_{X}\circ f)^{\prime}=T\circ(I_{\infty,2}\circ h)^{\prime}$, it
follows that $\gamma^{\mathcal{B}}_{2}(\iota_{X}\circ
f)\leq\left\|T\right\|p_{\mathcal{B}}(h)$, and taking infimum over all
factorizations of $\iota_{X}\circ f^{\prime}$ as above, we get
$\gamma^{\mathcal{B}}_{2}(\iota_{X}\circ f)\leq\pi_{2}^{\mathcal{B}}(f)$. In
this way, we have proved the following.

######  Corollary 1.5.

Every $2$-summing Bloch mapping $f\colon\mathbb{D}\to X$ with $f(0)=0$ factors
through a Hilbert space and
$\gamma_{2}^{\mathcal{B}}(f)\leq\pi_{2}^{\mathcal{B}}(f)$.$\hfill\qed$

###  1.3. Invariance by Möbius transformations

The Möbius group of $\mathbb{D}$, denoted $\mathrm{Aut}(\mathbb{D})$, is the
group of all biholomorphic bijections $\phi\colon\mathbb{D}\to\mathbb{D}$. If
$\mathbb{T}$ denotes the set of all unimodular complex numbers, then each
$\phi\in\mathrm{Aut}(\mathbb{D})$ has the form $\phi=\lambda\phi_{a}$ for some
$\lambda\in\mathbb{T}$ and $a\in\mathbb{D}$, where

$\phi_{a}(z)=\frac{a-z}{1-\overline{a}z}\qquad(z\in\mathbb{D}).$

Given a complex Banach space $X$, let us recall (see [2]) that a linear space
$\mathcal{A}(\mathbb{D},X)\subseteq\mathcal{H}(\mathbb{D},X)$ equipped with a
seminorm $p_{\mathcal{A}}$ is said to be Möbius-invariant if:

  1. (i)

$\mathcal{A}(\mathbb{D},X)\subseteq\mathcal{B}(\mathbb{D},X)$ and there exists
$c>0$ such that $p_{\mathcal{B}}(f)\leq cp_{\mathcal{A}}(f)$ for all
$f\in\mathcal{A}(\mathbb{D},X)$,

  2. (ii)

$f\circ\phi\in\mathcal{A}(\mathbb{D},X)$ with
$p_{\mathcal{A}}(f\circ\phi)=p_{\mathcal{A}}(f)$ for all
$\phi\in\mathrm{Aut}(\mathbb{D})$ and $f\in\mathcal{A}(\mathbb{D},X)$.

######  Proposition 1.6.

For any complex Banach space $X$,
$(\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X),\gamma^{\mathcal{B}}_{2})$ is Möbius-
invariant.

###### Proof.

By Lemma 1.2,
$\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)\subseteq\mathcal{B}(\mathbb{D},X)$
with $p_{\mathcal{B}}(f)\leq\gamma^{\mathcal{B}}_{2}(f)$ for all
$f\in\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$. Moreover, by an argument similar
to that of the proof of (N3) in Proposition 1.4, it is easy to prove that if
$f\in\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$ and
$\phi\in\mathrm{Aut}(\mathbb{D})$, then
$f\circ\phi\in\Gamma^{\mathcal{B}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f\circ\phi)\leq\gamma^{\mathcal{B}}_{2}(f)$, and
from this fact we also infer
$\gamma^{\mathcal{B}}_{2}(f)=\gamma^{\mathcal{B}}_{2}((f\circ\phi)\circ\phi^{-1})\leq\gamma^{\mathcal{B}}_{2}(f\circ\phi)$.
∎

###  1.4. Bloch domination

We first introduce a Bloch subordination/domination criterion of finite
sequences of elements of $\mathbb{C}\times\mathbb{D}$ (compare to [14,
Definition V.3]).

######  Definition 1.7.

Let $n,m\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$. We say that
$(\lambda_{i},z_{i})_{i=1}^{n}$ is _Bloch subordinated to_
$(\mu_{j},w_{j})_{j=1}^{m}$ or, equivalently, _Bloch dominated by_
$(\mu_{j},w_{j})_{j=1}^{m}$, and we write
$(\lambda_{i},z_{i})_{i=1}^{n}\prec(\mu_{j},w_{j})_{j=1}^{m}$, whenever

$\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|g^{\prime}(z_{i})\right|^{2}\leq\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|g^{\prime}(w_{j})\right|^{2}$

for all $g\in\widehat{\mathcal{B}}(\mathbb{D})$.

The following characterization of this criterion will be useful to us later.

######  Proposition 1.8.

Let $n,m\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$. The following
assertions are equivalent:

  1. (i)

$(\lambda_{i},z_{i})_{i=1}^{n}$ is Bloch dominated by
$(\mu_{j},w_{j})_{j=1}^{m}$.

  2. (ii)

There exists an operator $T\in\mathcal{L}(\ell_{2}^{m},\ell_{2}^{n})$ with
$\left\|T\right\|\leq 1$, represented by a $n\times m$ complex matrix
$(a_{ij})$, such that
$\lambda_{i}\gamma_{z_{i}}=\sum_{j=1}^{m}a_{ij}\mu_{j}\gamma_{w_{j}}$ for
every $1\leq i\leq n$.

###### Proof.

$(i)\Rightarrow(ii)$: We set

$S=\left\\{(\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m}\colon
g\in\widehat{\mathcal{B}}(\mathbb{D})\right\\}\subseteq\ell_{2}^{m}.$

Assume $(\lambda_{i},z_{i})_{i=1}^{n}\prec(\mu_{j},w_{j})_{j=1}^{m}$. Then the
mapping $T_{0}\colon S\to\ell_{2}^{n}$ given by

$T_{0}\left((\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m}\right)=(\lambda_{i}g^{\prime}(z_{i}))_{i=1}^{n},$

is well-defined. Clearly, $T_{0}$ is linear and since

$\displaystyle\left\|T_{0}\left((\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m}\right)\right\|_{\ell_{2}^{n}}$
$\displaystyle=\left\|(\lambda_{i}g^{\prime}(z_{i}))_{i=1}^{n}\right\|_{\ell_{2}^{n}}=\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|g^{\prime}(z_{i})\right|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|g^{\prime}(w_{j})\right|^{2}\right)^{\frac{1}{2}}=\left\|(\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m}\right\|_{\ell_{2}^{m}}$

for all $(\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m}\in S$, then $T_{0}$ is
continuous with $\left\|T_{0}\right\|\leq 1$. By Hahn–Banach Theorem, there
exists $T\in\mathcal{L}(\ell_{2}^{m},\ell_{2}^{n})$ with
$\left\|T\right\|=\left\|T_{0}\right\|$ such that $\left.T\right|_{S}=T_{0}$.
Let $(a_{ij})$ be the representation matrix of $T$ with respect to the
canonical bases of $\ell_{2}^{m}$ and $\ell_{2}^{n}$, that is,

$T((t_{j})_{j=1}^{m})=\left(\sum_{j=1}^{m}a_{1j}t_{j},\ldots,\sum_{j=1}^{m}a_{nj}t_{j}\right)^{t}$

for all $(t_{j})_{j=1}^{m}\in\ell_{2}^{m}$. Clearly, for each $1\leq i\leq n$,
we have

$\lambda_{i}\gamma_{z_{i}}(g)=\lambda_{i}g^{\prime}(z_{i})=\sum_{j=1}^{m}a_{ij}\mu_{j}g^{\prime}(w_{j})=\sum_{j=1}^{m}a_{ij}\mu_{j}\gamma_{w_{j}}(g)$

for all $g\in\widehat{\mathcal{B}}(\mathbb{D})$, and thus
$\lambda_{i}\gamma_{z_{i}}=\sum_{j=1}^{m}a_{ij}\mu_{j}\gamma_{w_{j}}$, as
required.

$(ii)\Rightarrow(i)$: Given $g\in\widehat{\mathcal{B}}(\mathbb{D})$, we have
$(\lambda_{i},g^{\prime}(z_{i}))_{i=1}^{n}=T((\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m})$,
hence

$\displaystyle\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|g^{\prime}(z_{i})\right|^{2}$
$\displaystyle=\left\|(\lambda_{i},g^{\prime}(z_{i}))_{i=1}^{n}\right\|^{2}_{\ell_{2}^{n}}\leq\left\|T\right\|^{2}\left\|(\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m}\right\|^{2}_{\ell_{2}^{m}}$
$\displaystyle\leq\left\|(\mu_{j}g^{\prime}(w_{j}))_{j=1}^{m}\right\|^{2}_{\ell_{2}^{m}}=\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|g^{\prime}(w_{j})\right|^{2}.$

∎

The following result is a Bloch version of a classical Kwapień’s theorem [13]
by means of Bloch dominated finite sequences of elements of
$\mathbb{C}\times\mathbb{D}$.

As usual, $B_{X}$ denotes the closed unit ball of a Banach space $X$. Given a
Hausdorff compact topological space $T$, $\mathscr{C}(T,\mathbb{K})$ stands
for the Banach space of all scalar-valued continuous functions on $T$, endowed
with the supremum norm.

######  Theorem 1.9.

Let $X$ be a complex Banach space and $f\in\mathcal{H}(\mathbb{D},X)$. The
following affirmations are equivalent:

  1. (i)

$f$ factors its derivative through a Hilbert space.

  2. (ii)

There exists a constant $c\geq 0$ such that

$\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}(z_{i})\right\|^{2}\leq
c^{2}\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}},$

whenever $(\lambda_{i},z_{i})_{i=1}^{n}\prec(\mu_{j},w_{j})_{j=1}^{m}$ with
$n,m\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$.

In this case, $\gamma^{\mathcal{B}}_{2}(f)$ is the minimum of all constants
$c$ satisfying the preceding inequality.

###### Proof.

$(i)\Rightarrow(ii)$: Assume $f^{\prime}=T\circ g^{\prime}$, where
$g\in\widehat{\mathcal{B}}(\mathbb{D},H)$ and $T\in\mathcal{L}(H,X)$ for some
Hilbert space $H$ with inner product $\langle\cdot\,|\,\cdot\rangle$. Let
$n,m\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$ such that
$(\lambda_{i},z_{i})_{i=1}^{n}\prec(\mu_{j},w_{j})_{j=1}^{m}$. We claim that
for each $v\in H$, the function $h_{v}\colon z\mapsto\langle
g(z)\,|\,v\rangle$ from $\mathbb{D}$ into $\mathbb{C}$ belongs to
$\widehat{\mathcal{B}}(\mathbb{D})$. Indeed, $h_{v}(0)=\langle
g(0)\,|\,v\rangle=\langle 0\,|\,v\rangle=0$ and $h_{v}$ is Bloch with
$p_{\mathcal{B}}(h_{v})\leq p_{\mathcal{B}}(g)\left\|v\right\|$ since

$(1-|z|^{2})\left|h_{v}^{\prime}(z)\right|=(1-|z|^{2})\left|\langle
g^{\prime}(z)\,|\,v\rangle\right|\leq(1-|z|^{2})\left\|g^{\prime}(z)\right\|\left\|v\right\|\leq
p_{\mathcal{B}}(g)\left\|v\right\|$

for all $z\in\mathbb{D}$. Consequently, we have

$\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|\langle
g^{\prime}(z_{i})\,|\,v\rangle\right|^{2}=\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|h_{v}^{\prime}(z_{i})\right|^{2}\leq\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|h_{v}^{\prime}(w_{j})\right|^{2}=\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|\langle
g^{\prime}(w_{j})\,|\,v\rangle\right|^{2}$

for all $v\in H$. Let $(v_{\alpha})_{\alpha\in A}$ be an orthonormal basis of
$H$. Since

$\left\|v\right\|^{2}=\sum_{\alpha\in A}\left|\langle
v\,|\,v_{\alpha}\rangle\right|^{2}$

for all $v\in H$, we deduce

$\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|g^{\prime}(z_{i})\right\|^{2}\leq\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left\|g^{\prime}(w_{j})\right\|^{2}.$

Therefore we have

$\displaystyle\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}(z_{i})\right\|^{2}$
$\displaystyle=\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|T(g^{\prime}(z_{i}))\right\|^{2}$
$\displaystyle\leq\left\|T\right\|^{2}\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|g^{\prime}(z_{i})\right\|^{2}\leq\left\|T\right\|^{2}\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left\|g^{\prime}(w_{j})\right\|^{2}$
$\displaystyle\leq\left\|T\right\|^{2}p_{\mathcal{B}}(g)^{2}\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}},$

and taking the infimum over all factorizations of $f^{\prime}$, we deduce

$\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}(z_{i})\right\|^{2}\leq\gamma^{\mathcal{B}}_{2}(f)\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}.$

$(ii)\Rightarrow(i)$: Assume that $(ii)$ holds. By Theorem 1.1, we can
consider the Hausdorff compact space
$(B_{\widehat{\mathcal{B}}(\mathbb{D})},w^{*})$. For each $z\in\mathbb{D}$,
define $\eta_{z}\colon B_{\widehat{\mathcal{B}}(\mathbb{D})}\to\mathbb{C}$ by
$\eta_{z}(k)=k^{\prime}(z)$ for all $k\in
B_{\widehat{\mathcal{B}}(\mathbb{D})}$. Clearly,
$\eta_{z}\in\mathscr{C}(B_{\widehat{\mathcal{B}}(\mathbb{D})},\mathbb{C})$
with $\left\|\eta_{z}\right\|_{\infty}=1/(1-|z|^{2})$. Consider the set
$K_{1}\subseteq\mathscr{C}(B_{\widehat{\mathcal{B}}(\mathbb{D})},\mathbb{R})$
formed by all functions of the form:

$\Phi=\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|\eta_{z_{i}}\right|^{2}-\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|\eta_{w_{j}}\right|^{2},$

where $(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$ satisfy

$\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}(z_{i})\right\|^{2}>c^{2}\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}$

By concatenation, it is easy to see that $K_{1}$ is convex. Condition in the
statement (ii) assures that $\sup\left\\{\Phi(k)\colon k\in
B_{\widehat{\mathcal{B}}(\mathbb{D})}\right\\}>0$ for each $\Phi\in K_{1}$.
Hence $K_{1}\cap K_{2}=\emptyset$, where

$K_{2}=\left\\{\Psi\in\mathscr{C}(B_{\widehat{\mathcal{B}}(\mathbb{D})},\mathbb{R})\colon\sup\left\\{\Psi(k)\colon
k\in B_{\widehat{\mathcal{B}}(\mathbb{D})}\right\\}<0\right\\}.$

Clearly, $K_{2}$ is an open convex subset of
$\mathscr{C}(B_{\widehat{\mathcal{B}}(\mathbb{D})},\mathbb{R})$. Therefore the
Hahn–Banach Separation Theorem and the Riesz Representation Theorem provide a
non-zero signed Borel measure $\nu$ on $B_{\widehat{\mathcal{B}}(\mathbb{D})}$
and a real number $\alpha$ such that

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\Psi(k)\
d\nu(k)<\alpha\leq\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\Phi(k)\
d\nu(k)\qquad(\Psi\in K_{2},\;\Phi\in K_{1}).$

Since $r\Psi\in K_{2}$ for all $r>0$, it follows that $\alpha\geq 0$. By the
definition of $K_{2}$, we obtain

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\Psi(k)\ d\nu(k)\leq
0\qquad(\Psi\in K_{2}).$

In particular,

$\nu(B_{\widehat{\mathcal{B}}(\mathbb{D})})=\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\
d\nu(k)\geq 0,$

(in fact, $\nu(B_{\widehat{\mathcal{B}}(\mathbb{D})})>0$ since $\nu\neq 0$),
and therefore $\nu$ is a positive measure. Moreover, we have

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\Phi(k)\ d\nu(k)\geq
0\qquad(\Phi\in K_{1}).$

Now, note that if $(\mu,w)\in\mathbb{C}\times\mathbb{D}$ and
$\left|\mu\right|/(1-|w|^{2})\leq 1$, we have

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\mu\right|^{2}\left|k^{\prime}(w)\right|^{2}\
d\nu(k)\leq\nu(B_{\widehat{\mathcal{B}}(\mathbb{D})}),$

and therefore there exists

$a:=\sup\left\\{\left(\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\mu\right|^{2}\left|k^{\prime}(w)\right|^{2}\
d\nu(k)\right)^{\frac{1}{2}}\colon(\mu,w)\in\mathbb{C}\times\mathbb{D},\,\frac{\left|\mu\right|}{1-|w|^{2}}\leq
1\right\\}.$

Furthermore, $a>0$. Otherwise, for all $(\mu,w)\in\mathbb{C}\times\mathbb{D}$
such that $\left|\mu\right|/(1-|w|^{2})\leq 1$, we would have

$\left\|\mu\eta_{w}\right\|_{L_{2}(\nu)}=\left(\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\mu\right|^{2}\left|k^{\prime}(w)\right|^{2}\
d\nu(k)\right)^{\frac{1}{2}}=0,$

that is, $\mu\eta_{w}=0$. In particular, $(1-|w|^{2})\eta_{w}=0$ or,
equivalently, $\eta_{w}=0$, but
$\eta_{w}(f_{w})=f^{\prime}_{w}(w)=1/(1-|w|^{2})\neq 0$, a contradiction.
Taking $\hat{\nu}:=(c^{2}/a^{2})\nu$ instead of $\nu$, we obtain

$\sup\left\\{\left(\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\mu\right|^{2}\left|k^{\prime}(w)\right|^{2}\
d\hat{\nu}(k)\right)^{\frac{1}{2}}\colon(\mu,w)\in\mathbb{C}\times\mathbb{D},\,\frac{\left|\mu\right|}{1-|w|^{2}}\leq
1\right\\}=c.$

Define $h\colon\mathbb{D}\to L_{2}(\hat{\nu})$ by $h(z)=j_{2}(\eta_{z})$ for
all $z\in\mathbb{D}$, where
$j_{2}\colon\mathscr{C}(B_{\widehat{\mathcal{B}}(\mathbb{D})},\mathbb{C})\to
L_{2}(\hat{\nu})$ is the formal inclusion operator. Note that
$h\in\mathcal{H}(\mathbb{D},L_{2}(\hat{\nu}))$. By [12, Lemma 2.9], there
exists a mapping $g\in\mathcal{H}(\mathbb{D},L_{2}(\hat{\nu}))$ with $g(0)=0$
such that $g^{\prime}=h$. In fact,
$g\in\widehat{\mathcal{B}}(\mathbb{D},L_{2}(\hat{\nu}))$ with
$p_{\mathcal{B}}(g)\leq c$ since

$(1-|z|^{2})\left\|g^{\prime}(z)\right\|_{L_{2}(\hat{\nu})}=(1-|z|^{2})\left\|h(z)\right\|_{L_{2}(\hat{\nu})}=(1-|z|^{2})\left(\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\right)^{\frac{1}{2}}\leq c$

for all $z\in\mathbb{D}$. Consider now the Hilbert space
$H:=\overline{\mathrm{lin}}(h(\mathbb{D}))\subseteq L_{2}(\hat{\nu})$ and the
linear operator $T\colon H\to X$ defined by $T(j_{2}(\eta_{z}))=f^{\prime}(z)$
for all $z\in\mathbb{D}$. We will now show that $T$ is continuous with
$\left\|T\right\|\leq 1$. First, we claim that

$(\lambda,z)\in\mathbb{C}\times\mathbb{D},\;\left|\lambda\right|^{2}\left\|f^{\prime}(z)\right\|^{2}>c^{2}\quad\Rightarrow\quad\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\lambda\right|^{2}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\geq c^{2}.$

To see this, if $(\mu,w)\in\mathbb{C}\times\mathbb{D}$ satisfies that
$\left|\mu\right|/(1-|w|^{2})\leq 1$, then we have

$\left|\lambda\right|^{2}\left\|f^{\prime}(z)\right\|^{2}>c^{2}\frac{\left|\mu\right|^{2}}{(1-|w|^{2})^{2}},$

hence the property of integration of the functions $\Phi\in K_{1}$ yields

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\lambda\right|^{2}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\geq\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\mu\right|^{2}\left|k^{\prime}(w)\right|^{2}\
d\hat{\nu}(k),$

and taking supremum over all such $(\mu,w)\in\mathbb{C}\times\mathbb{D}$, we
conclude that

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\lambda\right|^{2}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\geq c^{2}.$

Our claim implies that

$\left|\lambda\right|^{2}\left\|f^{\prime}(z)\right\|^{2}\leq\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\lambda\right|^{2}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k).$

Indeed, if $(\lambda,z)\in\mathbb{C}\times\mathbb{D}$,
$b:=\left|\lambda\right|^{2}\left\|f^{\prime}(z)\right\|^{2}$ and
$\varepsilon>0$, then

$\left|\lambda\right|^{2}\frac{c^{2}+\varepsilon}{b}\left\|f^{\prime}(z)\right\|^{2}=c^{2}+\varepsilon>c^{2},$

our claim yields

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\lambda\right|^{2}\frac{c^{2}+\varepsilon}{b}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\geq c^{2},$

that is,

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\lambda\right|^{2}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\geq\frac{c^{2}}{c^{2}+\varepsilon}b,$

and since $\varepsilon$ was arbitrary, we conclude that

$\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\lambda\right|^{2}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\geq b.$

We can adjust $\hat{\nu}$ so that
$\hat{\nu}(B_{\widehat{\mathcal{B}}(\mathbb{D})})=1$. Given $n\in\mathbb{N}$,
$\alpha_{1},\ldots,\alpha_{n}\in\mathbb{C}^{*}$ and
$z_{1},\ldots,z_{n}\in\mathbb{D}$, we have

$\displaystyle\left\|T\left(\sum_{i=1}^{n}\alpha_{i}j_{2}(\eta_{z_{i}})\right)\right\|$
$\displaystyle=\left\|\sum_{i=1}^{n}\alpha_{i}T(j_{2}(\eta_{z_{i}}))\right\|=\left\|\sum_{i=1}^{n}\alpha_{i}f^{\prime}(z_{i})\right\|$
$\displaystyle\leq\sum_{i=1}^{n}\left|\alpha_{i}\right|\left\|f^{\prime}(z_{i})\right\|\leq\sum_{i=1}^{n}\left|\alpha_{i}\right|\left(\int_{B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|k^{\prime}(z)\right|^{2}\
d\hat{\nu}(k)\right)^{\frac{1}{2}}$
$\displaystyle\leq\sum_{i=1}^{n}\frac{\left|\alpha_{i}\right|}{1-|z_{i}|^{2}}=\left|\sum_{i=1}^{n}\alpha_{i}\left(\frac{\overline{\alpha_{i}}}{\left|\alpha_{i}\right|}f_{z_{i}}\right)^{\prime}(z_{i})\right|$
$\displaystyle=\sup_{k\in
B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\sum_{i=1}^{n}\alpha_{i}k^{\prime}(z_{i})\right|=\sup_{k\in
B_{\widehat{\mathcal{B}}(\mathbb{D})}}\left|\sum_{i=1}^{n}\alpha_{i}\eta_{z_{i}}(k)\right|=\left\|\sum_{i=1}^{n}\alpha_{i}\eta_{z_{i}}\right\|_{\infty}$
$\displaystyle=\left\|j_{2}\left(\sum_{i=1}^{n}\alpha_{i}\eta_{z_{i}}\right)\right\|_{L_{2}(\hat{\nu})}=\left\|\sum_{i=1}^{n}\alpha_{i}j_{2}(\eta_{z_{i}})\right\|_{L_{2}(\hat{\nu})},$

and thus $\left\|T\right\|\leq 1$. Clearly, $f^{\prime}=T\circ g^{\prime}$.
Hence $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X)$ with
$\gamma^{\mathcal{B}}_{2}(f)\leq\left\|T\right\|p_{\mathcal{B}}(g)\leq c$. ∎

Theorem 1.9 admits the following remarkable reformulation.

######  Corollary 1.10.

Let $X$ be a complex Banach space and $f\in\mathcal{H}(\mathbb{D},X)$. The
following assertions are equivalent:

  1. (i)

$f$ factors its derivative through a Hilbert space.

  2. (ii)

There exists a constant $c\geq 0$ such that for every $n\in\mathbb{N}$, every
$n\times n$ unitary complex matrix $(a_{ij})$ and every
$(\mu_{j},w_{j})_{j=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$, we have

$\sum_{i=1}^{n}\left\|\sum_{j=1}^{n}\mu_{j}a_{ij}f^{\prime}(w_{j})\right\|^{2}\leq
c^{2}\sum_{j=1}^{n}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}.$

In this case, $\gamma^{\mathcal{B}}_{2}(f)$ is the minimum of all constants
$c$ satisfying the preceding inequality.

###### Proof.

$(i)\Rightarrow(ii)$: If $(i)$ holds, then Theorem 1.9 provides a constant
$c\geq 0$ such that

$\sum_{j=1}^{n}\left|\mu_{j}\right|^{2}\left\|f^{\prime}(w_{j})\right\|^{2}\leq
c^{2}\sum_{j=1}^{n}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}$

for all $n\in\mathbb{N}$ and
$(\mu_{j},w_{j})_{j=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$. Let $(a_{ij})$
be a $n\times n$ unitary complex matrix. Using the Cauchy–Schwarz Inequality,
we have

$\displaystyle\sum_{i=1}^{n}\left\|\sum_{j=1}^{n}\mu_{j}a_{ij}f^{\prime}(w_{j})\right\|^{2}$
$\displaystyle\leq\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\left|\mu_{j}\right|\left|a_{ij}\right|\left\|f^{\prime}(w_{j})\right\|\right)^{2}$
$\displaystyle\leq\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\left|\mu_{j}\right|^{2}\left\|f^{\prime}(w_{j})\right\|^{2}\right)\left(\sum_{j=1}^{n}\left|a_{ij}\right|^{2}\right)$
$\displaystyle=\left(\sum_{j=1}^{n}\left|\mu_{j}\right|^{2}\left\|f^{\prime}(w_{j})\right\|^{2}\right)\left(\sum_{i=1}^{n}\sum_{j=1}^{n}\left|a_{ij}\right|^{2}\right)$
$\displaystyle\leq\sum_{j=1}^{n}\left|\mu_{j}\right|^{2}\left\|f^{\prime}(w_{j})\right\|^{2}\leq
c^{2}\sum_{j=1}^{n}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}.$

$(ii)\Rightarrow(i)$: Assume there is a constant $c\geq 0$ such that

$\sum_{i=1}^{n}\left\|\sum_{j=1}^{n}\mu_{j}b_{ij}f^{\prime}(w_{j})\right\|^{2}\leq
c^{2}\sum_{j=1}^{n}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}$

for any $n\in\mathbb{N}$, any $n\times n$ unitary complex matrix $(b_{ij})$
and any $(\mu_{j},w_{j})_{j=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$. It
suffices if we show that the assertion $(ii)$ of Theorem 1.9 holds. For it,
let $n,m\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$ such that
$(\lambda_{i},z_{i})_{i=1}^{n}\prec(\mu_{j},w_{j})_{j=1}^{m}$. We can suppose
$m=n$ (otherwise we add some zeros). By Proposition 1.8, there is a $n\times
n$ complex matrix $A=(a_{ij})$ in $B_{\mathcal{L}(\ell_{2}^{n})}$ such that

$\lambda_{i}\gamma_{z_{i}}=\sum_{j=1}^{n}a_{ij}\mu_{j}\gamma_{w_{j}}$

for every $1\leq i\leq n$. Then, if $\phi\colon
B_{\mathcal{L}(\ell_{2}^{n})}\to\mathbb{R}^{+}$ is the convex functional
defined by

$\phi(B)=\sum_{i=1}^{n}\left\|\sum_{j=1}^{n}b_{ij}\mu_{j}f^{\prime}(w_{j})\right\|^{2}\qquad(B=(b_{ij})\in
B_{\mathcal{L}(\ell_{2}^{n})}),$

we have

$\phi(A)=\sum_{i=1}^{n}\left\|\sum_{j=1}^{n}a_{ij}\mu_{j}\gamma_{w_{j}}(f)\right\|^{2}=\sum_{i=1}^{n}\left\|\lambda_{i}\gamma_{z_{i}}(f)\right\|^{2}=\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}(z_{i})\right\|^{2}.$

Moreover, by hypothesis, we have

$\phi(B)\leq
c^{2}\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}$

for any $n\times n$ unitary complex matrix $B=(b_{ij})$. Since the extreme
points of $B_{\mathcal{L}(\ell_{2}^{n})}$ are precisely the unitary complex
matrices, then $B_{\mathcal{L}(\ell_{2}^{n})}$ coincides with the convex hull
of such matrices by the Minkowski–Carathéodory Theorem. Therefore we deduce
that

$\phi(A)\leq
c^{2}\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}},$

and this completes the proof. ∎

###  1.5. Duality

Our aim in this last subsection is to show that the space
$(\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*}),\gamma^{\mathcal{B}}_{2})$
can be canonically identified with the dual of the completion of the space
$\mathcal{G}(\mathbb{D})\otimes X$ with respect to a convenient norm.

We first recall some concepts introduced in [5]. In what follows, given a
complex Banach space $X$, we will sometimes write $\langle
x^{*},x\rangle=x^{*}(x)$ when $x\in X$ and $x^{*}\in X^{*}$. Given
$z\in\mathbb{D}$ and $x\in X$, the functional $\gamma_{z}\otimes
x\colon\widehat{\mathcal{B}}(\mathbb{D},X^{*})\to\mathbb{C}$ defined by

$\left(\gamma_{z}\otimes x\right)(f)=\left\langle
f^{\prime}(z),x\right\rangle\qquad\left(f\in\widehat{\mathcal{B}}(\mathbb{D},X^{*})\right),$

is linear and continuous with $\left\|\gamma_{z}\otimes
x\right\|=\left\|x\right\|/(1-|z|^{2})$. We set the linear space

$\mathrm{lin}(\Gamma(\mathbb{D}))\otimes
X=\mathrm{lin}\left\\{\gamma_{z}\otimes x\colon z\in\mathbb{D},\,x\in
X\right\\}\subseteq\widehat{\mathcal{B}}(\mathbb{D},X^{*})^{*},$

whose elements are called $X$-valued Bloch molecules on $\mathbb{D}$.

Our study of the duality of
$(\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*}),\gamma^{\mathcal{B}}_{2})$
requires to consider the following norm on these molecules.

######  Definition 1.11.

Let $X$ be a complex Banach space. Given
$\gamma\in\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$, define

$\left\|\gamma\right\|_{*}=\inf\left\\{\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}\right\\},$

where the infimum is taken over all such representations of $\gamma$ in the
form $\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes x_{i}$ such that
$(\lambda_{i},z_{i})_{i=1}^{n}\prec(\mu_{j},w_{j})_{j=1}^{m}$ with
$n,m\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$,
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$ and
$(x_{i})_{i=1}^{n}\in X^{n}$.

The concept of Bloch reasonable crossnorm was introduced in [5], inspired by
the analogue in the theory of tensor product (see, for example, [20]).

######  Theorem 1.12.

$\left\|\cdot\right\|_{*}$ is a Bloch reasonable crossnorm on
$\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$ for any complex Banach space $X$.

###### Proof.

Let $\gamma\in\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$ and let
$\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes x_{i}$ be a representation of
$\gamma$ with $n\in\mathbb{N}$, $(\lambda_{i})_{i=1}^{n}\in\mathbb{C}^{n}$,
$(z_{i})_{i=1}^{n}\in\mathbb{D}^{n}$ and $(x_{i})_{i=1}^{n}\in X^{n}$. Let
$(\mu_{j},w_{j})_{j=1}^{m}\succ(\lambda_{i},z_{i})_{i=1}^{n}$ with
$m\in\mathbb{N}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$.

Clearly, $\left\|\gamma\right\|_{*}\geq 0$. Given $\lambda\in\mathbb{C}$,
since $\lambda\gamma=\sum_{i=1}^{n}(\lambda\lambda_{i})\gamma_{z}\otimes
x_{i}$ and
$(\lambda\lambda_{i},z_{i})_{i=1}^{n}\prec(\lambda\mu_{j},w_{j})_{j=1}^{m}$,
we have

$\left\|\lambda\gamma\right\|_{*}\leq\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\frac{\left|\lambda\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}=\left|\lambda\right|\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}.$

If $\lambda=0$, we obtain
$\left\|\lambda\gamma\right\|_{*}=0=\left|\lambda\right|\left\|\gamma\right\|_{*}$.
For $\lambda\neq 0$, since the preceding inequality holds for every
representation of $\gamma$, we deduce that
$\left\|\lambda\gamma\right\|_{*}\leq\left|\lambda\right|\left\|\gamma\right\|_{*}$.
For the converse inequality, note that
$\left\|\gamma\right\|_{*}=\left\|\lambda^{-1}(\lambda\gamma)\right\|_{*}\leq|\lambda^{-1}|\left\|\lambda\gamma\right\|_{*}$
by the proved inequality, hence
$\left|\lambda\right|\left\|\gamma\right\|_{*}\leq\left\|\lambda\gamma\right\|_{*}$
and thus
$\left\|\lambda\gamma\right\|_{*}=\left|\lambda\right|\left\|\gamma\right\|_{*}$.

We now prove the triangular inequality of $\left\|\cdot\right\|_{*}$. Let
$\gamma_{1},\gamma_{2}\in\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$ and let
$\varepsilon>0$. We can choose a representation
$\gamma_{1}=\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes x_{i}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\succ(\lambda_{i},z_{i})_{i=1}^{n}$ such that

$\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\leq\left(\left\|\gamma_{1}\right\|_{*}+\varepsilon\right)^{\frac{1}{2}}\qquad\text{and}\qquad\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}\leq\left(\left\|\gamma_{1}\right\|_{*}+\varepsilon\right)^{\frac{1}{2}}.$

Similarly, take a representation
$\gamma_{2}=\sum_{i=n+1}^{n+k}\lambda_{i}\gamma_{z_{i}}\otimes x_{i}$ and
$(\mu_{j},w_{j})_{j=m+1}^{m+l}\succ(\lambda_{i},z_{i})_{i=n+1}^{n+k}$ such
that

$\left(\sum_{i=n+1}^{n+k}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\leq\left(\left\|\gamma_{2}\right\|_{*}+\varepsilon\right)^{\frac{1}{2}}\qquad\text{and}\qquad\left(\sum_{j=m+1}^{m+l}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}\leq\left(\left\|\gamma_{2}\right\|_{*}+\varepsilon\right)^{\frac{1}{2}}.$

Then $\gamma_{1}+\gamma_{2}=\sum_{i=1}^{n+k}\lambda_{i}\gamma_{z_{i}}\otimes
x_{i}$, $(\mu_{j},w_{j})_{j=1}^{m+l}\succ(\lambda_{i},z_{i})_{i=1}^{n+k}$ and

$\left(\sum_{i=n+1}^{n+k}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m+l}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}\leq\left\|\gamma_{1}\right\|_{*}+\left\|\gamma_{2}\right\|_{*}+2\varepsilon.$

hence
$\left\|\gamma_{1}+\gamma_{2}\right\|_{*}\leq\left\|\gamma_{1}\right\|_{*}+\left\|\gamma_{2}\right\|_{*}+2\varepsilon$,
and thus
$\left\|\gamma_{1}+\gamma_{2}\right\|_{*}\leq\left\|\gamma_{1}\right\|_{*}+\left\|\gamma_{2}\right\|_{*}$
by the arbitrariness of $\varepsilon$.

Hence $\left\|\cdot\right\|_{*}$ is a seminorm. To prove that it is a norm,
note first that

$\displaystyle\left|\sum_{i=1}^{n}\lambda_{i}g^{\prime}(z_{i})x^{*}(x_{i})\right|$
$\displaystyle\leq\sum_{i=1}^{n}\left|\lambda_{i}\right|\left|g^{\prime}(z_{i})\right|\left\|x_{i}\right\|$
$\displaystyle\leq\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|g^{\prime}(z_{i})\right|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|g^{\prime}(w_{j})\right|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}$

for any $g\in B_{\widehat{\mathcal{B}}(\mathbb{D})}$ and $x^{*}\in B_{X^{*}}$.
Since the first quantity
$\left|\sum_{i=1}^{n}\lambda_{i}g^{\prime}(z_{i})x^{*}(x_{i})\right|$ does not
depend on the representation of $\gamma$ because

$\sum_{i=1}^{n}\lambda_{i}g^{\prime}(z_{i})x^{*}(x_{i})=\left(\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes
x_{i}\right)(g\cdot x^{*})=\gamma(g\cdot x^{*}),$

taking the infimum over all representations of $\gamma$ we deduce that

$\left|\sum_{i=1}^{n}\lambda_{i}g^{\prime}(z_{i})x^{*}(x_{i})\right|\leq\left\|\gamma\right\|_{*}$

for any $g\in B_{\widehat{\mathcal{B}}(\mathbb{D})}$ and $x^{*}\in B_{X^{*}}$.
Now, if $\left\|\gamma\right\|_{*}=0$, the preceding inequality yields

$\left(\sum_{i=1}^{n}\lambda_{i}x^{*}(x_{i})\gamma_{z_{i}}\right)(g)=\sum_{i=1}^{n}\lambda_{i}x^{*}(x_{i})g^{\prime}(z_{i})=0$

for all $g\in B_{\widehat{\mathcal{B}}(\mathbb{D})}$ and $x^{*}\in B_{X^{*}}$.
For each $x^{*}\in B_{X^{*}}$, this implies that
$\sum_{i=1}^{n}\lambda_{i}x^{*}(x_{i})\gamma_{z_{i}}=0$, and since
$\Gamma(\mathbb{D})$ is a linearly independent subset of
$\mathcal{G}(\mathbb{D})$ by [12, Remark 2.8], it follows that
$x^{*}(x_{i})\lambda_{i}=0$ for all $i\in\\{1,\ldots,n\\}$, hence
$\lambda_{i}=0$ for all $i\in\\{1,\ldots,n\\}$ since $B_{X^{*}}$ separates the
points of $X$, and thus $\gamma=0$.

Finally, the norm $\left\|\cdot\right\|_{*}$ is a Bloch reasonable crossnorm
since it holds the two conditions of [5, Definition 2.5]:

  1. (i)

Given $z\in\mathbb{D}$ and $x\in X$, we have

$\left\|\gamma_{z}\otimes
x\right\|_{*}\leq\frac{\left\|x\right\|}{1-|z|^{2}}=\left\|\gamma_{z}\right\|\left\|x\right\|.$

  2. (ii)

For $g\in\widehat{\mathcal{B}}(\mathbb{D})$ and $x^{*}\in X^{*}$, we have

$\displaystyle\left|(g\otimes x^{*})(\gamma)\right|$
$\displaystyle=\left|\sum_{i=1}^{n}\lambda_{i}(g\otimes
x^{*})(\gamma_{z_{i}}\otimes
x_{i})\right|=\left|\sum_{i=1}^{n}\lambda_{i}g^{\prime}(z_{i})x^{*}(x_{i})\right|$
$\displaystyle\leq\left\|x^{*}\right\|\sum_{i=1}^{n}\left|\lambda_{i}\right|\left|g^{\prime}(z_{i})\right|\left\|x_{i}\right\|$
$\displaystyle\leq\left\|x^{*}\right\|\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left|g^{\prime}(z_{i})\right|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left\|x^{*}\right\|\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\left|\mu_{j}\right|^{2}\left|g^{\prime}(w_{j})\right|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left\|x^{*}\right\|p_{\mathcal{B}}(g)\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}},$

and taking infimum over all representations of $\gamma$, we obtain
$\left|(g\otimes x^{*})(\gamma)\right|\leq
p_{\mathcal{B}}(g)\left\|x^{*}\right\|\left\|\gamma\right\|_{*}$. Hence
$g\otimes
x^{*}\in(\mathrm{lin}(\Gamma(\mathbb{D}))\otimes_{\left\|\cdot\right\|_{*}}X)^{*}$
with $\left\|g\otimes x^{*}\right\|\leq
p_{\mathcal{B}}(g)\left\|x^{*}\right\|$.

∎

We are now ready to state the announced result.

######  Theorem 1.13.

Let $X$ be a complex Banach space. Then
$(\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*}),\gamma^{\mathcal{B}}_{2})$
is isometrically isomorphic to
$(\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X)^{*}$,
via the linear mapping
$\Lambda\colon\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})\to(\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X)^{*}$
defined by

$\Lambda(f)(\lambda\gamma_{z}\otimes x)=\lambda\left\langle
f^{\prime}(z),x\right\rangle$

for $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$,
$\lambda\in\mathbb{C}$, $z\in\mathbb{D}$ and $x\in X$, with inverse given by

$\left\langle\Lambda^{-1}(\varphi)(z),x\right\rangle=\left\langle\varphi,\gamma_{z}\otimes
x\right\rangle$

for
$\varphi\in(\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X)^{*}$,
$z\in\mathbb{D}$ and $x\in X$.

Moreover, on the unit ball of
$\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$ the weak* topology
coincides with the topology of pointwise $\sigma(X^{*},X)$-convergence.

###### Proof.

Let $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$ and let
$\Lambda_{0}(f)\colon\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X\to\mathbb{C}$
be the linear functional given by

$\Lambda_{0}(f)(\gamma)=\sum_{i=1}^{n}\lambda_{i}\left\langle
f^{\prime}(z_{i}),x_{i}\right\rangle$

for $\gamma=\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes
x_{i}\in\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$ with $n\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(x_{i})_{i=1}^{n}\in X^{n}$. Let
$(\mu_{j},w_{j})_{j=1}^{m}\succ(\lambda_{i},z_{i})_{i=1}^{n}$ with
$m\in\mathbb{N}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$. Note that
$\Lambda_{0}(f)\in(\mathrm{lin}(\Gamma(\mathbb{D}))\otimes_{\left\|\cdot\right\|_{*}}X)^{*}$
with $\left\|\Lambda_{0}(f)\right\|\leq\gamma^{\mathcal{B}}_{2}(f)$. Indeed,
by using Theorem 1.9, we have

$\displaystyle\left|\Lambda_{0}(f)(\gamma)\right|$
$\displaystyle=\left|\sum_{i=1}^{n}\lambda_{i}\left\langle
f^{\prime}(z_{i}),x_{i}\right\rangle\right|\leq\sum_{i=1}^{n}\left|\lambda_{i}\right|\left\|f^{\prime}(z_{i})\right\|\left\|x_{i}\right\|$
$\displaystyle\leq\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}(z_{i})\right\|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\gamma^{\mathcal{B}}_{2}(f)\left(\sum_{i=1}^{n}\left\|x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}$

and taking infimum over all the representations of $\gamma$, we deduce that
$\left|\Lambda_{0}(f)(\gamma)\right|\leq\gamma^{\mathcal{B}}_{2}(f)\left\|\gamma\right\|_{*}$,
and thus $\left\|\Lambda_{0}(f)\right\|\leq\gamma^{\mathcal{B}}_{2}(f)$.

Since $\mathrm{lin}(\Gamma(\mathbb{D}))$ is a norm-dense linear subspace of
$\mathcal{G}(\mathbb{D})$ and $\left\|\cdot\right\|_{*}$ is a norm on
$\mathcal{G}(\mathbb{D})\otimes X$, then $\mathcal{G}(\mathbb{D})\otimes X$ is
a dense linear subspace of
$\mathcal{G}(\mathbb{D})\otimes_{\left\|\cdot\right\|_{*}}X$ and therefore
also of its completion
$\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X$. Hence
there is a unique continuous mapping $\Lambda(f)$ from
$\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X$ into
$\mathbb{C}$ that extends $\Lambda_{0}(f)$. Further, $\Lambda(f)$ is linear
and $\left\|\Lambda(f)\right\|=\left\|\Lambda_{0}(f)\right\|$.

Let
$\Lambda\colon\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})\to(\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X)^{*}$
be the map so defined. It is immediate that
$\Lambda_{0}\colon\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})\to(\mathcal{G}(\mathbb{D})\otimes_{\left\|\cdot\right\|_{*}}X)^{*}$
is linear. If $f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$ and
$\Lambda_{0}(f)=0$, then $\left\langle
f^{\prime}(z),x\right\rangle=\Lambda_{0}(f)(\gamma_{z}\otimes x)=0$ for all
$z\in\mathbb{D}$ and $x\in X$, hence $f^{\prime}(z)=0$ for all
$z\in\mathbb{D}$ and therefore $f=0$. This proves that $\Lambda_{0}$ is
injective. It follows easily that $\Lambda$ is also linear and injective.

To prove that $\Lambda$ is a surjective isometry, let
$\varphi\in(\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X)^{*}$
and define $F_{\varphi}\colon\mathbb{D}\to X^{*}$ by

$\left\langle F_{\varphi}(z),x\right\rangle=\varphi(\gamma_{z}\otimes
x)\qquad\left(z\in\mathbb{D},\;x\in X\right).$

An argument similar to that in the proof of Proposition 2.4 in [5] assures
that there exists a mapping
$f_{\varphi}\in\widehat{\mathcal{B}}(\mathbb{D},X^{*})$ with
$p_{\mathcal{B}}(f_{\varphi})\leq\left\|\varphi\right\|$ such that
$f_{\varphi}^{\prime}=F_{\varphi}$.

We now see that
$f_{\varphi}\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$. Let
$n,m\in\mathbb{N}$,
$(\lambda_{i},z_{i})_{i=1}^{n}\in(\mathbb{C}\times\mathbb{D})^{n}$ and
$(\mu_{j},w_{j})_{j=1}^{m}\in(\mathbb{C}\times\mathbb{D})^{m}$ such that
$(\lambda_{i},z_{i})_{i=1}^{n}\prec(\mu_{j},w_{j})_{j=1}^{m}$. Let
$(x_{i})_{i=1}^{n}\in X^{n}$ and $\varepsilon>0$. For each
$i\in\\{1,\ldots,n\\}$, there exists $x_{i}\in X$ with
$\left\|x_{i}\right\|\leq 1+\varepsilon$ such that $\left\langle
f_{\varphi}^{\prime}(z_{i}),x_{i}\right\rangle=\left\|f_{\varphi}^{\prime}(z_{i})\right\|$.
Clearly, the function $T\colon\mathbb{C}^{n}\to\mathbb{C}$ given by

$T(t_{1},\ldots,t_{n})=\sum_{i=1}^{n}t_{i}\lambda_{i}\left\|f_{\varphi}^{\prime}(z_{i})\right\|,\quad\forall(t_{1},\ldots,t_{n})\in\mathbb{C}^{n},$

is linear and continuous on $(\mathbb{C}^{n},||\cdot||_{2})$ with
$\left\|T\right\|=\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}_{\varphi}(z_{i})\right\|^{2}\right)^{\frac{1}{2}}$.
For all $(t_{1},\ldots,t_{n})\in\mathbb{C}^{n}$ with
$||(t_{1},\ldots,t_{n})||_{2}\leq 1$, we have

$\displaystyle\left|T(t_{1},\ldots,t_{n})\right|$
$\displaystyle=\left|\varphi\left(\sum_{i=1}^{n}t_{i}\lambda_{i}\gamma_{z_{i}}\otimes
x_{i}\right)\right|\leq\left\|\varphi\right\|\left\|\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes
t_{i}x_{i}\right\|_{*}$
$\displaystyle\leq\left\|\varphi\right\|\left(\sum_{i=1}^{n}\left\|t_{i}x_{i}\right\|^{2}\right)^{\frac{1}{2}}\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}}$
$\displaystyle\leq(1+\varepsilon)\left\|\varphi\right\|\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}},$

therefore

$\left(\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}_{\varphi}(z_{i})\right\|^{2}\right)^{\frac{1}{2}}\leq(1+\varepsilon)\left\|\varphi\right\|\left(\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}\right)^{\frac{1}{2}},$

and the arbitrariness of $\varepsilon$ yields

$\sum_{i=1}^{n}\left|\lambda_{i}\right|^{2}\left\|f^{\prime}_{\varphi}(z_{i})\right\|^{2}\leq\left\|\varphi\right\|^{2}\sum_{j=1}^{m}\frac{\left|\mu_{j}\right|^{2}}{(1-|w_{j}|^{2})^{2}}.$

Therefore $f_{\varphi}\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$
with $\gamma^{\mathcal{B}}_{2}(f_{\varphi})\leq\left\|\varphi\right\|$ by
Theorem 1.9.

Finally, for any $\gamma=\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes
x_{i}\in\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$, we get

$\Lambda(f_{\varphi})(\gamma)=\sum_{i=1}^{n}\lambda_{i}\left\langle
f^{\prime}_{\varphi}(z_{i}),x_{i}\right\rangle=\sum_{i=1}^{n}\lambda_{i}\varphi(\gamma_{z_{i}}\otimes
x_{i})=\varphi\left(\sum_{i=1}^{n}\lambda_{i}\gamma_{z_{i}}\otimes
x_{i}\right)=\varphi(\gamma).$

Hence $\Lambda(f_{\varphi})=\varphi$ on a dense subspace of
$\mathcal{G}(\mathbb{D})\widehat{\otimes}_{\left\|\cdot\right\|_{*}}X$ and,
consequently, $\Lambda(f_{\varphi})=\varphi$. Moreover,
$\gamma^{\mathcal{B}}_{2}(f_{\varphi})\leq\left\|\varphi\right\|=\left\|\Lambda(f_{\varphi})\right\|$.

The proof is complete if we prove the last assertion of the statement. Let
$(f_{i})_{i\in I}$ be a net in
$\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$ and
$f\in\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$. Assume that
$(f_{i})_{i\in I}\to f$ weak* in
$\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$. This implies that

$(<f^{\prime}_{i}(z),x>)_{i\in I}=(\Lambda(f_{i})(\gamma_{z}\otimes x))_{i\in
I}\to\Lambda(f)(\gamma_{z}\otimes x)=<f^{\prime}(z),x>$

for every $z\in\mathbb{D}$ and $x\in X$. Given $z\in\mathbb{D}$ and $x\in X$,
we have

$\displaystyle\left|\left\langle f_{i}(z)-f(z),x\right\rangle\right|$
$\displaystyle=\left|\int_{[0,z]}\left\langle
f^{\prime}_{i}(w)-f^{\prime}(w),x\right\rangle\ dw\right|$
$\displaystyle\leq|z|\max\left\\{\left|\left\langle
f^{\prime}_{i}(w)-f^{\prime}(w),x\right\rangle\right|\colon
w\in[0,z]\right\\}$ $\displaystyle=|z|\left|\left\langle
f^{\prime}_{i}(w_{z})-f^{\prime}(w_{z}),x\right\rangle\right|$

for all $i\in I$ and some $w_{z}\in[0,z]$, and thus $(\left\langle
f_{i}(z),x\right\rangle)_{i\in I}\to\left\langle f(z),x\right\rangle$. Hence
$(f_{i})_{i\in I}\to f$ in the topology of pointwise
$\sigma(X^{*},X)$-convergence. Therefore the identity on
$\Gamma^{\widehat{\mathcal{B}}}_{2}(\mathbb{D},X^{*})$ is a continuous
bijection from the weak* topology to the topology of pointwise
$\sigma(X^{*},X)$-convergence. On the unit ball, the former topology is
compact and the latter is Hausdorff, and so they are equal. ∎

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