On (p, r, s)-summing Bloch maps and Lapresté norms

Abstract

The theory of (p, r, s)-summing and (p, r, s)-nuclear linear operators on Banach spaces was developed by Pietsch in his book on operator ideals (Pietsch in Operator ideals, North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980, Chapters 17 and 18) Due to recent advances in the theory of ideals of Bloch maps, we extend these concepts to Bloch maps from the complex open unit disc $\mathbb {D}$ into a complex Banach space X. Variants for (r, s)-dominated Bloch maps of classical Pietsch’s domination and Kwapień’s factorization theorems of (r, s)-dominated linear operators are presented. We define analogues of Lapresté’s tensor norms on the space of X-valued Bloch molecules on $\mathbb {D}$ to address the duality of the spaces of $(p^*,r,s)$-summing Bloch maps from $\mathbb {D}$ into $X^*$. The class of (p, r, s)-nuclear Bloch maps is introduced and analysed to give examples of (p, r, s)-summing Bloch maps.

1 Introduction

The concept of absolutely p-summing linear operators between Banach spaces for $0<p\le \infty$ was introduced by Pietsch [13] and the notion of absolutely (p, r)-summing operators for $0<r\le p\le \infty$ is due to Mitjagin and Pełczyński [11] though the famous factorization theorem for (p, r)-dominated operators was proved by Kwapień [9].

In his famous monograph about operator ideals [13], Pietsch introduced a more general multi-index concept with the definition of (p, r, s)-summing operators for $0<p,r,s\le \infty$ and $1/p\le 1/r+1/s$. The study of the duality of these operator spaces was addressed with the introduction of suitable norms on the tensor product of Banach spaces by Chevet [5], Saphar [15] and Lapresté [10].

In other settings, (p, r, s)-summing maps have been dealed by some authors as, for example, Chávez-Domínguez [4] for Lipschitz maps, and Achour [1], Bernardino, Pellegrino, Seoane-Sepúlveda and Souza [2] and Fernández-Unzueta and García-Hernández [7] for multilinear operators and polynomials.

Our main purpose in this paper is to introduce and establish the most notable properties of a notion of (p, r, s)-summing Bloch maps from the complex open unit disc $\mathbb {D}$ into a complex Banach space X.

Let $\mathcal {H}(\mathbb {D},X)$ be the space of all holomorphic maps from $\mathbb {D}$ into X. Let us recall that a map $f\in \mathcal {H}(\mathbb {D},X)$ is called Bloch if

$$\begin{aligned} \rho _{\mathcal {B}}\left( f\right) =\sup \left\{ (1-|z|^{2})\left\| f^{\prime }(z)\right\| :z\in \mathbb {D}\right\} <\infty . \end{aligned}$$

The linear space of all Bloch maps from $\mathbb {D}$ into X, under the Bloch seminorm $\rho _{\mathcal {B}}$, is denoted by $\mathcal {B}(\mathbb {D},X)$. The normalized Bloch space $\widehat{\mathcal {B}}(\mathbb {D},X)$ is the closed subspace of $\mathcal {B}(\mathbb {D},X)$ formed by all those maps f for which $f(0)=0$, under the Bloch norm $\rho _{\mathcal {B}}$. For simplicity, we write $\widehat{\mathcal {B}}(\mathbb {D})$ instead of $\widehat{\mathcal {B}}(\mathbb {D},\mathbb {C})$. We denote by $\widehat{\mathcal {B}}(\mathbb {D},\mathbb {D})$ the set of all holomorphic functions $h:\mathbb {D}\rightarrow \mathbb {D}$ for which $h(0)=0$.

We now introduce some notation. For Banach spaces X and Y, $\mathcal {L}(X,Y)$ denotes the Banach space of all continuous linear operators from X into Y, equipped with the operator canonical norm. As usual, $X^*$ denotes the dual space $\mathcal {L}(X,\mathbb {K})$, and $J_X$ the canonical injection of X into $X^{**}$. $B_{X}$ stands for the closed unit ball of X. Given $1\le p\le \infty$, $p^*$ denotes the conjugate index of p defined by $p^*=p/(p-1)$ if $p\ne 1$, $p^*=\infty$ if $p=1$, and $p^*=1$ if $p=\infty$.

Let X be a Banach space, $n\in \mathbb {N}$ and a finite set of vectors $(x_i)_{i=1}^n$ in X. For any $1\le p\le \infty$, the strong p-norm of $(x_i)_{i=1}^n$ is defined by

$$\begin{aligned} \left\| (x_i)_{i=1}^n\right\| _p=\left\{ \begin{array}{lll} \left( \displaystyle \sum _{i=1}^n\left\| x_i\right\| ^p\right) ^{\frac{1}{p}}&{} \text {if} &{} 1\le p<\infty , \\ &{} &{}\\ \displaystyle \max _{1\le i\le n}\left\| x_i\right\| &{} \text {if} &{} p=\infty , \end{array}\right. \end{aligned}$$

and the weak p-norm of $(x_i)_{i=1}^n$ by

$$\begin{aligned} \omega _p\left( (x_i)_{i=1}^n\right) =\sup _{x^*\in \mathcal {B}_{X^*}}\left\| (x^*(x_i))_{i=1}^n\right\| _p. \end{aligned}$$

According to Pietsch [14, 17.1.1], given Banach spaces X, Y and $0<p,r,s\le \infty$ such that $1/p\le 1/r+1/s$, an operator $T\in \mathcal {L}(X,Y)$ is (p, r, s)-summing if there exists a constant $C\ge 0$ such that

$$\begin{aligned} \left\| (y^*_i(T(x_i)))_{i=1}^n\right\| _p\le C \omega _r\left( (x_i)_{i=1}^n\right) \omega _s\left( (y^*_i)_{i=1}^n\right) \end{aligned}$$

for any $n\in \mathbb {N}$, $(x_i)_{i=1}^n$ in X and $(y^*_i)_{i=1}^n$ in $Y^*$. The least of all constants C for which such an inequality holds is denoted by $\pi _{(p,r,s)}(T)$, and the linear space of all such operators is represented by $\Pi _{(p,r,s)}(X,Y)$.

We now propose a Bloch version of the notion of (p, r, s)-summing linear operators. Towards this end, we introduce a third norm: given two finite sets of points $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$ and $(z_i)_{i=1}^n$ in $\mathbb {D}$, the weak Bloch p-norm of $(\lambda _i,z_i)_{i=1}^n$ is defined by

$$\begin{aligned} \omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right) =\sup _{g\in \mathcal {B}_{\widehat{\mathcal {B}}(\mathbb {D})}}\left\| (\lambda _i g'(z_i))_{i=1}^n\right\| _p. \end{aligned}$$

In particular, we write $\omega ^{\widehat{\mathcal {B}}}_p\left( (z_i)_{i=1}^n\right)$ instead of $\omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right)$ if $\lambda _i=1$ for all $i\in \{1,\ldots ,n\}$.

Definition 0.1

Let X be a complex Banach space and let $1\le p,r,s\le \infty$ such that $1/p\le 1/r+1/s$. We say that a map $f\in \mathcal {H}(\mathbb {D},X)$ is (p, r, s)-summing Bloch if there is a constant $C\ge 0$ such that for any $n\in \mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$, $(z_i)_{i=1}^n$ in $\mathbb {D}$ and $(x^*_i)_{i=1}^n$ in $X^*$, we have

$$\begin{aligned} \left\| (\lambda _ix^*_i(f'(z_i)))_{i=1}^n\right\| _p\le C \omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _s\left( (x^*_i)_{i=1}^n\right) . \end{aligned}$$

The smallest such constants C is denoted by $\pi _{(p,r,s)}^{\mathcal {B}}(f)$. The linear space of all such maps is denoted by $\Pi _{(p,r,s)}^{\mathcal {B}}(\mathbb {D},X)$, and $\Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ stands for its subspace formed by all those maps f for which $f(0)=0$. A map (p, r, s)-summing Bloch map f from $\mathbb {D}$ into X is called (r, s)-dominated Bloch whenever $1/p=1/r+1/s$.

We now describe the contents of this paper. In parallelism with the theory of absolutely (p, r, s)-summing operators, we prove that $[\Pi ^{\widehat{\mathcal {B}}}_{(p,r,s)},\pi ^{\mathcal {B}}_{(p,r,s)}]$ is a Banach ideal of normalized Bloch maps. We also show that the space $(\Pi ^{\mathcal {B}}_{(p,r,s)}(\mathbb {D},X),\pi ^{\mathcal {B}}_{(p,r,s)})$ is Möbius-invariant in an approach to Complex Analysis.

For $1\le p,r,s<\infty$ such that $1/p=1/r+1/s$, our main result in this paper gathers both variants for (r, s)-dominated Bloch maps of Pietsch’s domination and Kwapień’s factorization theorems for (r, s)-dominated linear operators (see [14, Theorems 7.4.2 and 7.4.3]).

In order to address the duality of the $\Pi ^{\widehat{\mathcal {B}}}_{(p,r,s)}$-spaces, we introduce Bloch analogues of Lapresté norms [10] on the space of X-valued Bloch molecules of $\mathbb {D}$, denoted by $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}$. For $1\le p,r,s\le \infty$ and $1/\theta :=1/p+1/r+1/s\ge 1$, we prove that $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}$ is a Bloch reasonable $\theta$-crossnorm on such a space so that, whenever $1/p^*\le 1/r+1/s$, $(\Pi _{(p^*,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X^*),\pi _{(p^*,r,s)}^{\mathcal {B}})$ is isometrically isomorphic to $(\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X,\mu _{(p,r,s)}^{\widehat{\mathcal {B}}})^*$, where $\mathcal {G}(\mathbb {D})$ is the Bloch-free Banach space of $\mathbb {D}$.

In order to give examples of (p, r, s)-summing Bloch maps, the concept of (p, r, s)-nuclear Bloch maps from $\mathbb {D}$ into X for $1\le p,r,s\le \infty$ such that $1+1/p\ge 1/r+1/s$ is introduced and it is proved that the space formed by such Bloch maps is an $\theta$-Banach normalized Bloch ideal where $1/\theta :=1/p+1/r^*+1/s^*$.

2 Results

From now on and unless otherwise stated, we will suppose that X is a complex Banach space and $1\le p,r,s\le \infty$ with $1/p\le 1/r+1/s$.

2.1 Inclusions

We first show that the new functions introduced are actually Bloch functions.

Given semi-normed spaces $(X,\rho _X)$ and $(Y,\rho _Y)$, we will write $(X,\rho _X)\le (Y,\rho _Y)$ to point out that $X\subseteq Y$ and $\rho _Y(x)\le \rho _X(x)$ for all $x\in X$.

Proposition 1.1

$(\Pi ^{\mathcal {B}}_{(p,r,s)}(\mathbb {D},X),\pi ^{\mathcal {B}}_{(p,r,s)})\le (\mathcal {B}(\mathbb {D},X),\rho _\mathcal {B})$.

Proof

Let $f\in \Pi ^{\mathcal {B}}_{(p,r,s)}(\mathbb {D},X)$. For each $z\in \mathbb {D}$, Hahn–Banach Theorem provides a functional $x^*_z\in B_{X^*}$ such that $\left| x^*_z(f'(z))\right| =\left\| f'(z)\right\|$. Taking $n=1$, $\lambda _1=(1-|z|^2)$, $z_1=z$ and $x^*_1=x^*_z$, we have

$$\begin{aligned} (1-|z|^2)\left\| f'(z)\right\|&=(1-|z|^2)\left| x^*_z(f'(z))\right| \\&\le \pi ^{\mathcal {B}}_{(p,r,s)}(f)\omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _s\left( (x^*_i)_{i=1}^n\right) \\&\le \pi ^{\mathcal {B}}_{(p,r,s)}(f). \end{aligned}$$

Hence $f\in \mathcal {B}(\mathbb {D},X)$ with $\rho _\mathcal {B}(f)\le \pi ^{\mathcal {B}}_{(p,r,s)}(f)$. $\square$

We now prove that the concept of (p, r, s)-summing Bloch maps extends that of p-summing Bloch maps introduced in [3].

For any $1\le p\le \infty$, let us recall that a map $f\in \mathcal {H}(\mathbb {D},X)$ is p-summing Bloch if there is a constant $C\ge 0$ such that

$$\begin{aligned} \left\| (\lambda _i f'(z_i))_{i=1}^n\right\| _p\le C \omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right) \end{aligned}$$

for any $n\in \mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$ and $(z_i)_{i=1}^n$ in $\mathbb {D}$. The infimum of all constants C for which such an inequality holds, denoted by $\pi ^{\mathcal {B}}_p(f)$, defines a seminorm on the linear space, denoted by $\Pi ^{\mathcal {B}}_p(\mathbb {D},X)$, of all p-summing Bloch maps from $\mathbb {D}$ into X. Furthermore, $\pi ^{\mathcal {B}}_p$ is a norm on the subspace $\Pi ^{\widehat{\mathcal {B}}}_p(\mathbb {D},X)$ formed by all those maps $f\in \Pi ^{\mathcal {B}}_p(\mathbb {D},X)$ for which $f(0)=0$.

Proposition 1.2

$(\Pi ^{\mathcal {B}}_{p,p,\infty }(\mathbb {D},X),\pi ^{\mathcal {B}}_{p,p,\infty })=(\Pi ^{\mathcal {B}}_p(\mathbb {D},X),\pi ^{\mathcal {B}}_p)$.

Proof

Let $f\in \Pi ^{\mathcal {B}}_{(p,p,\infty )}(\mathbb {D},X)$. Given $n\in \mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$ and $(z_i)_{i=1}^n$ in $\mathbb {D}$, we have

$$\begin{aligned} \left\| (\lambda _if'(z_i))_{i=1}^n\right\| _p&=\left\| (\lambda _ix^*_i(f'(z_i)))_{i=1}^n\right\| _p\\&\le \pi ^{\mathcal {B}}_{(p,p,\infty )}(f)\omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _\infty \left( (x^*_i)_{i=1}^n\right) \\&\le \pi ^{\mathcal {B}}_{(p,p,\infty )}(f)\omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right) \end{aligned}$$

where we have taken $x^{*}_i\in B_{X^*}$ such that $\left| x^*_i(f'(z_i))\right| =\left\| f'(z_i)\right\|$ for each $i\in \{1,\ldots ,n\}$ by Hahn–Banach Theorem. Hence $f\in \Pi ^{\mathcal {B}}_p(\mathbb {D},X)$ with $\pi ^{\mathcal {B}}_p(f)\le \pi ^{\mathcal {B}}_{(p,p,\infty )}(f)$.

Conversely, let $f\in \Pi ^{\mathcal {B}}_p(\mathbb {D},X)$. Let $n\in \mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$, $(z_i)_{i=1}^n$ in $\mathbb {D}$ and $(x^*_i)_{i=1}^n$ in $X^*$. For each $i\in \{1,\ldots ,n\}$, Hahn–Banach Theorem provides a functional $y^{**}_i\in B_{X^{**}}$ such that $\left| y_i^{**}(x^*_i)\right| =\left\| x^*_i\right\|$. We obtain

$$\begin{aligned} \left\| (\lambda _ix^*_i(f'(z_i)))_{i=1}^n\right\| _p&\le \left\| (\lambda _i\left\| x^*_i\right\| f'(z_i))_{i=1}^n\right\| _p\\&\le \left\| (\lambda _if'(z_i))_{i=1}^n\right\| _p\left\| (x^*_i)_{i=1}^n\right\| _\infty \\&\le \pi ^{\mathcal {B}}_p(f)\omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right) \left\| (x^*_i)_{i=1}^n\right\| _\infty \\&=\pi ^{\mathcal {B}}_p(f)\omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right) \left\| (y_i^{**}(x^*_i))_{i=1}^n\right\| _\infty \\&\le \pi ^{\mathcal {B}}_p(f)\omega ^{\widehat{\mathcal {B}}}_p\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _\infty \left( (x^*_i)_{i=1}^n\right) , \end{aligned}$$

and thus $f\in \Pi ^{\mathcal {B}}_{(p,p,\infty )}(\mathbb {D},X)$ with $\pi ^{\mathcal {B}}_{(p,p,\infty )}(f)\le \pi ^{\mathcal {B}}_p(f)$. $\square$

2.2 Banach Bloch ideal property

Given $\theta \in (0,1]$ and a linear space X over $\mathbb {K}$, recall that a $\theta$-norm on X is a function $\mu :X\rightarrow \mathbb {R}$ satisfying that $x=0$ whenever $\mu (x)=0$, $\mu (\lambda x)=|\lambda |\mu (x)$ for all $\lambda \in \mathbb {K}$ and $x\in X$, and $\mu (x+y)^\theta \le \mu (x)^\theta +\mu (y)^\theta$ for all $x,y\in X$. We say that $(X,\mu )$ is an $\theta$-normed space, and it is said that $(X,\mu )$ is an $\theta$-Banach space if every Cauchy sequence in $(X,\mu )$ converges in $(X,\mu )$.

Following [8, Definition 5.11], a $\theta$-normed ($\theta$-Banach) normalized Bloch ideal, denoted as $[\mathcal {I}^{\widehat{\mathcal {B}}},\left\| \cdot \right\| _{\mathcal {I}^{\widehat{\mathcal {B}}}}]$, is a subclass $\mathcal {I}^{\widehat{\mathcal {B}}}$ equipped with a $\theta$-norm $\left\| \cdot \right\| _{\mathcal {I}^{\widehat{\mathcal {B}}}}$ of the class of all normalized Bloch maps $\widehat{\mathcal {B}}$ endowed with the Bloch norm $\rho _{\mathcal {B}}$ such that for each complex Banach space X, the components $\mathcal {I}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ satisfy the following properties:

• (P1) $(\mathcal {I}^{\widehat{\mathcal {B}}}(\mathbb {D},X),\left\| \cdot \right\| _{\mathcal {I}^{\widehat{\mathcal {B}}}})$ is a $\theta$-normed ($\theta$-Banach) space and $\rho _{\mathcal {B}}(f)\le \left\| f\right\| _{\mathcal {I}^{\widehat{\mathcal {B}}}}$ for $f\in \mathcal {I}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$.

• (P2) For any $g\in \widehat{\mathcal {B}}(\mathbb {D})$ and $x\in X$, the map $g\cdot x:\mathbb {D}\rightarrow X$, given by $(g \cdot x)(z) = g(z)x$ if $z\in \mathbb {D}$, is in $\mathcal {I}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ and $\left\| g\cdot x\right\| _{\mathcal {I}^{\widehat{\mathcal {B}}}}=\rho _{\mathcal {B}}(g)\left\| x\right\|$.

• (P3) The ideal property: if $f\in \mathcal {I}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$, $h\in \widehat{\mathcal {B}}(\mathbb {D},\mathbb {D})$ and $T\in \mathcal {L}(X,Y)$ where Y is a complex Banach space, then $T\circ f\circ h$ belongs to $\mathcal {I}^{\widehat{\mathcal {B}}}(\mathbb {D},Y)$ and $\Vert T\circ f\circ h\Vert _{\mathcal {I}^{\widehat{\mathcal {B}}}}\le \Vert T\Vert \,\Vert f\Vert _{\mathcal {I}^{\widehat{\mathcal {B}}}}$.

In the case $\theta =1$, we remove any reference to $\theta$.

Proposition 1.3

$\left[ \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}},\pi _{(p,r,s)}^{\mathcal {B}}\right]$ is a Banach normalized Bloch ideal.

Proof

Let X be a complex Banach space and let $n\in \mathbb {N}$, $\lambda _i\in \mathbb {C}$, $z_i\in \mathbb {D}$ and $x^*_i\in X^*$ for all $i\in \{1,\ldots ,n\}$.

(P1) Given $f\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$, it is clear that $\pi _{(p,r,s)}^{\mathcal {B}}(f)\ge 0$. If $\pi ^{\mathcal {B}}_{(p,r,s)}(f)=0$, then $\rho _{\mathcal {B}}(f)=0$ by Proposition 1.1, and so $f=0$. For any $\lambda \in \mathbb {C}$, we have

$$\begin{aligned} \left\| (\lambda _ix^*_i((\lambda f)'(z_i)))_{i=1}^n\right\| _p&=\left| \lambda \right| \left\| (\lambda _ix^*_i(f'(z_i)))_{i=1}^n\right\| _p\\&\le \left| \lambda \right| \pi _{(p,r,s)}^{\mathcal {B}}(f)\omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _s\left( (x^*_i)_{i=1}^n\right) , \end{aligned}$$

and thus $\lambda f\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ with $\pi _{(p,r,s)}^{\mathcal {B}}(\lambda f)\le \left| \lambda \right| \pi _{(p,r,s)}^{\mathcal {B}}(f)$. If $\lambda =0$, this implies that $\pi _{(p,r,s)}^{\mathcal {B}}(\lambda f)=0=\left| \lambda \right| \pi _{(p,r,s)}^{\mathcal {B}}(f)$. If $\lambda \ne 0$, we have $\pi _{(p,r,s)}^{\mathcal {B}}(f)=\pi _{(p,r,s)}^{\mathcal {B}}(\lambda ^{-1}(\lambda f))\le \left| \lambda ^{-1}\right| \pi _{(p,r,s)}^{\mathcal {B}}(\lambda f)$, hence $\left| \lambda \right| \pi _{(p,r,s)}^{\mathcal {B}}(f)\le \pi _{(p,r,s)}^{\mathcal {B}}(\lambda f)$, and so $\pi _{(p,r,s)}^{\mathcal {B}}(\lambda f)=\left| \lambda \right| \pi _{(p,r,s)}^{\mathcal {B}}(f)$.

For any $f_1,f_2\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$, we have

$$\begin{aligned} \left\| (\lambda _ix^*_i((f_1+f_2)'(z_i)))_{i=1}^n\right\| _p&\le \left\| (\lambda _ix^*_i(f_1'(z_i)))_{i=1}^n\right\| _p+\left\| (\lambda _ix^*_i(f_2'(z_i)))_{i=1}^n\right\| _p\\&\le \left( \pi _{(p,r,s)}^{\mathcal {B}}(f_1)+\pi _{(p,r,s)}^{\mathcal {B}}(f_2)\right) \omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \\&\quad \times \omega _s\left( (x^*_i)_{i=1}^n\right) , \end{aligned}$$

and thus $f_1+f_2\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ with $\pi _{(p,r,s)}^{\mathcal {B}}(f_1+f_2)\le \pi _{(p,r,s)}^{\mathcal {B}}(f_1)+\pi _{(p,r,s)}^{\mathcal {B}}(f_2)$. Consequently, $(\Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X),\pi _{(p,r,s)}^{\mathcal {B}})$ is a normed space.

To show its completeness, let $(f_i)$ be a sequence in $\Pi _{(p,r,s)}^{\mathcal {B}}(\mathbb {D},X)$ such that $\sum _{i=1}^{\infty }\pi _{(p,r,s)}^{\mathcal {B}}(f_i)<\infty$. Since $\rho _{\mathcal {B}}\le \pi _{(p,r,s)}^{\mathcal {B}}$ on $\Pi ^{\mathcal {B}}_{(p,r,s)}(\mathbb {D},X)$ by Proposition 1.1, and $(\widehat{\mathcal {B}}(\mathbb {D},X),\rho _{\mathcal {B}})$ is a Banach space, there exists $f=\sum _{i=1}^\infty f_i\in \widehat{\mathcal {B}}(\mathbb {D},X)$ in the norm $\rho _{\mathcal {B}}$. We will prove that $\sum _{i=1}^\infty f_i=f$ in the norm $\pi _{(p,r,s)}^{\mathcal {B}}$. Given $m\in \mathbb {N}$, $\lambda _k\in \mathbb {C}$, $z_k\in \mathbb {D}$ and $x_k^*\in X^*$ for all $k\in \{1,\ldots ,m\}$, we have

$$\begin{aligned} \left\| \left( \lambda _kx^*_k\left( \left( \sum _{i=1}^nf_i\right) '(z_k)\right) \right) _{k=1}^m\right\| _p&\le \pi _{(p,r,s)}^{\mathcal {B}}\left( \sum _{i=1}^nf_i\right) \omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _k,z_k)_{k=1}^m\right) \\&\quad \times \omega _s\left( (x^*_k)_{k=1}^m\right) \\&\le \left( \sum _{i=1}^n\pi _{(p,r,s)}^{\mathcal {B}}(f_i)\right) \omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _k,z_k)_{k=1}^m\right) \\&\quad \times \omega _s\left( (x^*_k)_{k=1}^m\right) \end{aligned}$$

for all $n\in \mathbb {N}$, and by taking limits with $n\rightarrow \infty$ yields

$$\begin{aligned} \left\| \left( \lambda _kx^*_k\left( \left( \sum _{i=1}^\infty f_i\right) '(z_k)\right) \right) _{k=1}^m\right\| _p&\le \left( \sum _{i=1}^\infty \pi _{(p,r,s)}^{\mathcal {B}}(f_i)\right) \omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _k,z_k)_{k=1}^m\right) \\&\quad \times \omega _s\left( (x^*_k)_{k=1}^m\right) . \end{aligned}$$

Hence $f\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ with $\pi _{(p,r,s)}^{\mathcal {B}}(f)\le \sum _{n=1}^{\infty }\pi _{(p,r,s)}^{\mathcal {B}}(f_n)$. Moreover,

$$\begin{aligned} \pi _{(p,r,s)}^{\mathcal {B}}\left( f-\sum _{i=1}^{n}f_{i}\right) =\pi _{(p,r,s)}^{\mathcal {B}}\left( \sum _{i=n+1}^{\infty }f_{i}\right) \le \sum _{i=n+1}^{\infty }\pi _{(p,r,s)}^{\mathcal {B}}\left( f_{i}\right) \end{aligned}$$

for all $n\in \mathbb {N}$, and therefore $\sum _{i=1}^{\infty }f_i=f$ in the norm $\pi _{(p,r,s)}^{\mathcal {B}}$.

(P2) Let $g\in \widehat{\mathcal {B}}(\mathbb {D})$ and $x\in X$. It is immediate that $g\cdot x\in \widehat{\mathcal {B}}(\mathbb {D},X)$ with $\rho _{\mathcal {B}}(g\cdot x)=\rho _{\mathcal {B}}(g)\left\| x\right\|$. For $g=0$ or $x=0$, (P2) is clear. If $g\ne 0$ and $x\ne 0$, the generalized Hölder’s inequality gives

$$\begin{aligned}&||(\lambda _ix^*_i((g\cdot x)'(z_i)))_{i=1}^n||_p=\rho _{\mathcal {B}}(g)\left\| x\right\| \left\| \left( \lambda _i\left( \frac{g}{\rho _{\mathcal {B}}(g)}\right) '(z_i)x^*_i\left( \frac{x}{\left\| x\right\| }\right) \right) _{i=1}^n\right\| _p\\&\quad =\rho _{\mathcal {B}}(g)\left\| x\right\| \left\| \left( \lambda _i\left( \frac{g}{\rho _{\mathcal {B}}(g)}\right) '(z_i)J_X\left( \frac{x}{\left\| x\right\| }\right) (x^*_i)\right) _{i=1}^n\right\| _p\\&\quad \le \rho _{\mathcal {B}}(g)\left\| x\right\| \left\| \left( \lambda _i\left( \frac{g}{\rho _{\mathcal {B}}(g)}\right) '(z_i)\right) _{i=1}^n\right\| _r\left\| \left( J_X\left( \frac{x}{\left\| x\right\| }\right) (x^*_i)\right) _{i=1}^n\right\| _s\\&\quad \le \rho _{\mathcal {B}}(g)\left\| x\right\| \omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _s\left( (x^*_i)_{i=1}^n\right) , \end{aligned}$$

and thus $g\cdot x\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ with $\pi _{(p,r,s)}^{\mathcal {B}}(g\cdot x)\le \rho _{\mathcal {B}}(g)\left\| x\right\|$. Conversely,

$$\begin{aligned} \rho _{\mathcal {B}}(g)\left\| x\right\| =\rho _{\mathcal {B}}(g\cdot x)\le \pi _{(p,r,s)}^{\mathcal {B}}(g\cdot x) \end{aligned}$$

by using Proposition 1.1.

(P3) Let $f\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$, $T\in \mathcal {L}(X,Y)$ and $h\in \widehat{\mathcal {B}}(\mathbb {D},\mathbb {D})$. Clearly, $(T\circ f\circ h)(0)=0$ and $T\circ f\circ h\in \mathcal {H}(\mathbb {D},Y)$ with

$$\begin{aligned} (T\circ f\circ h)'=T\circ (f\circ h)'=T\circ [h'\cdot (f'\circ h)]. \end{aligned}$$

Let $n\in \mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$, $(z_i)_{i=1}^n$ in $\mathbb {D}$ and $(y^*_i)_{i=1}^n$ in $Y^*$. We have

$$\begin{aligned}&\Vert (\lambda _iy^*_i((T\circ f\circ h)'(z_i)))_{i=1}^n\Vert _p=\left\| (\lambda _iy^*_i(T(h'(z_i)f'(h(z_i)))))_{i=1}^n\right\| _p\\&\quad \le \left\| T\right\| \left\| (\lambda _ih'(z_i)y^*_i(f'(h(z_i))))_{i=1}^n\right\| _p\\&\quad \le \left\| T\right\| \pi _{(p,r,s)}^{\mathcal {B}}(f)\omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _ih'(z_i),h(z_i))_{i=1}^n\right) \omega _s\left( (y^*_i)_{i=1}^n\right) \\&\quad \le \left\| T\right\| \pi _{(p,r,s)}^{\mathcal {B}}(f)\omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _s\left( (y^*_i)_{i=1}^n\right) , \end{aligned}$$

where it is applied that $\rho _{\mathcal {B}}(g\circ h)\le \rho _{\mathcal {B}}(g)$ for all $g\in \widehat{\mathcal {B}}(\mathbb {D})$ by the Pick–Schwarz Lemma. So $T\circ f\circ h\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ with $\pi _{(p,r,s)}^{\mathcal {B}}(T\circ f\circ h)\le \left\| T\right\| \pi _{(p,r,s)}^{\mathcal {B}}(f)$. $\square$

2.3 Pietsch’s domination and Kwapień’s factorization

For $1\le p,r,s<\infty$ such that $1/p=1/r+1/s$, we present a result gathering both variants for (p, r, s)-summing Bloch maps of Pietsch’s domination theorem [14, Theorem 7.4.2] and Kwapień’s factorization theorem [14, Theorem 7.4.3] for (r, s)-dominated linear operators.

Given a Banach space X, we will denote by $\mathcal {P}(B_{X^{*}})$ the set of all regular Borel probability measures $\mu$ on $B_{X^{*}}$ with the topology $w^*$.

Theorem 1.4

Let $1\le p,r,s<\infty$ be with $1/p=1/r+1/s$ and $f\in \widehat{\mathcal {B}}(\mathbb {D},X)$. The following statements are equivalent:

• (i) $f\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$.

• (ii) (Pietsch’s domination). There exist a constant $C>0$ and measures $\mu \in \mathcal {P}(B_{\widehat{\mathcal {B}}(\mathbb {D})})$ and $\nu \in \mathcal {P}(B_{X^{**}})$ such that

  $$\begin{aligned} \left| x^*(f'(z))\right| \le C\left( \int _{B_{\widehat{\mathcal {B}}(\mathbb {D})}}\left| g'(z)\right| ^{r}d\mu (g)\right) ^{\frac{1}{r}}\left( \int _{B_{X^{**}}}\left| x^{**}(x^*)\right| ^{s}d\nu (x^{**})\right) ^{\frac{1}{s}} \end{aligned}$$

  for all $z\in \mathbb {D}$ and $x^*\in X^*$.

• (iii) (Kwapień’s factorization). There exist a Banach space Z, a closed subspace $Y\subseteq Z$, a map $h\in \Pi ^{\widehat{\mathcal {B}}}_r(\mathbb {D},Z)$ with $h'(\mathbb {D})\subseteq Y$ and an operator $T\in \mathcal {L}(Y,X)$ with $T^*\in \Pi _s(X^*,Y^*)$ such that $f'=T\circ h'$.

In this case,

$$\begin{aligned} \pi _{(p,r,s)}^{\mathcal {B}}(f)=\inf \{C :C\text { as in }(ii)\}=\inf \left\{ \pi _s(T^*)\pi _{r}^{\mathcal {B}}(h):f'=T\circ h'\right\} \end{aligned}$$

and, in addition, both infimums are attained.

Proof

$(i)\Leftrightarrow (ii)$: We will apply a general Pietsch domination theorem (see [12, Theorem 4.6]). Define the functions

$$\begin{aligned} R_{1}&:B_{\widehat{\mathcal {B}}(\mathbb {D})}\times \mathbb {D}\times \mathbb {C}\rightarrow [0,\infty [,\qquad R_1(g,z,\lambda )=\left| \lambda \right| \left| g'(z)\right| ,\\ R_2&:B_{X^{**}}\times X^*\rightarrow [0,\infty [,\qquad R_2(x^{**},x^*)=\left| x^{**}(x^*)\right| ,\\ S&:\widehat{\mathcal {B}}(\mathbb {D},X)\times \mathbb {D}\times \mathbb {C}\times X^*\rightarrow [0,\infty [,\qquad S(f,z,\lambda ,x^*)=\left| \lambda \right| \left| x^*(f'(z))\right| . \end{aligned}$$

Note that $R_1$, $R_2$ and S satisfy the properties (1)–(2) preceding to [12, Definition 4.4]:

• 1. For each $z\in \mathbb {D}$, $\lambda \in \mathbb {C}$ and $x^*\in X^*$, the maps

  $$\begin{aligned} (R_1)_{z,\lambda }&:B_{\widehat{\mathcal {B}}(\mathbb {D})}\rightarrow [0,\infty [\qquad (R_1)_{z,\lambda }(g)=R_1(g,z,\lambda ),\\ (R_2)_{x^*}&:B_{X^{**}}\rightarrow [0,\infty [\qquad (R_2)_{x^*}(x^{**})=R_2(x^{**},x^*), \end{aligned}$$

  are continuous.

• 2. The equalities

  $$\begin{aligned} R_1(g,z,\beta _1\lambda )&=\beta _1 R_1(g,z,\lambda ),\\ R_2(x^{**},\beta _2 x^*)&=\beta _2 R_2(x^{**},x^*),\\ S(f,z,\beta _1\lambda ,\beta _2x^*)&=\overline{\beta }_1 \beta _2 S(f,z,\lambda ,x^*), \end{aligned}$$

  hold for all $g\in B_{\widehat{\mathcal {B}}(\mathbb {D})}$, $x^{**}\in B_{X^{**}}$, $z\in \mathbb {D}$, $\lambda \in \mathbb {C}$, $x^*\in X^*$, $\beta _1,\beta _2\in [0,1]$ and $f\in \widehat{\mathcal {B}}(\mathbb {D},X)$.

Now, in view of Definition 4.4 and Theorem 4.6 in [12], we have that f is (p, r, s)-summing Bloch if and only if f is $R_1,R_2$-S abstract (r, s)-summing if and only if there is a constant $C>0$ and measures $\mu \in \mathcal {P}(B_{\widehat{\mathcal {B}}(\mathbb {D})})$ and $\nu \in \mathcal {P}(B_{X^{**}})$ such that

$$\begin{aligned} S(f,z,\lambda ,x^*)\le C\left( \int _{B_{\widehat{\mathcal {B}}(\mathbb {D})}}R_1(g,z,\lambda )^rd\mu (g)\right) ^{\frac{1}{r}}\left( \int _{B_{X^{**}}}R_2(x^{**},x^*)^{s}d\nu (x^{**})\right) ^{\frac{1}{s}} \end{aligned}$$

for all $z\in \mathbb {D}$, $\lambda \in \mathbb {C}$ and $x^*\in X^*$, and this means that

$$\begin{aligned} \left| x^*(f'(z))\right| \le C\left( \int _{B_{\widehat{\mathcal {B}}(\mathbb {D})}}\left| g'(z)\right| ^rd\mu (g)\right) ^{\frac{1}{r}}\left( \int _{B_{X^{**}}}\left| x^{**}(x^*)\right| ^{s}d\nu (x^{**})\right) ^{\frac{1}{s}} \end{aligned}$$

for all $z\in \mathbb {D}$ and $x^*\in X^*$. In this case, $\pi _{(p,r,s)}^{\mathcal {B}}(f)=\min \{C :C\text { as in }(ii)\}$.

$(ii)\Rightarrow (iii)$: Let $\iota _\mathbb {D}:\mathbb {D}\rightarrow C(B_{\widehat{\mathcal {B}}(\mathbb {D})})$ be defined by $\iota _{\mathbb {D}}(z)(g)=g'(z)$ for all $z\in \mathbb {D}$ and $g\in B_{\widehat{\mathcal {B}}(\mathbb {D})}$, and let $j_r:C(B_{\widehat{\mathcal {B}}(\mathbb {D})})\rightarrow L_r(\mu )$ be the canonical injection. In the light of [3, Lemma 1.5], we can find a map $h\in \widehat{\mathcal {B}}(\mathbb {D},L_r(\mu ))$ with $\rho _{\mathcal {B}}(h)=1$ such that $h'=j_{r}\circ \iota _\mathbb {D}$. Moreover, $h\in \Pi ^{\widehat{\mathcal {B}}}_r(\mathbb {D},L_r(\mu ))$ with $\pi _{r}^{\mathcal {B}}(h)=1$. Consider the linear subspace $Y=\overline{\textrm{lin}}(h'(\mathbb {D}))\subseteq L_r(\mu )$ and the operator $T\in \mathcal {L}(Y,X)$ defined by $T(h'(z))=f'(z)$ for all $z\in \mathbb {D}$. Using (ii), we have

$$\begin{aligned} \left\| T^*(x^*)\right\|&=\sup \left\{ \left| T^*(x^*)(h'(z))\right| :z\in \mathbb {D},\; \left\| h'(z)\right\| \le 1\right\} \\&=\sup \left\{ \left| x^*(T(h'(z)))\right| :z\in \mathbb {D},\;\left\| h'(z)\right\| \le 1\right\} \\&=\sup \left\{ \left| x^*(f'(z))\right| :z\in \mathbb {D},\;\left\| h'(z)\right\| \le 1\right\} \\&\le C\left( \int _{B_{X^{**}}}\left| x^{**}(x^*)\right| ^{s}d\nu (x^{**})\right) ^{\frac{1}{s}} \end{aligned}$$

for all $x^*\in X^*$, and thus $T^*\in \Pi _s(X^*,Y^*)$ with $\pi _s(T^*)\le C$. Hence (iii) holds and $\pi _s(T^*)\pi _{r}^{\mathcal {B}}(h)\le C$. Taking the infimum over all such constants C, it follows that $\pi _s(T^*)\pi _{r}^{\mathcal {B}}(h)\le \inf \{C :C\text { as in }(ii)\}$.

$(iii)\Rightarrow (ii)$: Suppose there exist maps h and T as in (iii). For any $z\in \mathbb {D}$ and $x^*\in X^*$, we have

$$\begin{aligned} \left| x^*(f'(z))\right| =\left| x^*((T\circ h')(z))\right| =\left| T^*(x^*)(h'(z))\right| \le \left\| T^*(x^*)\right\| \left\| h'(z)\right\| . \end{aligned}$$

By both Pietsch domination theorems for p-summing linear operators [14, Theorem 7.3.2] and p-summing Bloch maps [3, Theorem 1.4], there are measures $\nu \in \mathcal {P}(B_{X^{**}})$ and $\mu \in \mathcal {P}(B_{\widehat{\mathcal {B}}(\mathbb {D})})$ such that

$$\begin{aligned} \left\| T^*(x^*)\right\| \le \pi _s(T^*)\left( \int _{B_{X^{**}}}\left| x^{**}(x^*)\right| ^{s}d\nu (x^{**})\right) ^{\frac{1}{s}} \end{aligned}$$

and

$$\begin{aligned} \left\| h'(z)\right\| \le \pi ^{\mathcal {B}}_r(h)\left( \int _{B_{\widehat{\mathcal {B}}(\mathbb {D})}}\left| g'(z)\right| ^{r}d\mu (g)\right) ^{\frac{1}{r}}. \end{aligned}$$

Hence we have

$$\begin{aligned} \left| x^*(f'(z))\right|&\le \pi _s(T^*)\pi ^{\mathcal {B}}_r(h)\\&\quad \times \left( \int _{B_{\widehat{\mathcal {B}}(\mathbb {D})}}\left| g'(z)\right| ^{r}d\mu (g)\right) ^{\frac{1}{r}}\left( \int _{B_{X^{**}}}\left| x^{**}(x^*)\right| ^{s}d\nu (x^{**})\right) ^{\frac{1}{s}}, \end{aligned}$$

and this proves (ii) with $\pi _s(T^*)\pi ^{\mathcal {B}}_r(h)\in \{C :C\text { as in }(ii)\}$. It follows that $\inf \{C :C\text { as in }(ii)\}\le \inf \left\{ \pi _s(T^*)\pi _{r}^{\mathcal {B}}(h):f'=T\circ h'\right\}$. $\square$

2.4 Möbius invariance

The Möbius group of $\mathbb {D}$, denoted by $\textrm{Aut}(\mathbb {D})$, consists of all biholomorphic bijections $\phi :\mathbb {D}\rightarrow \mathbb {D}$. Each $\phi \in \textrm{Aut}(\mathbb {D})$ has the form $\phi =\lambda \phi _a$ with $\lambda \in \mathbb {T}$ and $a\in \mathbb {D}$, where

$$\begin{aligned} \phi _a(z)=\frac{a-z}{1-\overline{a}z}\qquad (z\in \mathbb {D}). \end{aligned}$$

Given a complex Banach space X, let us recall that a linear space $\mathcal {A}(\mathbb {D},X)\subseteq \mathcal {H}(\mathbb {D},X)$ endowed with a seminorm $p_\mathcal {A}$ is Möbius-invariant if:

• (i) $\mathcal {A}(\mathbb {D},X)\subseteq \mathcal {B}(\mathbb {D},X)$ and there exists $C\ge 0$ such that $\rho _\mathcal {B}(f)\le Cp_\mathcal {A}(f)$ for all $f\in \mathcal {A}(\mathbb {D},X)$,

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

We have the following interesting fact.

Proposition 1.5

The space $(\Pi ^{\mathcal {B}}_{(p,r,s)}(\mathbb {D},X),\pi ^{\mathcal {B}}_{(p,r,s)})$ is Möbius-invariant.

Proof

By Proposition 1.1, $\Pi ^\mathcal {B}_{(p,r,s)}(\mathbb {D},X)\subseteq \mathcal {B}(\mathbb {D},X)$ and $\rho _\mathcal {B}(f)\le \pi ^{\mathcal {B}}_{(p,r,s)}(f)$ for all $f\in \Pi ^\mathcal {B}_{(p,r,s)}(\mathbb {D},X)$. On the other hand, a proof similar to that of (P3) in Proposition 1.3 yields that if $f\in \Pi ^\mathcal {B}_{(p,r,s)}(\mathbb {D},X)$ and $\phi \in \textrm{Aut}(\mathbb {D})$, then $f\circ \phi \in \Pi ^\mathcal {B}_{(p,r,s)}(\mathbb {D},X)$ with $\pi ^{\mathcal {B}}_p(f\circ \phi )\le \pi ^{\mathcal {B}}_p(f)$, and from this fact it is inferred that $\pi ^{\mathcal {B}}_p(f)=\pi ^{\mathcal {B}}_p((f\circ \phi )\circ \phi ^{-1})\le \pi ^{\mathcal {B}}_p(f\circ \phi )$. $\square$

2.5 Lapresté norms on Bloch molecules

Our approach on the duality of the spaces $(\Pi ^{\widehat{\mathcal {B}}}_{(p,r,s)},\pi ^{\mathcal {B}}_{(p,r,s)})$ requires the introduction of Bloch analogues of Lapresté norms [10] on the tensor product of Banach spaces (a generalization of the Chevet–Saphar norms [5, 15] on such tensor products). Given two linear spaces E and F, the tensor product space $E\otimes F$ equipped with a norm $\alpha$ will be denoted by $E\otimes _\alpha F$, and the completion of $E\otimes _\alpha F$ by $E\widehat{\otimes }_\alpha F$.

Towards this end, we first recall some concepts and results borrowed from [8]. For each $z\in \mathbb {D}$, a Bloch atom of $\mathbb {D}$ is the functional $\gamma _{z}\in \widehat{\mathcal {B}}(\mathbb {D})^*$ given by $\gamma _{z}(f)=f'(z)$ for all $f\in \widehat{\mathcal {B}}(\mathbb {D})$. The called Bloch molecules of $\mathbb {D}$ are the elements of the space

$$\begin{aligned} \textrm{lin}(\{\gamma _{z}:z\in \mathbb {D}\})\subseteq \widehat{\mathcal {B}}(\mathbb {D})^*, \end{aligned}$$

and the Bloch-free Banach space of $\mathbb {D}$ is the space

$$\begin{aligned} \mathcal {G}(\mathbb {D})=\overline{\textrm{lin}}(\{\gamma _{z}:z\in \mathbb {D}\})\subseteq \widehat{\mathcal {B}}(\mathbb {D})^*. \end{aligned}$$

Theorem 1.6

[8]

• (i) The map $\Gamma :z\in \mathbb {D}\mapsto \gamma _{z}\in \mathcal {G}(\mathbb {D})$ is holomorphic with $\left\| \gamma _{z}\right\| =1/(1-|z|^{2})$.

• (ii) The map $\Lambda :\widehat{\mathcal {B}}(\mathbb {D})\rightarrow \mathcal {G}(\mathbb {D})^*$, given by $\Lambda (g)(\gamma _z)=g'(z)$ for all $z\in \mathbb {D}$ and $g\in \widehat{\mathcal {B}}(\mathbb {D})$, is an isometric isomorphism.

• (iii) For each $h\in \widehat{\mathcal {B}}(\mathbb {D},\mathbb {D})$, there exists a unique $\widehat{h}\in \mathcal {L}(\mathcal {G}(\mathbb {D}),\mathcal {G}(\mathbb {D}))$ such that $\widehat{h}\circ \Gamma =h^{\prime }\cdot (\Gamma \circ h)$. Furthermore, $||\widehat{h}||\le 1$.

• (iv) For each complex Banach space X and each $f\in \widehat{\mathcal {B}}(\mathbb {D},X)$, there is a unique $S_f\in \mathcal {L}(\mathcal {G}(\mathbb {D}),X)$ such that $S_f\circ \Gamma =f'$ and $\Vert S_f\Vert = p_\mathcal {B}(f)$.

• (v) $f \mapsto S_f$ is an isometric isomorphism of $\widehat{\mathcal {B}}(\mathbb {D},X)$ onto $\mathcal {L}(\mathcal {G}(\mathbb {D}),X)$. $\square$

Given a complex Banach space X, the space of X-valued Bloch molecules of $\mathbb {D}$ is defined as

$$\begin{aligned} \textrm{lin}(\Gamma (\mathbb {D}))\otimes X=\textrm{lin}(\left\{ \gamma _{z}\otimes x:z\in \mathbb {D},\,x\in X\right\} ) \subseteq \widehat{\mathcal {B}}(\mathbb {D},X^{*})^{*}, \end{aligned}$$

where $\gamma _{z}\otimes x:\widehat{\mathcal {B}}(\mathbb {D},X^{*})\rightarrow \mathbb {C}$ is the functional given by

$$\begin{aligned} \left( \gamma _{z}\otimes x\right) (f)=\left\langle f^{\prime }(z),x\right\rangle \qquad \left( f\in \widehat{\mathcal {B}}(\mathbb {D},X^{*})\right) . \end{aligned}$$

Each element $\gamma \in \textrm{lin}(\Gamma (\mathbb {D}))\otimes X$ can be expressed as $\gamma =\sum _{i=1}^{n}\lambda _{i}\gamma _{z_{i}}\otimes x_{i}$ for some n in $\mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$, $(z_i)_{i=1}^n$ in $\mathbb {D}$ and $(x_i)_{i=1}^n$ in X, and its action is

$$\begin{aligned} \gamma (f)=\sum _{i=1}^{n}\lambda _i\left\langle f'(z_i),x_i\right\rangle . \end{aligned}$$

Definition 1.7

Let $1\le p,r,s\le \infty$ and $\gamma \in \textrm{lin}(\Gamma (\mathbb {D}))\otimes X$. Define

$$\begin{aligned} \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )=\inf \left\{ \left\| (\lambda _i)_{i=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) \right\} , \end{aligned}$$

the infimum being taken over all the representations of $\gamma$ as $\sum _{i=1}^{n}\lambda _{i}\gamma _{z_i}\otimes x_i$ with n in $\mathbb {N}$, $\left( \lambda _i\right) _{i=1}^{n}$ in $\mathbb {C}$, $\left( z_{i}\right) _{i=1}^{n}$ in $\mathbb {D}$ and $\left( x_i\right) _{i=1}^{n}$ in X.

Following [3, Definition 2.5], we say that a $\theta$-norm $\alpha$ on $\textrm{lin}(\Gamma (\mathbb {D}))\otimes X$ with $\theta \in (0,1]$ is a Bloch reasonable crossnorm if:

• (i) $\alpha (\gamma _z\otimes x)\le \left\| \gamma _z\right\| \left\| x\right\|$ for all $z\in \mathbb {D}$ and $x\in X$,

• (ii) For $g\in \widehat{\mathcal {B}}(\mathbb {D})$ and $x^*\in X^*$, the linear functional $g\otimes x^*$ on $\textrm{lin}(\Gamma (\mathbb {D}))\otimes X$ given by $(g\otimes x^*)(\gamma _z\otimes x)=g'(z)x^*(x)$ is bounded on $\textrm{lin}(\Gamma (\mathbb {D}))\otimes _\alpha X$ with $\left\| g\otimes x^*\right\| \le \rho _\mathcal {B}(g)\left\| x^*\right\|$.

The proof of the following result is based on [10, Theorem 1.1].

Theorem 1.8

Let $1\le p,r,s\le \infty$ and $1/\theta :=1/p+1/r+1/s\ge 1$. Then $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}$ is a Bloch reasonable $\theta$-crossnorm on $\textrm{lin}(\Gamma (\mathbb {D}))\otimes X$.

Proof

Let $\gamma \in \textrm{lin}(\Gamma (\mathbb {D}))\otimes X$ and let $\sum _{i=1}^n\lambda _i\gamma _{z_i}\otimes x_i$ be a representation of $\gamma$. Clearly, $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )\ge 0$. Given $\lambda \in \mathbb {C}$, we have

$$\begin{aligned} \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\lambda \gamma )&\le \left\| (\lambda \lambda _i)_{i=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) \\&=\left| \lambda \right| \left\| (\lambda _i)_{i=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) \end{aligned}$$

If $\lambda =0$, we obtain $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\lambda \gamma )=0=\left| \lambda \right| \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )$. For $\lambda \ne 0$, since the preceding inequality holds for every representation of $\gamma$, we deduce that $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\lambda \gamma )\le \left| \lambda \right| \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )$. For the converse inequality, note that

$$\begin{aligned} \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )=\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\lambda ^{-1}(\lambda \gamma ))\le |\lambda ^{-1}|\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\lambda \gamma ), \end{aligned}$$

thus $\left| \lambda \right| \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )\le \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\lambda \gamma )$ and hence $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\lambda \gamma )=\left| \lambda \right| \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )$.

We now prove that $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )=0$ implies $\gamma =0$. Applying that $\theta \le 1$ and the generalized Hölder’s inequality, we obtain

$$\begin{aligned} \left| \sum _{i=1}^n \lambda _i h'(z_i)y^*(x_i)\right|&\le \left| \sum _{i=1}^n \left| \lambda _i\right| ^\theta \left| h'(z_i)\right| ^\theta \left| y^*(x_i)\right| ^\theta \right| ^{\frac{1}{\theta }}\\&\le \left( \sum _{i=1}^{n}\left| \lambda _i\right| ^p\right) ^{\frac{1}{p}}\left( \sum _{i=1}^{n}\left| h'(z_i)\right| ^r\right) ^{\frac{1}{r}}\left( \sum _{i=1}^{n}\left| y^*(x_i)\right| ^s\right) ^{\frac{1}{s}}\\&\le \left\| (\lambda _i)_{i=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) \end{aligned}$$

for any $h\in B_{\widehat{\mathcal {B}}(\mathbb {D})}$ and $y^*\in B_{X^*}$. Since the quantity $\left| \sum _{i=1}^n \lambda _i h'(z_i)y^*(x_i)\right|$ does not depend on the representation of $\gamma$ since

$$\begin{aligned} \sum _{i=1}^n \lambda _i h'(z_i)y^*(x_i)=\left( \sum _{i=1}^n\lambda _i\gamma _{z_i}\otimes x_i\right) (h\cdot y^*)=\gamma (h\cdot y^*), \end{aligned}$$

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

$$\begin{aligned} \left| \sum _{i=1}^n \lambda _i h'(z_i)y^*(x_i)\right| \le \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma ) \end{aligned}$$

for any $h\in B_{\widehat{\mathcal {B}}(\mathbb {D})}$ and $y^*\in B_{X^*}$. Now, if $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )=0$, we have

$$\begin{aligned} \left( \sum _{i=1}^n \lambda _i y^*(x_i)\gamma _{z_i}\right) (h)=\sum _{i=1}^n \lambda _i h'(z_i)y^*(x_i)=0 \end{aligned}$$

for all $h\in B_{\widehat{\mathcal {B}}(\mathbb {D})}$ and $y^*\in B_{X^*}$. For each $y^*\in B_{X^*}$, it is $\sum _{i=1}^n\lambda _i y^*(x_i)\gamma _{z_i}=0$, and since $\Gamma (\mathbb {D})$ is linearly independent in $\mathcal {G}(\mathbb {D})$ by [8, Remark 2.8], it follows that $\lambda _iy^*(x_i)=0$ for all $i\in \{1,\ldots ,n\}$, hence $\lambda _ix_i=0$ for all $i\in \{1,\ldots ,n\}$ since $B_{X^*}$ separates the points of X, and thus $\gamma =\sum _{i=1}^n\lambda _i\gamma _{z_i}\otimes x_i=0$.

To prove the triangular inequality of $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}$, let $\gamma _j\in \textrm{lin}(\Gamma (\mathbb {D}))\otimes X$ for $j=1,2$ and $\varepsilon >0$. For $j=1,2$, by homogeneity we can choose a representation

$$\begin{aligned} \gamma _j=\sum _{i=1}^n\lambda _{j,i}\gamma _{z_{j,i}}\otimes x_{j,i} \end{aligned}$$

for some n in $\mathbb {N}$, $\left( \lambda _{j,i}\right) _{i=1}^{n}$ in $\mathbb {C}$, $\left( z_{j,i}\right) _{i=1}^{n}$ in $\mathbb {D}$ and $\left( x_{j,i}\right) _{i=1}^{n}$ in X, so that

$$\begin{aligned} \left\| (\lambda _{j,i})_{i=1}^n\right\| _p&\le \left( \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _j)^\theta +\varepsilon \right) ^{\frac{1}{p}},\\ \omega ^{\widehat{\mathcal {B}}}_r\left( (z_{j,i})_{i=1}^n\right)&\le \left( \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _j)^\theta +\varepsilon \right) ^{\frac{1}{r}},\\ \omega _s\left( (x_{j,i})_{i=1}^n\right)&\le \left( \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _j)^\theta +\varepsilon \right) ^{\frac{1}{s}}. \end{aligned}$$

We can joint these representations of $\gamma _1$ and $\gamma _2$ to obtain a representation of $\gamma _1+\gamma _2$ in the form $\sum _{i,j=1}^n\lambda _{j,i}\gamma _{z_{j,i}}\otimes x_{j,i}$ such that

$$\begin{aligned} \left\| (\lambda _{j,i})_{i,j=1}^n\right\| _p&\le \left( \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _1)^\theta +\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _2)^\theta +2\varepsilon \right) ^{\frac{1}{p}},\\ \omega ^{\widehat{\mathcal {B}}}_r\left( (z_{j,i})_{i,j=1}^n\right)&\le \left( \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _1)^\theta +\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _2)^\theta +2\varepsilon \right) ^{\frac{1}{r}},\\ \omega _s\left( (x_{j,i})_{i,j=1}^n\right)&\le \left( \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _1)^\theta +\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _2)^\theta +2\varepsilon \right) ^{\frac{1}{s}}. \end{aligned}$$

Hence

$$\begin{aligned} \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _1+\gamma _2)&\le \left\| (\lambda _{j,i})_{i,j=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (z_{j,i})_{i,j=1}^n\right) \omega _s\left( (x_{j,i})_{i,j=1}^n\right) \\&\le \left( \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _1)^\theta +\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _2)^\theta +2\varepsilon \right) ^{\frac{1}{\theta }}, \end{aligned}$$

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

$$\begin{aligned} \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _1+\gamma _2)^\theta \le \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _1)^\theta +\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _2)^\theta . \end{aligned}$$

To finish, we will show that $\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}$ is a Bloch reasonable crossnorm on $\textrm{lin}(\Gamma (\mathbb {D}))\otimes X$. First, given $z\in \mathbb {D}$ and $x\in X$, taking $n=1$, $\lambda _1=1$, $z_1=z$ and $x_1=x$, we have

$$\begin{aligned} \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma _z\otimes x)&\le \left\| (\lambda _i)_{i=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) \\&\le \frac{1}{1-|z|^2}\left\| x\right\| =\left\| \gamma _z\right\| \left\| x\right\| . \end{aligned}$$

Second, given $g\in \widehat{\mathcal {B}}(\mathbb {D})$ and $x^*\in X^*$ with $g\ne 0\ne x^*$, using that $\theta \le 1$ and the generalized Hölder’s inequality, one has

$$\begin{aligned}&\left| (g\otimes x^*)(\gamma )\right| =\left| \sum _{i=1}^n\lambda _i(g\otimes x^*)(\gamma _{z_i}\otimes x_i)\right| =\left| \sum _{i=1}^n\lambda _i g'(z_i)x^*(x_i)\right| \\&\quad \le \rho _\mathcal {B}(g)\left\| x^*\right\| \left\| \left( \lambda _i\left( \frac{g}{\rho _\mathcal {B}(g)}\right) '(z_i)\left( \frac{x^*}{\left\| x^*\right\| }\right) (x_i)\right) _{i=1}^n\right\| _1\\&\quad \le \rho _\mathcal {B}(g)\left\| x^*\right\| \left\| \left( \lambda _i\left( \frac{g}{\rho _\mathcal {B}(g)}\right) '(z_i)\left( \frac{x^*}{\left\| x^*\right\| }\right) (x_i)\right) _{i=1}^n\right\| _{\theta }\\&\quad \le \rho _\mathcal {B}(g)\left\| x^*\right\| \left\| \left( \lambda _i\right) _{i=1}^n\right\| _p\left\| \left( \left( \frac{g}{\rho _\mathcal {B}(g)}\right) '(z_i)\right) _{i=1}^n\right\| _r\left\| \left( \left( \frac{x^*}{\left\| x^*\right\| }\right) (x_i)\right) _{i=1}^n\right\| _s\\&\quad \le \rho _\mathcal {B}(g)\left\| x^*\right\| \left\| (\lambda _i)_{i=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) . \end{aligned}$$

It follows that $\left| (g\otimes x^*)(\gamma )\right| \le \rho _\mathcal {B}(g)\left\| x^*\right\| \mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma )$ by taking infimum over all the representations of $\gamma$. Hence $g\otimes x^*\in (\textrm{lin}(\Gamma (\mathbb {D}))\otimes _{\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}} X)^*$ with $\left\| g\otimes x^*\right\| \le \rho _\mathcal {B}(g)\left\| x^*\right\|$. $\square$

2.6 Duality

We will prove that the dual of $\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X$ can be canonically identified as the space $\Pi _{(p^*,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X^*)$ with the norm $\pi _{(p^*,r,s)}^{\mathcal {B}}$ whenever $1\le p,r,s\le \infty$ such that $1/p^*\le 1/r+1/s$.

The following easy lemma will be needed.

Lemma 1.9

Let X be a Banach space, $n\in \mathbb {N}$, $(x_i^*)_{i=1}^n$ in $X^*$ and $1\le p\le \infty$. Then

$$\begin{aligned} \sup _{x^{**}\in \mathcal {B}_{X^{**}}}\left\| (x^{**}(x^*_i))_{i=1}^n\right\| _p=\sup _{x\in \mathcal {B}_X}\left\| (x^*_i(x))_{i=1}^n\right\| _p. \end{aligned}$$

Proof

Since $x^*_i(x)=J_X(x)(x^*_i)$ for $i=1,\ldots ,n$, the inequality $\ge$ is immediate. Conversely, let $\varepsilon >0$. For each $x^{**}\in \mathcal {B}_{X^{**}}$, Helly’s Lemma gives an $y\in X$ such that $\left\| y\right\| \le 1+\varepsilon$ and $x^*_i(y)=x^{**}(x_i^*)$ for all $i\in \{1,\ldots ,n\}$, and therefore

$$\begin{aligned} \left\| (x^{**}(x_i^*))_{i=1}^n\right\| _p&=(1+\varepsilon )\left\| \left( x^*_i\left( \frac{y}{1+\varepsilon }\right) \right) _{i=1}^n\right\| _p\\&\le (1+\varepsilon )\sup _{x\in \mathcal {B}_X}\left\| (x^*_i(x))_{i=1}^n\right\| _p. \end{aligned}$$

It follows that

$$\begin{aligned} \sup _{x^{**}\in \mathcal {B}_{X^{**}}}\left\| (x^{**}(x^*_i))_{i=1}^n\right\| _p\le (1+\varepsilon )\sup _{x\in \mathcal {B}_X}\left\| (x^*_i(x))_{i=1}^n\right\| _p \end{aligned}$$

and since $\varepsilon$ was arbitrary, we obtain the inequality $\le$. $\square$

Theorem 1.10

Let $1\le p,r,s\le \infty$ such that $1/p^*\le 1/r+1/s$. Then the spaces $(\Pi _{(p^*,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X^*),\pi _{(p^*,r,s)}^{\mathcal {B}})$ and $(\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X)^*$ are isometrically isomorphic via the canonical pairing

$$\begin{aligned} \Lambda (f)(\gamma )=\sum _{i=1}^{n}\lambda _{i}\left\langle f'(z_{i}),x_{i}\right\rangle \end{aligned}$$

for all $f\in \Pi _{p^{*},r,s}^{\widehat{\mathcal {B}}}(\mathbb {D},X^*)$ and $\gamma =\sum _{i=1}^{n}\lambda _{i}\gamma _{z_i}\otimes x_i\in \textrm{lin}(\Gamma (\mathbb {D}))\otimes X$. Moreover, on the closed unit ball of $(\Pi _{(p^*,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X^*),\pi _{(p^*,r,s)}^{\mathcal {B}})$, the weak* topology coincides with the topology of pointwise $\sigma (X^*,X)$-convergence.

Proof

We will only prove the result whenever $1<p<\infty$, and the other cases can be proved similarly.

Let $f\in \Pi _{(p^*,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X^*)$ and let $\Lambda _{0}(f):\textrm{lin}(\Gamma (\mathbb {D}))\otimes X\rightarrow \mathbb {C}$ be the linear functional given by

$$\begin{aligned} \Lambda _{0}(f)(\gamma )=\sum _{i=1}^{n}\lambda _i\left\langle f'(z_i),x_i\right\rangle \end{aligned}$$

for any $\gamma =\sum _{i=1}^{n}\lambda _i\gamma _{z_i}\otimes x_i\in \textrm{lin}(\Gamma (\mathbb {D}))\otimes X$. We claim that $\Lambda _{0}(f)\in (\textrm{lin}(\Gamma (\mathbb {D}))\otimes _{\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}}X)^*$ with $\left\| \Lambda _0(f)\right\| \le \pi _{(p^*,r,s)}^{\mathcal {B}}(f)$. Indeed, Hölder’s inequality and an application of Lemma 1.9 yield

$$\begin{aligned} \left| \Lambda _0(f)(\gamma )\right|&\le \sum _{i=1}^{n}\left| \lambda _i\right| \left| \left\langle f'(z_i),x_i\right\rangle \right| \\&\le \left( \sum _{i=1}^{n}\left| \lambda _i\right| ^{p}\right) ^{\frac{1}{p}}\left( \sum _{i=1}^{n}\left| \left\langle f'(z_i),x_i\right\rangle \right| ^{p^*}\right) ^{\frac{1}{p^*}}\\&=\left( \sum _{i=1}^{n}\left| \lambda _i\right| ^{p}\right) ^{\frac{1}{p}}\left( \sum _{i=1}^{n}\left| \left\langle J_X(x_i),f'(z_i)\right\rangle \right| ^{p^*}\right) ^{\frac{1}{p^*}}\\&\le \left\| (\lambda _i)_{i=1}^n\right\| _p\pi _{(p^*,r,s)}^{\mathcal {B}}(f)\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (J_X(x_i))_{i=1}^n\right) \\&=\left\| (\lambda _i)_{i=1}^n\right\| _p\pi _{(p^*,r,s)}^{\mathcal {B}}(f)\omega ^{\widehat{\mathcal {B}}}_r\left( (z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) . \end{aligned}$$

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

$$\begin{aligned} \left| \Lambda _{0}(f)(\gamma )\right| \le \pi _{(p^*,r,s)}^{\mathcal {B}}(f)\mu _{(p,r,s)}^{\widehat{\mathcal {B}}}(\gamma ), \end{aligned}$$

and since $\gamma$ was arbitrary, this proves our claim.

Since $\textrm{lin}(\Gamma (\mathbb {D}))$ is a norm-dense linear subspace of $\mathcal {G}(\mathbb {D})$ and $\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}$ is a $\theta$-norm on $\textrm{lin}(\Gamma (\mathbb {D}))\otimes X$, then $\textrm{lin}(\Gamma (\mathbb {D}))\otimes X$ is a norm-dense linear subspace of $\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X$. Hence there is a unique continuous map $\Lambda (f)$ from $\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X$ into $\mathbb {C}$ extending $\Lambda _0(f)$. Further, $\Lambda (f)$ is linear and $\left\| \Lambda (f)\right\| =\left\| \Lambda _0(f)\right\|$.

Let $\Lambda :\Pi ^{\widehat{\mathcal {B}}}_{(p^*,r,s)}(\mathbb {D},X^*)\rightarrow (\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X)^*$ be so defined. In view of [3, Corollary 2.3], $\Lambda _0$ is injective and linear from $\Pi ^{\widehat{\mathcal {B}}}_{(p^*,r,s)}(\mathbb {D},X^*)$ into $(\mathcal {G}(\mathbb {D})\otimes X)^*$, and therefore so is $\Lambda$. To prove that $\Lambda$ is a surjective isometry, let $\varphi \in (\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X)^*$ and define $F_\varphi :\mathbb {D}\rightarrow X^*$ by

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

As in the proof of [3, Proposition 2.4], there exists $f_\varphi \in \widehat{\mathcal {B}}(\mathbb {D},X^*)$ with $\rho _{\mathcal {B}}(f_\varphi )\le \left\| \varphi \right\|$ such that $f_\varphi '=F_\varphi$.

We now prove that $f_\varphi \in \Pi ^{\widehat{\mathcal {B}}}_{(p^*,r,s)}(\mathbb {D},X^*)$. Fix $n\in \mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$, $(z_i)_{i=1}^n$ in $\mathbb {D}$ and $(x^{**}_i)_{i=1}^n$ in $X^{**}$. Let $\varepsilon >0$. By Helly’s Lemma, for each $i\in \{1,\ldots ,n\}$, we can find $x_i\in X$ with $\left\| x_i\right\| \le (1+\varepsilon )\left\| x^{**}_i\right\|$ and $\left\langle f_\varphi '(z_i),x_i\right\rangle =\left\langle x^{**}_i,f_\varphi '(z_i)\right\rangle$. Clearly, the map $T:\mathbb {C}^n\rightarrow \mathbb {C}$, defined by

$$\begin{aligned} T(t_1,\ldots ,t_n)=\sum _{i=1}^n t_i \lambda _i\left\langle x^{**}_i,f_\varphi '(z_i)\right\rangle ,\quad \forall (t_1,\ldots ,t_n)\in \mathbb {C}^n, \end{aligned}$$

is linear and continuous on $(\mathbb {C}^n,||\cdot ||_p)$ with

$$\begin{aligned} \left\| T\right\| =\left( \sum _{i=1}^n\left| \lambda _i\right| ^{p^*}\left| \left\langle x^{**}_i,f_\varphi '(z_i)\right\rangle \right| ^{p^*}\right) ^{\frac{1}{p^*}}. \end{aligned}$$

For any $(t_1,\ldots ,t_n)\in \mathbb {C}^n$ with $||(t_1,\ldots ,t_n)||_p\le 1$, we have

$$\begin{aligned} \left| T(t_1,\ldots ,t_n)\right|&=\left| \varphi \left( \sum _{i=1}^n t_i\lambda _i\gamma _{z_i}\otimes x_i\right) \right| \\&\le \left\| \varphi \right\| \mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}\left( \sum _{i=1}^nt_i\lambda _i\gamma _{z_i}\otimes x_i\right) \\&\le \left\| \varphi \right\| \left\| (t_i)_{i=1}^n\right\| _p\omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _s\left( (x_i)_{i=1}^n\right) . \end{aligned}$$

For each $i\in \{1,\ldots ,n\}$, Hahn–Banach Theorem provides $x^{***}_i\in B_{X^{***}}$ such that $\left| x^{***}_i(x^{**}_i)\right| =\left\| x^{**}_i\right\|$. Note that $\omega _s\left( (x_i)_{i=1}^n\right) \le \omega _s\left( (x^{**}_i)_{i=1}^n\right)$ because

$$\begin{aligned} \left\| (x^*(x_i))_{i=1}^n\right\| _s&\le (1+\varepsilon )\left\| (x_i^{**})_{i=1}^n\right\| _s=(1+\varepsilon )\left\| (x_i^{***}(x_i^{**}))_{i=1}^n\right\| _s\\&\le (1+\varepsilon )\omega _s\left( (x^{**}_i)_{i=1}^n\right) \end{aligned}$$

for all $x^*\in B_{X^*}$. Therefore we can write

$$\begin{aligned} \left( \sum _{i=1}^n\left| \lambda _i\right| ^{p^*}\left| \left\langle x^{**}_i,f_\varphi '(z_i)\right\rangle \right| ^{p^*}\right) ^{\frac{1}{p^*}} \le (1+\varepsilon )\left\| \varphi \right\| \omega ^{\widehat{\mathcal {B}}}_r\left( (\lambda _i,z_i)_{i=1}^n\right) \omega _s\left( (x^{**}_i)_{i=1}^n\right) . \end{aligned}$$

By letting $\varepsilon$ tend to zero gives $f_\varphi \in \Pi ^{\widehat{\mathcal {B}}}_{(p^*,r,s)}(\mathbb {D},X^*)$ with $\pi ^{\mathcal {B}}_{(p^*,r,s)}(f_\varphi )\le \left\| \varphi \right\|$.

Finally, for any $\gamma =\sum _{i=1}^n \lambda _i\gamma _{z_i}\otimes x_i\in \textrm{lin}(\Gamma (\mathbb {D}))\otimes X$, we get

$$\begin{aligned} \Lambda (f_\varphi )(\gamma )&=\sum _{i=1}^n\lambda _i\left\langle f'_\varphi (z_i),x_i\right\rangle =\sum _{i=1}^n\lambda _i\varphi (\gamma _{z_i}\otimes x_i)\\&=\varphi \left( \sum _{i=1}^n\lambda _i\gamma _{z_i}\otimes x_i\right) =\varphi (\gamma ). \end{aligned}$$

Hence $\Lambda (f_\varphi )=\varphi$ on a dense subspace of $\mathcal {G}(\mathbb {D})\widehat{\otimes }_{\mu ^{\widehat{\mathcal {B}}}_{(p,r,s)}} X$ and, consequently, $\Lambda (f_\varphi )=\varphi$. Moreover, $\pi ^{\mathcal {B}}_{(p^*,r,s)}(f_\varphi )\le \left\| \varphi \right\| =\left\| \Lambda (f_\varphi )\right\|$.

The assertion about the weak* topology can be proved with the same argument as in the proof of Theorem 2.8 in [3]. $\square$

3 (p, r, s)-Nuclear Bloch maps

In order to present examples of (p, r, s)-summing Bloch maps, we introduce the class of (p, r, s)-nuclear Bloch maps.

Let X be a complex Banach space and $1\le p\le \infty$. Let $\ell _p(X)$ be the Banach space of all p-summable sequences $(x_n)_{n=1}^\infty$ in X, with the norm

$$\begin{aligned} \left\| (x_n)_{n=1}^\infty \right\| _p=\left\{ \begin{array}{lll} \left( \displaystyle \sum _{n=1}^\infty \left\| x_n\right\| ^p\right) ^{\frac{1}{p}}&{} \text {if} &{} 1\le p<\infty , \\ &{} &{}\\ \displaystyle \max _{n\in \mathbb {N}}\left\| x_n\right\| &{} \text {if} &{} p=\infty , \end{array}\right. \end{aligned}$$

and let $\ell _p^\omega (X)$ be the Banach space of all weakly p-summable sequences $(x_n)_{n=1}^\infty$ in X, with the norm

$$\begin{aligned} \omega _p\left( (x_n)_{n=1}^\infty \right) =\sup _{x^*\in \mathcal {B}_{X^*}}\left\| (x^*(x_n))_{n=1}^\infty \right\| _p. \end{aligned}$$

As usual, we will write $\ell _p$ and $\ell _p^\omega$ instead of $\ell _p(\mathbb {C})$ and $\ell _p^\omega (\mathbb {C})$, respectively.

By [14, Definition 18.1.1], given Banach spaces X, Y and $0<p,r,s\le \infty$ with $1+1/p\ge 1/r+1/s$, an operator $T\in \mathcal {L}(X,Y)$ is (p, r, s)-nuclear if $T=\sum _{n=1}^\infty \lambda _n x_n^*\cdot y_n$ in the operator canonical norm of $\mathcal {L}(X,Y)$, where $(\lambda _n)_{n=1}^\infty \in \ell _p$, $(x_n^*)_{n=1}^\infty \in \ell _{s^*}^\omega (X^*)$ and $(y_n)_{n=1}^\infty \in \ell _{r^*}^\omega (Y)$. In the case $p=\infty$, we take $(\lambda _n)_{n=1}^\infty \in c_0$. It is said that $\sum _{n=1}^\infty \lambda _n x_n^* \cdot y_n$ is a (p, r, s)-nuclear representation of T. Define

$$\begin{aligned} \nu _{(p,r,s)}(T)=\inf \{\Vert (\lambda _n)_{n=1}^\infty \Vert _p\omega _{s^*}((x^*_n)_{n=1}^\infty )\omega _{r^*}((y_n)_{n=1}^\infty )\}, \end{aligned}$$

where the infimum is taken over all (p, r, s)-nuclear representations of T. Let $\mathcal {N}_{(p,r,s)}(X,Y)$ be the set of all (p, r, s)-nuclear operators from X into Y.

The corresponding version for Bloch maps could be the following.

Definition 2.1

Let $1\le p,r,s\le \infty$ such that $1+1/p\ge 1/r+1/s$. A map $f\in \mathcal {H}(\mathbb {D},X)$ is said to be (p, r, s)-nuclear Bloch if $f=\sum _{n=1}^\infty \lambda _n g_n\cdot x_n$ in the Bloch norm $\rho _\mathcal {B}$, where $(\lambda _n)_{n=1}^\infty \in \ell _p$, $(g_n)_{n=1}^\infty \in \ell _{s^*}^\omega (\widehat{\mathcal {B}}(\mathbb {D}))$ and $(x_n)_{n=1}^\infty \in \ell _{r^*}^\omega (X)$. For $p=\infty$, we choose $(\lambda _n)_{n=1}^\infty \in c_0$. We say that $\sum _{n=1}^\infty \lambda _n g_n\cdot x_n$ is a (p, r, s)-nuclear Bloch representation of f and we set

$$\begin{aligned} \nu _{(p,r,s)}^\mathcal {B}(f)=\inf \{\Vert (\lambda _n)_{n=1}^\infty \Vert _p\omega _{s^*}((g_n)_{n=1}^\infty )\omega _{r^*}((x_n)_{n=1}^\infty )\}, \end{aligned}$$

where the infimum is taken over all (p, r, s)-nuclear Bloch representations of f. Let $\mathcal {N}_{(p,r,s)}^\mathcal {B}(\mathbb {D},X)$ be the set of all (p, r, s)-nuclear Bloch maps from $\mathbb {D}$ into X, and let $\mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ be its subset formed by all those maps f for which $f(0)=0$.

Putting $1/\theta :=1/p+1/r^*+1/s^*$, $\mathcal {N}_{(p,r,s)}(X,Y)$ is a $\theta$-Banach operator ideal under the norm

$$\begin{aligned} \nu _{(p,r,s)}(T)=\inf \{\Vert (\lambda _n)_{n=1}^\infty \Vert _p\omega _{s^*}((x_n^*)_{n=1}^\infty )\omega _{r^*}((y_n)_{n=1}^\infty )\}, \end{aligned}$$

by taking the infimum is taken over all (p, r, s)-nuclear representations of T (see [14, Theorem 18.1.2]).

In order to establish a Bloch variant of this result, we first study the linearization of (p, r, s)-summing Bloch maps and (p, r, s)-nuclear Bloch maps.

Proposition 2.2

Let $f\in \widehat{\mathcal {B}}(\mathbb {D},X)$ and assume that $S_f\in \Pi _{(p,r,s)}(\mathcal {G}(\mathbb {D}),X)$. Then $f\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ and $\pi _{(p,r,s)}^{\mathcal {B}}(f)\le \pi _{(p,r,s)}(S_f)$.

Proof

Given $n\in \mathbb {N}$, $(\lambda _i)_{i=1}^n$ in $\mathbb {C}$, $(z_i)_{i=1}^n$ in $\mathbb {D}$ and $(x^*_i)_{i=1}^n$ in $X^*$, using Theorem 1.6 we have

$$\begin{aligned} \left\| (\lambda _i x^*_i(f'(z_i)))_{i=1}^n\right\| _p&=\left\| (x^*_i(S_f(\lambda _i\gamma _{z_i})))_{i=1}^n\right\| _p\\&\le \pi _{(p,r,s)}(S_f)\omega _r\left( (\lambda _i\gamma _{z_i})_{i=1}^n\right) \omega _s\left( (x^*_i)_{i=1}^n\right) \end{aligned}$$

and since

$$\begin{aligned} \omega _r\left( (\lambda _i\gamma _{z_i})_{i=1}^n\right)&=\sup _{\phi \in B_{\mathcal {G}(\mathbb {D})^*}}\left\| (\phi (\lambda _i\gamma _{z_i}))_{i=1}^n\right\| _r\\&=\sup _{g\in B_{\widehat{\mathcal {B}}(\mathbb {D})}}\left\| (\Lambda (g)(\lambda _i\gamma _{z_i}))_{i=1}^n\right\| _r\\&=\sup _{g\in B_{\widehat{\mathcal {B}}(\mathbb {D})}}\left\| (\lambda _i g'(z_i))_{i=1}^n\right\| _r=\omega ^{\widehat{\mathcal {B}}}_r((\lambda _i,z_i)_{i=1}^n), \end{aligned}$$

the result is proven. $\square$

Theorem 2.3

Let $1\le p,r,s\le \infty$ such that $1+1/p\ge 1/r+1/s$ and let $f\in \widehat{\mathcal {B}}(\mathbb {D},X)$. The following assertions are equivalent:

• (i) $f:\mathbb {D}\rightarrow X$ is a (p, r, s)-nuclear Bloch map.

• (ii) $S_f:\mathcal {G}(\mathbb {D})\rightarrow X$ is a (p, r, s)-nuclear linear operator.

In this case, $\nu _{(p,r,s)}^{\mathcal {B}}(f)=\nu _{(p,r,s)}(S_f)$.

Proof

$(i)\Rightarrow (ii)$: Assume that $f\in \mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ and let $\sum _{n=1}^\infty \lambda _n g_n\cdot x_n$ be a (p, r, s)-nuclear Bloch representation of f. First, note that if $g\in \widehat{\mathcal {B}}(\mathbb {D})$ and $x\in X$, we have that $\Lambda (g)\cdot x\in \mathcal {L}(\mathcal {G}(\mathbb {D}),X)$ and

$$\begin{aligned} (g\cdot x)'(z)=g'(z)x=\Lambda (g)(\gamma _z)x=(\Lambda (g)\cdot x)(\gamma _z)=(\Lambda (g)\cdot x\circ \Gamma )(z) \end{aligned}$$

for all $z\in \mathbb {D}$, and thus Theorem 1.6 gives $S_{g\cdot x}=\Lambda (g)\cdot x$. Since

$$\begin{aligned} \rho _\mathcal {B}\left( f-\sum _{k=1}^n\lambda _kg_k\cdot x_k\right) =\left\| S_f-\sum _{k=1}^n\lambda _kS_{g_k\cdot x_k}\right\| =\left\| S_f-\sum _{k=1}^n\lambda _k\Lambda (g_k)\cdot x_k\right\| \end{aligned}$$

for all $n\in \mathbb {N}$, it follows that $S_f=\sum _{n=1}^\infty \lambda _n\Lambda (g_n)\cdot x_n$ in the operator norm. Moreover, note that

$$\begin{aligned} \omega _{s^*}((\Lambda (g_n))_{n=1}^\infty )&=\sup _{\phi \in B_{\mathcal {G}(\mathbb {D})^{**}}}\left\| (\phi (\Lambda (g_n)))_{n=1}^\infty \right\| _{s^*}\\&=\sup _{\phi \in B_{\mathcal {G}(\mathbb {D})^{**}}}\left\| (\Lambda ^*(\phi )(g_n))_{n=1}^\infty \right\| _{s^*}\\&=\sup _{\varphi \in B_{\widehat{\mathcal {B}}(\mathbb {D})^*}}\left\| (\varphi (g_n))_{n=1}^\infty \right\| _{s^*}=\omega _{s^*}((g_n)_{n=1}^\infty ), \end{aligned}$$

where $\Lambda ^*:\mathcal {G}(\mathbb {D})^{**}\rightarrow \widehat{\mathcal {B}}(\mathbb {D})^*$ is the adjoint operator of $\Lambda :\widehat{\mathcal {B}}(\mathbb {D})\rightarrow \mathcal {G}(\mathbb {D})^*$. Hence $S_f\in \mathcal {N}_{(p,r,s)}(\mathcal {G}(\mathbb {D}),X)$ with

$$\begin{aligned} \nu _{(p,r,s)}(S_f)\le \Vert (\lambda _n)_{n=1}^\infty \Vert _p\omega _{s^*}((g_n)_{n=1}^\infty )\omega _{r^*}((x_n)_{n=1}), \end{aligned}$$

and passing to the infimum over all (p, r, s)-nuclear Bloch decompositions of f, we conclude that $\nu _{(p,r,s)}(S_f)\le \nu _{(p,r,s)}^{\mathcal {B}}(f)$.

$(ii)\Rightarrow (i)$ is proven with a reasoning similar to the previous one. $\square$

We are ready to establish a Bloch version of Theorem 18.1.2 in [14].

Corollary 2.4

Let $1\le p,r,s\le \infty$ such that $1+1/p\ge 1/r+1/s$ and let $1/\theta :=1/p+1/r^*+1/s^*$. Then $[\mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}},\nu _{(p,r,s)}^{\mathcal {B}}]$ is a $\theta$-Banach normalized Bloch ideal.

Proof

Let X be a complex Banach space.

(P1): Let $\lambda \in \mathbb {C}$ and $f,g\in \mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$. Using Theorems 1.6, 2.3 and [14, Theorem 18.1.2], we obtain that $\nu _{(p,r,s)}^{\mathcal {B}}$ is a norm on $\mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$:

$$\begin{aligned} \nu _{(p,r,s)}^{\mathcal {B}}(\lambda f)&=\nu _{(p,r,s)}(S_{\lambda f})=\nu _{(p,r,s)}(\lambda S_f)=\left| \lambda \right| \nu _{(p,r,s)}(S_f)=\left| \lambda \right| \nu _{(p,r,s)}^{\mathcal {B}}(f),\\ \nu _{(p,r,s)}^{\mathcal {B}}(f+g)^\theta&=\nu _{(p,r,s)}(S_{f+g})^\theta =\nu _{(p,r,s)}(S_f+S_g)^\theta \\&\le \nu _{(p,r,s)}(S_f)^\theta +\nu _{(p,r,s)}(S_g)^\theta =\nu _{(p,r,s)}^{\mathcal {B}}(f)^\theta +\nu _{(p,r,s)}^{\mathcal {B}}(g)^\theta ,\\ \nu _{(p,r,s)}^{\mathcal {B}}(f)&=0\Rightarrow \nu _{(p,r,s)}(S_f)=0 \Rightarrow S_f=0\Rightarrow f'=S_f\circ \Gamma =0\Rightarrow f=0. \end{aligned}$$

To see that the norm $\nu _{(p,r,s)}^{\mathcal {B}}$ is complete, note that another application of those theorems assures that $f\mapsto S_f$ is an isometric isomorphism of $(\mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X),\nu _{(p,r,s)}^{\mathcal {B}})$ onto $(\mathcal {N}_{(p,r,s)}(\mathcal {G}(\mathbb {D}),X),\nu _{(p,r,s)})$, and moreover

$$\begin{aligned} \rho _{\mathcal {B}}(f)=\left\| S_f\right\| \le \nu _{(p,r,s)}(S_f)=\nu _{(p,r,s)}^{\mathcal {B}}(f). \end{aligned}$$

(P2): Let $g\in \widehat{\mathcal {B}}(\mathbb {D})$ and $x\in X$. By the operator ideal property of $[\mathcal {N}_{(p,r,s)},\nu _{(p,r,s)}]$ and Theorem 1.6, $S_{g\cdot x}=\Lambda (g)\cdot x\in \mathcal {N}_{(p,r,s)}(\mathcal {G}(\mathbb {D}),X)$ with $\nu _{(p,r,s)}(S_{g\cdot x})=\left\| \Lambda (g)\right\| \left\| x\right\| =\rho _{\mathcal {B}}(g)\left\| x\right\|$. Hence $g\cdot x\in \mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ with $\nu _{(p,r,s)}^{\mathcal {B}}(g\cdot x)=\rho _{\mathcal {B}}(g)\left\| x\right\|$ by Theorem 2.3.

(P3): Let $f\in \mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$, $T\in \mathcal {L}(X,Y)$ and $h\in \widehat{\mathcal {B}}(\mathbb {D},\mathbb {D})$. Since $T\circ S_f\circ \widehat{h}\in \mathcal {L}(\mathcal {G}(\mathbb {D}),Y)$ and

$$\begin{aligned} (T\circ f\circ h)'&=T\circ [h'\cdot (f'\circ h)]=T\circ [h'\cdot (S_f\circ \Gamma \circ h)]\\&=T\circ [S_f(h'\cdot (\Gamma \circ h))]=T\circ [S_f\circ (\widehat{h}\circ \Gamma )]\\&=(T\circ S_f\circ \widehat{h})\circ \Gamma , \end{aligned}$$

one has that $S_{T\circ f\circ h}=T\circ S_f\circ \widehat{h}$ by Theorem 1.6. Since $S_f\in \mathcal {N}_{(p,r,s)}(\mathcal {G}(\mathbb {D}),X)$ by Theorem 2.3, we get that $S_{T\circ f\circ h}\in \mathcal {N}_{(p,r,s)}(\mathcal {G}(\mathbb {D}),Y)$ with $\nu _{(p,r,s)}(S_{T\circ f\circ h})\le \left\| T\right\| \nu _{(p,r,s)}(S_f)||\widehat{h}||$ by the operator ideal property of $[\mathcal {N}_{(p,r,s)},\nu _{(p,r,s)}]$, and thus $T\circ f\circ h\in \mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},Y)$ with $\nu _{(p,r,s)}^{\mathcal {B}}(T\circ f\circ h)\le \left\| T\right\| \nu _{(p,r,s)}^{\mathcal {B}}(f)$ by Theorems 1.6 and 2.3. $\square$

We conclude arriving at the objective of this section.

Corollary 2.5

Let $1\le p,r,s\le \infty$ such that $1/p\le 1/r+1/s\le 1+1/p$. Then $(\mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X),\nu _{(p,r,s)}^{\mathcal {B}})\le (\Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X),\pi _{(p,r,s)}^\mathcal {B})$.

Proof

Let $f\in \mathcal {N}_{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$. Then $S_f\in \mathcal {N}_{(p,r,s)}(\mathcal {G}(\mathbb {D}),X)$ with $\nu _{(p,r,s)}(S_f)=\nu _{(p,r,s)}^{\mathcal {B}}(f)$ by Theorem 2.3. Since

$$\begin{aligned} (\mathcal {N}_{(p,r,s)}(\mathcal {G}(\mathbb {D}),X),\nu _{(p,r,s)})\le (\Pi _{(p,r,s)}(\mathcal {G}(\mathbb {D}),X),\pi _{(p,r,s)}), \end{aligned}$$

it follows that $S_f\in \Pi _{(p,r,s)}(\mathcal {G}(\mathbb {D}),X)$ with $\pi _{(p,r,s)}(S_f)\le \nu _{(p,r,s)}(S_f)$. By Proposition 2.2, $f\in \Pi _{(p,r,s)}^{\widehat{\mathcal {B}}}(\mathbb {D},X)$ and $\pi _{(p,r,s)}^\mathcal {B}(f)\le \pi _{(p,r,s)}(S_f)\le \nu _{(p,r,s)}^{\mathcal {B}}(f)$. $\square$