Generating Ideals of Bloch Mappings via Pietsch’s Quotients

Abstract

In this paper, we introduce the notion of the normalized Bloch left-hand quotient ideal ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$, where $\mathcal{A}$ is an operator ideal and ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ is a normalized Bloch ideal, as a nonlinear extension of the concept of the left-hand quotient of operator ideals. We show that these quotients constitute a new method for generating normalized Bloch ideals, complementing the existing methods of generation by composition and transposition. In fact, if ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the linearization property in a linear operator ideal $\mathcal{J}$, then ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$ is a composition ideal of the form $({\mathcal{A}}^{-1}\circ \mathcal{J})\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$. We conclude this work by introducing two important subclasses of Bloch maps; these are Bloch maps with the Grothendieck and Rosenthal range. We focus on showing that they form normalized Bloch ideals which can be seen as normalized Bloch left-hand quotients ideals. In addition, we pose an open problem concerning when a Bloch quotient without the linearization property in an operator ideal cannot be related to a normalized Bloch ideal of the composition type, for which we will use the subclass of p-summing Bloch maps.

1. Introduction

Let $\mathbb{D}$ be the open complex unit disc in the complex plane $\mathbb{C}$, X be a complex Banach space, and $\mathcal{H}(\mathbb{D},X)$ be the space of all holomorphic mappings from $\mathbb{D}$ into X. A mapping $f\in \mathcal{H}(\mathbb{D},X)$ is called Bloch if

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

We denote $\mathcal{B}(\mathbb{D},X)$ to the space of all Bloch mappings from $\mathbb{D}$ into X, and $\widehat{\mathcal{B}}(\mathbb{D},X)$ to the closed subspace formed by all zero-preserving Bloch mappings, that is, such that $f\left(0\right)=0$, endowed with the Bloch norm $p_{\mathcal{B}}$. In addition, we become used to the abbreviation $\widehat{\mathcal{B}}\left(\mathbb{D}\right)=\widehat{\mathcal{B}}(\mathbb{D},\mathbb{C})$.

Bloch-type spaces are of significant interest in Complex Analysis and Functional Analysis because of their rich structure and applications in some mathematical fields. Bloch spaces play a crucial role in studying the geometric properties of holomorphic mappings, operator theory, and the interplay between function theory and hyperbolic geometry. Their connections with other function spaces, such as Hardy, Bergman, Sobolev, or Besov spaces, further highlight their importance. These kinds of spaces can be deeply studied with the book [1] by Zhou.

Let us recall by [2] that an operator ideal $\mathcal{I}$ is a subclass of the class of all continuous linear operators $\mathcal{L}$ such that for all Banach spaces X and Y, the components

$$\mathcal{I}(X,Y):=\mathcal{I}\cap \mathcal{L}(X,Y)$$

satisfy the following properties:

(i)

$I_{\mathbb{K}}\in \mathcal{I}$, where $\mathbb{K}$ is the one-dimensional Banach space.

(ii)

$\mathcal{I}(X,Y)$ is a subspace of $\mathcal{L}(X,Y)$.

(iii)

The ideal property: if $T\in \mathcal{L}(W,X)$, $S\in \mathcal{I}(X,Y)$ and $R\in \mathcal{L}(Y,Z)$, then $R\circ S\circ T\in \mathcal{I}(W,Z)$, with W and Z being Banach spaces.

In addition, consider that for all Banach spaces X and Y, we endow $\mathcal{I}$ with a (complete) norm ${∥·∥}_{\mathcal{I}}$ so that the following hold:

(iv)

$(\mathcal{I}(X,Y),{∥·∥}_{\mathcal{I}})$ is a Banach space.

(v)

If W and Z are Banach spaces and $T\in \mathcal{L}(W,X)$, $S\in \mathcal{I}(X,Y)$ and $R\in \mathcal{L}(Y,Z)$ such that

$${\parallel R\circ S\circ T\parallel }_{\mathcal{I}}\le {\parallel R\parallel \parallel S\parallel }_{\mathcal{I}}\parallel T\parallel$$

then we will say that $[\mathcal{I},{∥·∥}_{\mathcal{I}}]$ is a (Banach) normed operator ideal.

Let $\mathcal{I},\mathcal{J}$ be operator ideals and let $X,Y$ be Banach spaces. Following [3] (p. 132), a bounded linear operator $T:X\to Y$ is said to belong to the left-hand quotient ${\mathcal{I}}^{-1}\circ \mathcal{J}$, and we write $T\in {\mathcal{I}}^{-1}\circ \mathcal{J}(X,Y)$, if $S\circ T\in \mathcal{J}(X,Z)$ for all $S\in \mathcal{I}(Y,Z)$, with Z being an arbitrary Banach space. The right-hand quotient$\mathcal{I}\circ {\mathcal{J}}^{-1}$ is defined in a similar way. Of course, the symbols ${\mathcal{I}}^{-1}$ and ${\mathcal{J}}^{-1}$ have no meaning. It is well known that ${\mathcal{I}}^{-1}\circ \mathcal{J}$ and $\mathcal{I}\circ {\mathcal{J}}^{-1}$ are operator ideals (see Section 3.2.2 of [2]).

Moreover, if $[\mathcal{I},{∥·∥}_{\mathcal{I}}]$ and $[\mathcal{J},{∥·∥}_{\mathcal{J}}]$ are Banach operator ideals and $T\in {\mathcal{I}}^{-1}\circ \mathcal{J}(X,Y)$, we define

$${\parallel T\parallel }_{{\mathcal{I}}^{-1}\circ \mathcal{J}}={sup\{\parallel S\circ T\parallel }_{\mathcal{J}}:S\in {\mathcal{I}(Y,Z),\parallel S\parallel }_{\mathcal{I}}\le 1\},$$

with Z covering all Banach spaces. In such a case, $[{\mathcal{I}}^{-1}\circ \mathcal{J},{∥·∥}_{{\mathcal{I}}^{-1}\circ \mathcal{J}}]$ is a Banach operator ideal by Section 7.2.2 of [2].

Left-hand quotients and right-hand quotients have been studied by many authors over time. For example, Causey and Navoyan generalized in [4] a result from Pietsch’s manuscript (Section 3.2.3 of [2]) proving that the class of $\xi$-completely continuous operators can be seen as a right-hand quotient generated by the classes of compact operators and $\xi$-w-compact operators; Johnson, Lillemets and Oja showed in [5] that completely continuous operators can be represented through w-∞-compact operators via a right-hand quotient, and they used it to prove that only in Schur spaces is the w-Grothendieck compactness principle satisfied; and Kim showed in [6] that the class of operators which sends w-p-summable sequences to unconditionally p-summable sequences is a right-hand quotient induced by the ideals of unconditionally p-compact operators and w-p-compact operators. However, the most prolific application of quotients of operator ideals was introduced by Carl and Defant in [7]. There, they defined the ideal of $(q,p)$-mixing operators, showing which is expressible as a left-hand quotient induced by the ideals of q-summing operators and p-summing operators. This work led to important generalizations of the quotients of operator ideals to the nonlinear setting, with papers such as [8,9], where Chávez-Domínguez introduced the notions of Lipschitz $(q,p)$-mixing operator and completely $(q,p)$-mixing maps, respectively. Our goal in this work is to present a Bloch version of the concept of the left-hand quotient of operator ideals through the use of the concept of the normalized Bloch ideal introduced in Definition 5.11 of [10]. Some work of this nature is currently underway in different settings, such as in the bounded holomorphic context.

We divide our work into three sections, which are discussed below. Section 2 is devoted to recall and introduce some basic concepts such as the notion of the (Banach) normed normalized Bloch ideal, the linearization theorem for Banach-valued Bloch maps and the novel definition of the normalized Bloch left-hand quotient ideal. Next, in Section 3, we present the first properties of the left-hand quotients ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$, where $\mathcal{A}$ is an operator ideal and ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ is a normalized Bloch ideal. If both ideals are endowed with Banach norms, then we will prove that ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$ endowed with the norm ${∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$ is a Banach normalized Bloch ideal which becomes surjective whenever ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ is surjective. Thus, normalized Bloch left-hand quotients ideals prove to be an interesting method of generating bounded-holomorphic ideals. There are already two well-known ways to produce bounded-holomorphic ideals: by composition and by transposition (see Proposition 2.2 and Theorem 4.7 of [11]).

We show that if ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the linearization property in an operator ideal $\mathcal{A}$, then ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}=\widehat{\mathcal{B}}$, while if ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the linearization property in another operator ideal $\mathcal{J}\ne \mathcal{A}$, then a map $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ belongs to the normalized Bloch left-hand quotient ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ if and only if its linearization $S_f\in \mathcal{L}(\mathcal{G}\left(\mathbb{D}\right),X)$ belongs to the operator left-hand quotient ${\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$, where the Bloch-free space $\mathcal{G}\left(\mathbb{D}\right)$ is defined by

$$\mathcal{G}\left(\mathbb{D}\right):=\overline{\mathrm{lin}}\left(\{{\gamma }_z:z\in \mathbb{D}\}\right)\subseteq \widehat{\mathcal{B}}{\left(\mathbb{D}\right)}^{*},$$

with ${\gamma }_z:\widehat{\mathcal{B}}\left(\mathbb{D}\right)\to \mathbb{C}$ being the functional given as ${\gamma }_z\left(f\right)=f^{\prime }\left(z\right)$ for all $f\in \widehat{\mathcal{B}}\left(\mathbb{D}\right)$. In this case, we also prove that ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$ is a composition ideal of the form $({\mathcal{A}}^{-1}\circ \mathcal{J})\circ \widehat{\mathcal{B}}$.

Finally, Section 4 is dedicated to introducing two relevant examples of normalized Bloch left-hand quotient ideals generated by an operator ideal and a normalized Bloch ideal: the spaces of holomorphic mappings which have the Grothendieck Bloch range and Rosenthal Bloch range.

The following is the notation we will use throughout this paper. We shall use the symbol $\mathbb{D}$ to denote the open unit disc of the complex plane $\mathbb{C}$. $\mathcal{L}(X,Y)$ stands for the space of all linear continuous operators from X into Y endowed with the operator canonical norm, where X and Y are normed spaces. We will write $X^{*}$ to denote the dual space of X, that is, $X^{*}=\mathcal{L}(X,\mathbb{C})$. As usual, $B_X$ denotes the closed unit ball of X. On the other hand, for a set $A\subseteq X$, we denote the linear hull and the norm-closed absolutely convex hull of A in X by $\mathrm{lin}\left(A\right)$ and $\overline{\mathrm{abco}}\left(A\right)$, respectively.

2. Preliminaries

Firstly, recall by Definition 5.11 of [10] that a (Banach) normed normalized Bloch ideal, denoted as $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$, is a subset of the class of all zero-preserving Bloch maps $[\widehat{\mathcal{B}},p_{\mathcal{B}}]$ which is characterized by the condition that the components ${\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ verify the properties that follow:

(P1)

$({\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X),{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}})$ is a (Banach) normed space and $p_{\mathcal{B}}\left(f\right)\le {∥f∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$ for $f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$.

(P2)

The function $g·x:\mathbb{D}\to X$, defined as

$$(g·x)\left(z\right)=g\left(z\right)x(z\in \mathbb{D}),$$

is in ${\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ with ${∥g·x∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}=p_{\mathcal{B}}\left(g\right)∥x∥$ for any $g\in \widehat{\mathcal{B}}\left(\mathbb{D}\right)$ and $x\in X$.

(P3)

The ideal property: Given $f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, $S\in \mathcal{L}(X,Y)$ and $h\in \widehat{\mathcal{B}}(\mathbb{D},\mathbb{D})$, where Y is a complex Banach space, then the map $S\circ f\circ h$ is in ${\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ and ${∥S\circ f\circ h∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le ∥S∥{∥f∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$.

A normed normalized Bloch ideal $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ is

(C)

Closed if every component ${\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ is a closed subspace of $\widehat{\mathcal{B}}(\mathbb{D},X)$ endowed with the Bloch norm topology.

(S)

Surjective if for any complex Banach space X, $f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ with ${\parallel f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}={\parallel f\circ \pi \parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$, whenever $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$, $\pi \in \widehat{\mathcal{B}}(\mathbb{D},\mathbb{D})$ is a metric surjection, and $f\circ \pi \in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$.

Our main tool in this paper is the method of linearization of Bloch mappings gathered in the following result.

Theorem 1 

([10]).

(i)

The map $\Gamma :\mathbb{D}\to \mathcal{G}\left(\mathbb{D}\right)$, given by

$$\Gamma \left(z\right)={\gamma }_z(z\in \mathbb{D}),$$

is in $\mathcal{H}(\mathbb{D},\mathcal{G}(\mathbb{D}\left)\right)$ and $\parallel {\gamma }_z{\parallel =1/(1-|z|}^2)$ for any $z\in \mathbb{D}$.

(ii)

$B_{\mathcal{G}\left(\mathbb{D}\right))}=\overline{\mathrm{abco}}\left({\mathcal{M}}_{\mathcal{B}}\left(\mathbb{D}\right)\right)$, where ${\mathcal{M}}_{\mathcal{B}}\left(\mathbb{D}\right)=\left\{\right(1-{\left|z\right|}^2){\gamma }_z:z\in \mathbb{D}\}$.

(iii)

For each complex Banach space X and each map $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$, there exists a unique operator $S_f\in \mathcal{L}(\mathcal{G}\left(\mathbb{D}\right),X)$ such that $S_f\circ \Gamma =f^{\prime }$. Moreover, $∥S_f∥=p_{\mathcal{B}}\left(f\right)$. In addition, the map $f\mapsto S_f$ is an isometric isomorphism from $\widehat{\mathcal{B}}(\mathbb{D},X)$ onto $\mathcal{L}\left(\mathcal{G}\right(\mathbb{D}),X)$.

Our research will mainly be based on a certain linearization property for the functions of the normalized Bloch ideal ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ in a suitable operator ideal $\mathcal{A}$.

Definition 1. 

For a normed operator ideal $[\mathcal{J},{∥·∥}_{\mathcal{J}}]$ and a normed normalized Bloch ideal $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$, ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ is said to have the linearization property (hereafter, we simply write LP for short) in $\mathcal{J}$ if for any $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$, it is satisfied that $f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ if and only if $S_f\in \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$. In such a case, ${∥f∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}={∥S_f∥}_{\mathcal{J}}$.

Motivated by the concept of the left-hand quotient of operator ideals studied in Section 3.2.1 of [2]), we introduce the notion of the left-hand quotient of an operator ideal and a normalized Bloch ideal, focusing on analyzing its fundamental properties.

Definition 2. 

Let ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ be a normalized Bloch ideal and let $\mathcal{A}$ be an operator ideal. A map $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ belongs to the normalized Bloch left-hand quotient ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$, and shall be written as $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, if $T\circ f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ for all $T\in \mathcal{A}(X,Y)$, with Y being a complex Banach space.

If the operator ideal $\mathcal{A}$ is equipped with a Banach norm ${∥·∥}_{\mathcal{A}}$ and ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ with a norm ${∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$, we define the Bloch left-hand quotient norm as follows:

$${\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}={sup\{\parallel T\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in {\mathcal{A}(X,Y),\parallel T\parallel }_{\mathcal{A}}\le 1\}.$$

The first goal is to demonstrate the existence of ${∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$. Toward this end, let us recall Cartesian ${\ell }_p$-products of Banach spaces (see C.4 of [2]). Let $\left(X_i\right)$ with $i\in I$ be a family of Banach spaces. For $1\le p<\infty$, the Cartesian ${\ell }_p$-product ${\ell }_p(I,X_i)$ is defined as the set of all families $\mathbf{x}=\left(x_i\right)$, where $x_i\in X_i$ for $i\in I$, such that $(\parallel x_i\parallel )\in {\ell }_p\left(I\right)$, being that ${\ell }_p\left(I\right)$ is the Banach space comprising all scalar families $x=\left({\xi }_i\right)$ that are absolutely p-summable, with $i\in I$. It is well known by C.4.1 of [2] that ${\ell }_p(I,X_i)$ is a Banach space equipped with the following norm:

$${\parallel \mathbf{x}\parallel }_p={\left(\sum _{i\in I}{\parallel x_i\parallel }^p\right)}^{1/p}.$$

If the underlying index set is $\mathbb{N}$, we just write ${\ell }_p\left(X_n\right)$.

Proposition 1. 

Let $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$ be a Banach operator ideal and let $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ be a normed normalized Bloch ideal. If $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, then

$$sup\{\parallel T\circ f{\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}<\infty .$$

Proof. 

Let us assume that the preceding supremum is not finite. Then, for each $n\in \mathbb{N}$, we are in a position to find a complex Banach space $Y_n$ and an operator $S_n\in \mathcal{A}(X,Y_n)$ satisfying

$$\parallel S_n{\parallel }_{\mathcal{A}}\le 1/2^n,{\parallel S_n\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\ge n(n\in \mathbb{N}).$$

Consider the sequence of Banach spaces $\left(Y_i\right)$ with $i\in \mathbb{N}$, and the Cartesian ${\ell }_1$-product ${\ell }_1(\mathbb{N},Y_i)$. For each $n\in \mathbb{N}$, let us define the continuous linear operators $J_n:Y_n\to {\ell }_1(\mathbb{N},Y_i)$ and $Q_n:{\ell }_1(\mathbb{N},Y_i)\to Y_n$ by

$$\begin{array}{ccc}\hfill & J_n\left(y\right)={\left({\delta }_{in}y\right)}_i\hfill & \hfill (y\in Y_n),\\ \hfill & Q_n\left(\left(y_i\right)\right)=y_n\hfill & \hfill (\left(y_i\right)\in {\ell }_1(\mathbb{N},Y_i)),\end{array}$$

with ${\delta }_{in}$ being the Kronecker delta. It is easy to check $∥J_n∥=1$ and $∥Q_n∥\le 1$. Notice that $(J_n\circ S_n)$ is a sequence of elements of the Banach space $(\mathcal{A}(X,{\ell }_1(\mathbb{N},Y_i)),{∥·∥}_{\mathcal{A}})$, and

$$\begin{array}{cc}\hfill {∥\sum _{n=j+1}^{j+m}{J}_n\circ S_n∥}_{\mathcal{A}}& \le \sum _{n=j+1}^{j+m}\parallel J_n\circ S_n{\parallel }_{\mathcal{A}}=\sum _{k=1}^m{\parallel J_{j+k}\circ S_{j+k}\parallel }_{\mathcal{A}}\hfill \\ \hfill & \le \sum _{k=1}^m\parallel S_{j+k}{\parallel }_{\mathcal{A}}\le \sum _{k=1}^{\infty }{\parallel S_{j+k}\parallel }_{\mathcal{A}}\le \sum _{k=1}^{\infty }{\displaystyle \frac{1}{2^{j+k}}}={\displaystyle \frac{1}{2^j}},\hfill \end{array}$$

for all $j,m\in \mathbb{N}$. Thus, the sequence of partial sums ${({\sum }_{n=1}^m{J}_n\circ S_n)}_m$ forms a ${∥·∥}_{\mathcal{A}}$-Cauchy sequence. Hence, $S:={\sum }_{n=1}^{\infty }{J}_n\circ S_n\in \mathcal{A}(X,{\ell }_1(\mathbb{N},Y_i))$ and we have

$$n\le \parallel S_n{\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}=\parallel Q_n{\circ S\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {\parallel S\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}},$$

which is a contradiction, where we use that

$$\begin{array}{cc}\hfill Q_n\circ T\circ f\left(z\right)& =Q_n\circ \left(\sum _{n=1}^{\infty }{J}_n\circ T_n\right)\circ f\left(z\right)\hfill \\ \hfill & =Q_n\left(\sum _{n=1}^{\infty }{\left({\delta }_{in}{T}_n\left(f\left(z\right)\right)\right)}_i\right)\hfill \\ \hfill & =Q_n\left({\left(T_i\left(f\left(z\right)\right)\right)}_i\right)=T_n\left(f\left(z\right)\right)(z\in \mathbb{D}).\hfill \end{array}$$

□

3. Normalized Bloch Left-Hand Quotient Ideals

Our first objective is to study the relationship between two normed normalized Bloch left-hand quotient ideals when the corresponding normed normalized Bloch ideals are related. Before doing so, let us establish the following notation: if $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$ and $[\mathcal{J},{∥·∥}_{\mathcal{J}}]$ are normed operator ideals, we write $[\mathcal{A},{∥·∥}_{\mathcal{A}}]\le [\mathcal{J},{∥·∥}_{\mathcal{J}}]$ in case $\mathcal{A}\subseteq \mathcal{J}$ and ${\parallel f\parallel }_{\mathcal{J}}\le {\parallel f\parallel }_{\mathcal{A}}$ for all $f\in \mathcal{A}$.

Proposition 2. 

Let $[{\mathcal{I}}_1^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}_1^{\widehat{\mathcal{B}}}}]$ and $[{\mathcal{I}}_2^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}_2^{\widehat{\mathcal{B}}}}]$ be normed normalized Bloch ideals satisfying

$$[{\mathcal{I}}_1^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}_1^{\widehat{\mathcal{B}}}}]\le [{\mathcal{I}}_2^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}_2^{\widehat{\mathcal{B}}}}].$$

Then,

$$[{\mathcal{A}}^{-1}\circ {\mathcal{I}}_1^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}_1^{\widehat{\mathcal{B}}}}]\le [{\mathcal{A}}^{-1}\circ {\mathcal{I}}_2^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}_2^{\widehat{\mathcal{B}}}}],$$

for any Banach operator ideal $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$.

It is well known that $\widehat{\mathcal{B}}$ is a normalized Bloch ideal which becomes Banach if we endow it with the Bloch norm $p_{\mathcal{B}}$. The following result is an immediate consequence of the previous proposition. In this, we ensure that ${\mathcal{A}}^{-1}\circ \widehat{\mathcal{B}}$ is the largest normalized Bloch left-hand quotient for any Banach operator ideal $\mathcal{A}$ in the following sense.

Corollary 1. 

Let $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ be a normed normalized Bloch ideal. Then

$$[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]\le [{\mathcal{A}}^{-1}\circ \widehat{\mathcal{B}},{∥·∥}_{{\mathcal{A}}^{-1}\circ \widehat{\mathcal{B}}}]$$

for any Banach operator ideal $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$.

Under a suitable assumption over the normalized Bloch ideal ${\mathcal{I}}^{\widehat{\mathcal{B}}}$, and in relation to Corollary 1, we have the following useful result.

Proposition 3. 

Let $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$ be a Banach operator ideal and $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ be a normed normalized Bloch ideal. Then,

$$[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]\le [\widehat{\mathcal{B}},p_{\mathcal{B}}].$$

Furthermore, if ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the LP in $\mathcal{A}$, then

$$[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]=[\widehat{\mathcal{B}},p_{\mathcal{B}}].$$

Proof. 

Firstly, for $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, we claim that $p_{\mathcal{B}}\left(f\right)\le {\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$. Indeed, notice that $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ and $T\circ f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ for all $T\in \mathcal{A}(X,Y)$. Let $z\in \mathbb{D}$. Following [12] (p. 131), we may consider a functional $x^{*}\in B_{X^{*}}$ so that $∥f^{\prime }\left(z\right)∥=\left|x^{*}\left(f^{\prime }\left(z\right)\right)\right|$. Taking into account that $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$ is a Banach operator ideal, we have that the function $x^{*}\otimes 1:X\to \mathbb{C}$, given by

$$(x^{*}\otimes 1)\left(x\right)=x^{*}\left(x\right)(x\in X),$$

is in the component $\mathcal{A}(X,\mathbb{C})$ with ${∥x^{*}\otimes 1∥}_{\mathcal{A}}=∥x^{*}∥\le 1$. Note that, for $z\in \mathbb{D}$, we have

$$\begin{array}{cc}\hfill {(1-|z|}^2\left)\right|{((x^{*}\otimes 1)\circ f)}^{\prime }\left(z\right)|& ={(1-|z|}^2\left)\right|((x^{*}\otimes 1)\circ f^{\prime })\left(z\right)|\hfill \\ \hfill & \le {(1-|z|}^2)\parallel f^{\prime }\left(z\right)\parallel \parallel x^{*}\otimes 1\parallel \hfill \\ \hfill & \le p_{\mathcal{B}}\left(f\right){\parallel x^{*}\otimes 1\parallel }_{\mathcal{A}}\le p_{\mathcal{B}}\left(f\right).\hfill \end{array}$$

Taking the supremum over all $z\in \mathbb{D}$, we have $(x^{*}\otimes 1)\circ f\in \mathcal{B}\left(\mathbb{D}\right)$. Furthermore, $((x^{*}\otimes 1)\circ f)\left(0\right)=0$ and then $(x^{*}\otimes 1)\circ f\in \widehat{\mathcal{B}}\left(\mathbb{D}\right)$ with $p_{\mathcal{B}}((x^{*}\otimes 1)\circ f)\le p_{\mathcal{B}}\left(f\right)$. Finally,

$$\begin{array}{cc}\hfill {(1-|z|}^2)∥f^{\prime }\left(x\right)∥& ={(1-|z|}^2)\left|((x^{*}\otimes 1)\circ f^{\prime })\left(z\right)\right|\hfill \\ \hfill & ={(1-|z|}^2)\left|{((x^{*}\otimes 1)\circ f)}^{\prime }\left(z\right)\right|\hfill \\ \hfill & \le p_{\mathcal{B}}((x^{*}\otimes 1)\circ f)\le {\parallel (x^{*}\otimes 1)\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\hfill \\ \hfill & \le {\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}},\hfill \end{array}$$

and taking the supremum over all $z\in \mathbb{D}$, we conclude that $p_{\mathcal{B}}\left(f\right)\le {∥f∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{J}}^{\widehat{\mathcal{B}}}}$.

Let us now assume that ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the LP in the operator ideal $\mathcal{A}$. Let $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ and let $T\in \mathcal{A}(X,Y)$, for a complex Banach space Y. It is not difficult to see that $T\circ f\in \widehat{\mathcal{B}}(\mathbb{D},Y)$. By Theorem 1, we can ensure the existence and uniqueness of the linearizations $S_{T\circ f}\in \mathcal{L}(\mathcal{G}\left(\mathbb{D}\right),Y)$ and $S_f\in \mathcal{L}(\mathcal{G}\left(\mathbb{D}\right),X)$ with $∥S_{T\circ f}∥=p_{\mathcal{B}}(T\circ f)$ and $∥S_f∥=p_{\mathcal{B}}\left(f\right)$ satisfying

$$S_{T\circ f}\circ \Gamma ={(T\circ f)}^{\prime }=T\circ f^{\prime }=T\circ S_f\circ \Gamma .$$

Hence, $S_{T\circ f}=T\circ S_f$ by Remarks 2.8 (2) of [10]. By the ideal property of $\mathcal{A}$, we have $S_{T\circ f}\in \mathcal{A}(\mathcal{G}\left(\mathbb{D}\right),Y)$ with

$${∥S_{T\circ f}∥}_{\mathcal{A}}={∥T\circ S_f∥}_{\mathcal{A}}\le {∥T∥}_{\mathcal{A}}∥S_f∥=\le {∥T∥}_{\mathcal{A}}{p}_{\mathcal{B}}\left(f\right)\le p_{\mathcal{B}}\left(f\right).$$

By hypothesis, ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the LP in $\mathcal{A}$, and then we have $T\circ f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ with ${∥T\circ f∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}={∥S_{T\circ f}∥}_{\mathcal{A}}$. Therefore, we are in a position to conclude that $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ with ${∥f∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}\le p_{\mathcal{B}}\left(f\right)$ as desired. □

Next, we show that normalized Bloch left-hand quotients are a method for generating normalized Bloch ideals. There are already two well-known ways to generate normalized Bloch ideals: by composition and by transposition (see Proposition 2.2 and Theorem 4.7 of [11]).

Theorem 2. 

Let $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$ be a Banach operator ideal and let $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ be a (Banach) normed normalized Bloch ideal. Then, $[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ is a (Banach) normed normalized Bloch ideal.

Proof. 

(P1): Let ${\gamma }_i\in \mathbb{C}$ and $g_i\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ for $i=1,2$. Let $T\in \mathcal{A}(X,Y)$, where Y is a complex Banach space and note that $T\circ ({\gamma }_1{g}_1+{\gamma }_2{g}_2)={\gamma }_1(T\circ g_1)+{\gamma }_2(T\circ g_2)$ by the linearity of T. Since $T\circ g_i\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ for $i=1,2$ and ${\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ is a linear space, we obtain ${\gamma }_1(T\circ g_1)+{\gamma }_2(T\circ g_2)\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$, and thus ${\gamma }_1{f}_1+{\gamma }_2{f}_2\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$.

We will now demonstrate that ${∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$ is a norm on ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$. Let $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ and assume that ${\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}=0$. Consequently, $T\circ f=0$ for all $T\in \mathcal{A}(X,Y)$, with ${\parallel T\parallel }_{\mathcal{A}}\le 1$ and Y being a complex Banach space. In particular, $x^{*}\circ f=(x^{*}\otimes 1)\circ f=0$ for all $x^{*}\in B_{X^{*}}$, and then $f=0$ since $B_{X^{*}}$ separates the points of X.

For $\alpha \in \mathbb{C}$ and $f,g\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, we claim that ${\parallel \alpha f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}={\left|\alpha \right|\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$ and ${\parallel f+g\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}+{\parallel g\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$. Indeed, note that

$$\begin{array}{cc}\hfill {\parallel \alpha f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}& =sup\{\parallel T\circ \left(\alpha f\right){\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & =sup\{\parallel \alpha (T\circ f){\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & ={\left|\alpha \right|\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\parallel f+g\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}& =sup\{\parallel T\circ (f+g){\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & =sup\{\parallel (T\circ f)+(T\circ g){\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & \le {\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}+{\parallel g\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}.\hfill \end{array}$$

Now, we are going to study the structure, as the Banach space of the pair $({\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X),{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}})$ when ${∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$ is a complete norm on ${\mathcal{I}}^{\widehat{\mathcal{B}}}$. Let $\left(f_n\right)$ be a ${∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$-Cauchy sequence. For a complex Banach space Y, let $T\in \mathcal{A}(X,Y)$. By Proposition 3, we know $p_{\mathcal{B}}(·)\le {\parallel ·\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$ on ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, and then there exists a map $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ such that $p_{\mathcal{B}}(f_n-f)\to 0$ as $n\to \infty$. As a consequence, $p_{\mathcal{B}}(T\circ f_n-T\circ f)\to 0$ as $n\to \infty$ because $p_{\mathcal{B}}(T\circ f_n-T\circ f)\le ∥T∥p_{\mathcal{B}}(f_n-f)$ for all $n\in \mathbb{N}$. On the other hand, for all $p,q\in \mathbb{N}$, the inequality

$${∥T\circ f_p-T\circ f_q∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}={∥T\circ (f_p-f_q)∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {∥T∥}_{\mathcal{A}}{∥f_p-f_q∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$$

shows that $(T\circ f_n)$ is a Cauchy sequence in $({\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y),{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}})$. Hence, we could take a mapping $g\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ so that ${∥T\circ f_n-g∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\to 0$ as $n\to \infty$. Bearing in mind the fact that $p_{\mathcal{B}}(·)\le {\parallel ·\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$ on ${\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$, we obtain that $T\circ f=g$, and then $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ with ${∥T\circ f_n-T\circ f∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\to 0$ as $n\to \infty$.

Next, we prove that $\left(f_n\right)$ converges to f in $({\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X),{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}})$. For this purpose, let $\epsilon >0$. Then, there exists $m\in \mathbb{N}$ such that, for all $p,q\ge m$, ${∥f_p-f_q∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}<\epsilon /2$. Thus,

$${∥T\circ f_p-T\circ f_{p+n}∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}<{\displaystyle \frac{\epsilon }{2}}$$

for all $p\ge m$, $n\in \mathbb{N}$ and $T\in \mathcal{A}(X,Y)$ with ${\parallel T\parallel }_{\mathcal{A}}\le 1$. Taking limits with $n\to \infty$, it follows that

$${∥T\circ f_p-T\circ f∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {\displaystyle \frac{\epsilon }{2}}$$

for all $p\ge m$ and $T\in \mathcal{A}(X,Y)$ with ${\parallel T\parallel }_{\mathcal{A}}\le 1$. Taking the supremum over all such T, we obtain that the sequence $\left(f_n\right)$ is ${∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$-convergent to f, as we wanted.

(P2): Let $g\in \widehat{\mathcal{B}}\left(\mathbb{D}\right)$ and $x\in X$. Since ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ is a normed normalized Bloch ideal, we have that $g·x\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ with ${\parallel g·x\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}=p_{\mathcal{B}}\left(g\right)\parallel x\parallel$. Let $T\in \mathcal{A}(X,Y)$ and do not lose sight of the fact that $T\circ (g·x)=g·T\left(x\right)$. Hence, $T\circ (g·x)\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(X,Y)$ with ${\parallel T\circ (g·x)\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}=p_{\mathcal{B}}\left(g\right)∥T\left(x\right)∥$. Therefore, $g·x\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ with

$$\begin{array}{cc}\hfill {\parallel g·x\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}& =p_{\mathcal{B}}\left(g\right)sup\{\parallel T\left(x\right)\parallel :T\in {\mathcal{A}(X,Y),\parallel T\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & \ge p_{\mathcal{B}}\left(g\right)sup\left\{\right|x^{*}\left(x\right)|:x^{*}\in B_{X^{*}}\}=p_{\mathcal{B}}\left(g\right)\parallel x\parallel .\hfill \end{array}$$

To obtain the reciprocal inequality, see that

$${\parallel T\circ (g·x)\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {\parallel T\parallel \parallel g·x\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {\parallel T\parallel }_{\mathcal{A}}{\parallel g·x\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le p_{\mathcal{B}}\left(g\right)\parallel x\parallel$$

for all $T\in \mathcal{A}(X,Y)$ with ${\parallel T\parallel }_{\mathcal{A}}\le 1$, and then ${\parallel g·x\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}\le p_{\mathcal{B}}\left(g\right)\parallel x\parallel$.

(P3): Let Y be a complex Banach space, $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, $h\in \widehat{\mathcal{B}}(\mathbb{D},\mathbb{D})$ and $S\in \mathcal{L}(X,Y)$. Let $T\in \mathcal{A}(Y,Z)$, with Z being a complex Banach space. Then, $T\circ S\in \mathcal{A}(X,Z)$ and ${∥T\circ S∥}_{\mathcal{A}}\le {∥T∥}_{\mathcal{A}}∥S∥$ by the ideal property of $\mathcal{A}$. By the definitions of ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$ and ${∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$, we can ensure that $T\circ S\circ f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Z)$ with ${\parallel T\circ S\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {∥T\circ S∥}_{\mathcal{A}}{∥f∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$. Hence, $T\circ S\circ f\circ h\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Z)$ with ${\parallel T\circ S\circ f\circ h\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le {\parallel T\circ S\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$ by the ideal property of ${\mathcal{I}}^{\widehat{\mathcal{B}}}$. As a result, $S\circ f\circ h\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ and since ${\parallel T\circ S\circ f\circ h\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}\le ∥S∥{∥f∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$ for all $T\in \mathcal{A}(Y,Z)$ with ${\parallel T\parallel }_{\mathcal{A}}\le 1$, we deduce that ${\parallel S\circ f\circ h\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}\le ∥S∥{∥f∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$.

Consequently, $[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ becomes a (Banach) normed normalized Bloch ideal. □

In the following result, we show that normed normalized Bloch quotients inherit the surjectivity of the associated normed normalized Bloch ideal.

Theorem 3. 

Let $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ be a surjective normed normalized Bloch ideal and $[\mathcal{A},\parallel ·{\parallel }_{\mathcal{A}}]$ be a normed operator ideal. Then, $[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ is surjective.

Proof. 

Let $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ and assume that $f\circ \pi \in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, where and $\pi \in \widehat{\mathcal{B}}(\mathbb{D},\mathbb{D})$ is a metric surjection. Then, $T\circ f\circ \pi \in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ for all $T\in \mathcal{A}(X,Y)$, being Y a complex Banach space. Since the normed normalized Bloch ideal $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ is surjective and $T\circ f\in \widehat{\mathcal{B}}(\mathbb{D},Y)$, it follows that $T\circ f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ with ${\parallel T\circ f\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}={\parallel T\circ f\circ \pi \parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}$. By the arbitrariness of $T\in \mathcal{A}(X,Y)$, we can ensure that $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$. Moreover, notice that

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}& =sup\{\parallel T\circ f{\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & =sup\{\parallel T\circ f\circ \pi {\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & ={\parallel f\circ \pi \parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}.\hfill \end{array}$$

Hence, $[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ is surjective. □

The following theorem will allow us to establish a relationship between normalized Bloch quotients and left-hand quotients of operator ideals for those normalized Bloch ideals with the LP.

Theorem 4. 

Let $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$ be a Banach operator ideal, $[\mathcal{J},{∥·∥}_{\mathcal{J}}]$ be a normed operator ideal, and $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ be a normed normalized Bloch ideal with the LP in $\mathcal{J}$. For every $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$, we have $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ if and only if $S_f\in {\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$. In such a case, ${\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}={\parallel S_f\parallel }_{{\mathcal{A}}^{-1}\circ \mathcal{J}}$. Moreover, the correspondence function $f\mapsto S_f$ verifies that it is an isometric isomorphism from $({\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X),{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}})$ onto $({\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X),{∥·∥}_{{\mathcal{A}}^{-1}\circ \mathcal{J}})$.

Proof. 

(i) ⇒ (ii): Let $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$. Then, for all $T\in \mathcal{A}(X,Y)$ we have that $T\circ f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$. Using an analogous reasoning to that of the proof of Proposition 3, Theorem 1 allows us to guarantee the existence of two operators $S_f\in \mathcal{L}(\mathcal{G}\left(\mathbb{D}\right),X)$ and $S_{T\circ f}\in \mathcal{L}(\mathcal{G}\left(\mathbb{D}\right),Y)$ such that $S_{T\circ f}=T\circ S_f$. Since ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the LP in $\mathcal{J}$, we deduce that $S_{T\circ f}\in \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),Y)$ simply taking into account the ideal property of $\mathcal{J}$. The arbitrariness of $T\in \mathcal{A}(X,Y)$ leads us to conclude that $S_f\in {\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),Y)$.

(ii) ⇒ (i): Let us suppose that $S_f\in {\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$. Then, $T\circ S_f\in \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),Y)$ for all $T\in \mathcal{A}(X,Y)$, and thus $S_{T\circ f}\in \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),Y)$ due to $S_{T\circ f}=T\circ S_f$. Since ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the LP in $\mathcal{J}$, it follows that $T\circ f\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ and then $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$.

We now show the equality between the associated norms

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}& =sup\{\parallel T\circ f{\parallel }_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & =sup\{\parallel S_{T\circ f}{\parallel }_{\mathcal{J}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & =sup\{\parallel T\circ S_f{\parallel }_{\mathcal{J}}:T\in \mathcal{A}(X,Y),\parallel T{\parallel }_{\mathcal{A}}\le 1\}\hfill \\ \hfill & =\parallel S_f{\parallel }_{{\mathcal{A}}^{-1}\circ \mathcal{J}},\hfill \end{array}$$

where the second equality is because of the LP of ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ in $\mathcal{J}$.

Finally, to guarantee the map $f\mapsto S_f$ is an isometric isomorphism, it remains to prove the surjectivity of such a map from ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ into ${\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$. With such a goal in mind, let $T\in {\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$. We have $S\circ T\in \mathcal{J}\left(\mathcal{G}\right(\mathbb{D}),Y)$ for all $S\in \mathcal{A}(X,Y)$, with Y being a complex Banach space. Since $S\circ T$ is concretely a bounded linear operator, we could apply Theorem 1 to ensure that $S\circ T=S_g$ for some $g\in \widehat{\mathcal{B}}(\mathbb{D},Y)$. Hence, $g\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$ by the LP of ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ in $\mathcal{J}$. Let us define the mapping $f:\mathbb{D}\to X$ given by

$$f\left(z\right)={\int }_0^z(T\circ \Gamma )\left(s\right)ds(z\in \mathbb{D}).$$

Clearly, $f\in \mathcal{H}(\mathbb{D},X)$ with $f\left(0\right)=0$ and $f^{\prime }=T\circ \Gamma$. Moreover, note that

$${(1-|z|}^2)\parallel f^{\prime }{\left(z\right)\parallel =(1-|z|}^2)\parallel (T\circ \Gamma )\left(z\right)\parallel \le {(1-|z|}^2)\parallel T\parallel \parallel \Gamma \left(z\right)\parallel =\parallel T\parallel$$

for all $z\in \mathbb{D}$. Thus $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ and ${(S\circ f)}^{\prime }=S\circ f^{\prime }=S\circ T\circ \Gamma =S_g\circ \Gamma =g^{\prime }$. Now for all ${\varphi }^{*}\in X^{*}$, we have

$${({\varphi }^{*}\circ S\circ f)}^{\prime }={\varphi }^{*}\circ {(S\circ f)}^{\prime }={\varphi }^{*}\circ g^{\prime }={({\varphi }^{*}\circ g)}^{\prime }.$$

Then, ${\varphi }^{*}\circ S\circ f={\varphi }^{*}\circ g$ for all ${\varphi }^{*}\in X^{*}$ due to ${\varphi }^{*}\circ S\circ f,{\varphi }^{*}\circ g\in \mathcal{H}\left(\mathbb{D}\right)$ with $({\varphi }^{*}\circ S\circ f)\left(0\right)=({\varphi }^{*}\circ g)\left(0\right)=0$. Finally, taking into account that $X^{*}$ separates points of X, we obtain $S\circ f=g\in {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},Y)$. Hence, $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ and $S_f=T$. □

Let us now recall the composition method for generating normalized Bloch ideals which was introduced in [11]. A Bloch map $f:\mathbb{D}\to X$ with $f\left(0\right)=0$ is in the composition ideal $\mathcal{A}\circ \widehat{\mathcal{B}}$, with $\mathcal{A}$ being an operator ideal, and is denoted as $f\in \mathcal{A}\circ \widehat{\mathcal{B}}(\mathbb{D},X)$, if there are a complex Banach space Y, an operator $T\in \mathcal{A}(Y,X)$, and a map $g\in \widehat{\mathcal{B}}(\mathbb{D},Y)$ such that it is verified that $f=T\circ g$. Furthermore, if $\mathcal{A}$ is endowed with a norm ${∥·∥}_{\mathcal{A}}$ and $f\in \mathcal{A}\circ \widehat{\mathcal{B}}(\mathbb{D},X)$, we define

$${∥f∥}_{\mathcal{A}\circ \mathcal{B}}=inf\left\{{∥T∥}_{\mathcal{A}}{p}_{\mathcal{B}}\left(g\right)\right\},$$

with the infimum being taken over all factorizations of f as above.

As a final result of this section, we will prove the relationship between normalized Bloch left-hand quotients ideals and normalized Bloch ideals generated by composition, where it exists. More precisely, we show that every normalized Bloch left-hand quotient ideal where the associated normalized Bloch ideal has the LP can be seen as a composition ideal.

Proposition 4. 

Let $[\mathcal{A},{∥·∥}_{\mathcal{A}}]$ be a Banach operator ideal, $[\mathcal{J},{∥·∥}_{\mathcal{J}}]$ be a normed operator ideal, and $[{\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{I}}^{\widehat{\mathcal{B}}}}]$ be a normed normalized Bloch ideal with the LP in $\mathcal{J}$. Then,

$$[{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}},{∥·∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}]=[({\mathcal{A}}^{-1}\circ \mathcal{J})\circ \widehat{\mathcal{B}},{∥·∥}_{({\mathcal{A}}^{-1}\circ \mathcal{J})\circ \mathcal{B}}].$$

Proof. 

In order to prove this result, let $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$. Note that $f\in ({\mathcal{A}}^{-1}\circ \mathcal{J})\circ \widehat{\mathcal{B}}(\mathbb{D},X)$ if and only if $S_f\in {\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$ by Theorem 2.2 of [11]. At the same time, applying Theorem 4, we have $S_f\in {\mathcal{A}}^{-1}\circ \mathcal{J}(\mathcal{G}\left(\mathbb{D}\right),X)$ if and only if $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ due to the fact that ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ has the LP in $\mathcal{J}$. Furthermore, by the above theorems, we have

$${∥f∥}_{({\mathcal{A}}^{-1}\circ \mathcal{J})\circ \mathcal{B}}={∥S_f∥}_{{\mathcal{A}}^{-1}\circ \mathcal{J}}={∥f∥}_{{\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}}$$

for all $f\in {\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}(\mathbb{D},X)$. □

4. Examples of Normalized Bloch Left-Hand Quotient Ideals

In this very last section, we provide two examples of normalized Bloch left-hand quotient ideals generated by a linear operator ideal and a normalized Bloch ideal; these are the subclasses of Bloch maps with the Grothendieck range and Rosenthal range. In addition to this, we raise an open problem concerning normalized Bloch left-hand quotient ideals and normalized Bloch ideals of the composition type.

First, let us recall some important notions related to the theory of linear operators. We stand $\mathcal{K}(X,Y)$, $\mathcal{W}(X,Y)$, $\mathcal{S}(X,Y)$, $\mathcal{R}(X,Y)$, and $\mathfrak{G}(X,Y)$ for the spaces of compact operators, w-compact operators, separable bounded operators, Rosenthal operators, and Grothendieck operators from X into Y, respectively. Keep in mind that compactness, w-compactness, separability, Rosenthal, and Grothendieck are topological properties, and then an operator $T\in \mathcal{L}(X,Y)$ is said to belong to any of the above classes if $T\left(B_X\right)$ has the corresponding associated property. The following inclusions between the above-mentioned operator spaces are classical and could be found in the monograph [2] by Pietsch and [13]:

$$\begin{array}{cc}\hfill & \mathcal{K}(X,Y)\subseteq \mathcal{W}(X,Y)\subseteq \mathcal{R}(X,Y),\hfill \\ \hfill & \mathcal{W}(X,Y)\subseteq \mathfrak{G}(X,Y)\hfill \\ \hfill & \mathcal{K}(X,Y)\subseteq \mathcal{S}(X,Y).\hfill \end{array}$$

4.1. Grothendieck and Rosenthal Bloch Maps as Normalized Bloch Left-Hand Quotient Ideals

In the following, we introduce a nonlinear version of the Grothendieck operator concept in the Banach-valued Bloch setting.

Definition 3. 

We will say that a map $f\in \mathcal{H}(\mathbb{D},X)$ is a Grothendieck Bloch if

$${\mathrm{rang}}_{\mathcal{B}}\left(f\right)=\left\{\right(1-{\left|z\right|}^2)f^{\prime }\left(z\right):z\in \mathbb{D}\}$$

is a Grothendieck subset of X. Let ${\widehat{\mathcal{B}}}_{\mathfrak{G}}(\mathbb{D},X)$ denote the space of all zero-preserving Grothendieck Bloch maps from $\mathbb{D}$ into X.

According to [10], ${\widehat{\mathcal{B}}}_{\mathcal{K}}(\mathbb{D},X)$ and ${\widehat{\mathcal{B}}}_{\mathcal{W}}(\mathbb{D},X)$ represent the spaces of all zero-preserving Bloch maps from $\mathbb{D}$ into X with a relatively compact Bloch range and relatively w-compact Bloch range, respectively. By Theorems 5.4, 5.6 and Proposition 5.14 of [10], ${\widehat{\mathcal{B}}}_{\mathcal{K}}$ and ${\widehat{\mathcal{B}}}_{\mathcal{W}}$ are normed normalized Bloch ideals with the LP in $\mathcal{K}$ and $\mathcal{W}$, respectively. Next, we show that ${\widehat{\mathcal{B}}}_{\mathfrak{G}}$ has the LP in $\mathfrak{G}$.

Theorem 5. 

The following statements are equivalent for a Bloch map $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$:

(i) 

$f:\mathbb{D}\to X$ is a Grothendieck Bloch.

(ii) 

$S_f:\mathcal{G}\left(\mathbb{D}\right)\to X$ is a Grothendieck operator.

(iii) 

$f=T\circ g$, with Y being a complex Banach space, $g\in \widehat{\mathcal{B}}(\mathbb{D},Y)$ and $T\in \mathfrak{G}(Y,X)$.

In such a case, $p_{\mathcal{B}}\left(f\right)=∥S_f∥={∥f∥}_{\mathfrak{G}\circ \mathcal{B}}$ and the correspondence $f\mapsto S_f$ is an isometric isomorphism from $({\widehat{\mathcal{B}}}_{\mathfrak{G}}(\mathbb{D},X),p_{\mathcal{B}})$ onto $(\mathfrak{G}(\mathcal{G}\left(\mathbb{D}\right),X),∥·∥)$ and from $(\mathfrak{G}\circ \widehat{\mathcal{B}}(\mathbb{D},X),{∥·∥}_{\mathfrak{G}\circ \mathcal{B}})$ onto $(\mathfrak{G}(\mathcal{G}\left(\mathbb{D}\right),X),∥·∥)$.

Proof. 

(i) ⇒ (ii): If $f\in {\widehat{\mathcal{B}}}_{\mathfrak{G}}(\mathbb{D},X)$, then we have that

$${\mathrm{rang}}_{\mathcal{B}}\left(f\right)=S_f\left({\mathcal{M}}_{\mathcal{B}}\left(\mathbb{D}\right)\right)\subseteq \overline{\mathrm{abco}}\left({\mathrm{rang}}_{\mathcal{B}}\left(f\right)\right)$$

is Grothendieck in X. It is not difficult to see that the norm-closed absolutely convex hull of a Grothendieck set is itself Grothendieck since the norm-closed absolutely convex hull of a relatively w-compact set is relatively w-compact by Theorem 2.8.14 of [14]. Thus, $\overline{\mathrm{abco}}\left({\mathrm{rang}}_{\mathcal{B}}\left(f\right)\right)$ is Grothendieck in X. Do not lose sight of the fact that

$$S_f\left(B_{\mathcal{G}\left(\mathbb{D}\right)}\right)=S_f\left(\overline{\mathrm{abco}}\left({\mathcal{M}}_{\mathcal{B}}\left(\mathbb{D}\right)\right)\right)\subseteq \overline{\mathrm{abco}}\left(S_f\left({\mathcal{M}}_{\mathcal{B}}\left(\mathbb{D}\right)\right)\right),$$

and then $S_f\left(B_{\mathcal{G}\left(\mathbb{D}\right)}\right)$ is a Grothendieck subset of X. Hence, $S_f\in \mathfrak{G}(\mathcal{G}\left(\mathbb{D}\right),X)$ by Proposition 6.1.1 of [13].

(ii) ⇒ (i): Let us suppose that $S_f\in \mathfrak{G}(\mathcal{G}\left(\mathbb{D}\right),X)$. Then, $S_f\left(B_{\mathcal{G}\left(\mathbb{D}\right)}\right)$ is a Grothendieck subset of X. Since ${\mathrm{rang}}_{\mathcal{B}}\left(f\right)\subseteq S_f\left(B_{\mathcal{G}\left(\mathbb{D}\right)}\right)$, it follows that ${\mathrm{rang}}_{\mathcal{B}}\left(f\right)=S_f\left({\mathcal{M}}_{\mathcal{B}}\left(\mathbb{D}\right)\right)$ is Grothendieck in X by Lemma 1.3 of [15].

(ii) ⇔ (iii): A direct application of Theorem 2.2 of [11] gives us this equivalence.

Finally, the equality of the norms follows easily by using Theorem 1 and (ii) ⇒ (i), and the last statement could easily be verified from Theorem 2.2 of [11]. □

For our next purpose, we will need to make use of the following auxiliary result, which is an elementary consequence of Theorem 2.2 of [11].

Lemma 1. 

Let $\mathcal{A}$ be a closed operator ideal. Then, $\mathcal{A}\circ \widehat{\mathcal{B}}$ is a closed normalized Bloch ideal.

Proof. 

Let $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ and consider a sequence in $\mathcal{A}\circ \widehat{\mathcal{B}}(\mathbb{D},X)$, namely, $\left(f_n\right)$ so that $p_{\mathcal{B}}(f_n-f)\to 0$ as $n\to \infty$. A simple application of Theorem 2.2 of [11] allows us to guarantee that $S_{f_n}\in \mathcal{A}(\mathcal{G}\left(\mathbb{D}\right),X)$ with

$$\parallel S_{f_n}-S_f\parallel =\parallel S_{f_n-f}\parallel =p_{\mathcal{B}}(f_n-f)(n\in \mathbb{N}).$$

Hence, $S_f\in \mathcal{A}(\mathcal{G}\left(\mathbb{D}\right),X)$ and again by Theorem 2.2 of [11], we conclude that $f\in \mathcal{A}\circ \widehat{\mathcal{B}}(\mathbb{D},X)$. □

Let us not forget that $\mathfrak{G}$ is a closed operator ideal (the norm-limit of a convergent sequence of Grothendieck operators is Grothendieck) and ${\widehat{\mathcal{B}}}_{\mathfrak{G}}=\mathfrak{G}\circ \widehat{\mathcal{B}}$ by Theorem 5. Thus, Lemma 4.1 yields the following straightforward result.

Corollary 2. 

$[{\widehat{\mathcal{B}}}_{\mathfrak{G}},p_{\mathcal{B}}]$ is a closed normalized Bloch ideal. □

In this way, we can characterize Grothendieck Bloch maps through a normalized Bloch left-hand quotient ideal as follows.

Theorem 6. 

$[{\widehat{\mathcal{B}}}_{\mathfrak{G}},p_{\mathcal{B}}]=[{\mathcal{S}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{W}},{∥·∥}_{{\mathcal{S}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{W}}}]$.

Proof. 

Let $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$. Just using Theorem 5, [2] (3.2.6), Definition 2, Theorem 4, and Theorem 5.6 of [10], respectively, we obtain

$$\begin{array}{cc}\hfill f\in {\widehat{\mathcal{B}}}_{\mathfrak{G}}(\mathbb{D},X)& \iff S_f\in \mathfrak{G}(\mathcal{G}\left(\mathbb{D}\right),X)\hfill \\ \hfill & \iff T\circ S_f\in \mathcal{W}(\mathcal{G}\left(\mathbb{D}\right),Y)(T\in \mathcal{S}(X,Y))\hfill \\ \hfill & \iff S_f\in {\mathcal{S}}^{-1}\circ \mathcal{W}(\mathcal{G}\left(\mathbb{D}\right),X)\hfill \\ \hfill & \iff f\in {\mathcal{S}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{W}}(\mathbb{D},X).\hfill \end{array}$$

Furthermore, notice that $p_{\mathcal{B}}\left(f\right)=∥S_f∥={∥S_f∥}_{{\mathcal{S}}^{-1}\circ \mathcal{W}}={∥f∥}_{{\mathcal{S}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{W}}}$. □

Let us take a look back and recall that a subset R of X is said to be conditionally w-compact (i.e., Rosenthal) if every sequence in R has a weak Cauchy subsequence. An operator $T\in \mathcal{L}(X,Y)$ will be called a Rosenthal operator if it sends the closed unit ball of X to a Rosenthal subset of Y. By [16], it is well known that Rosenthal operators could be characterized as those that admit a factorization by means of a Banach space that does not contain an isomorphic copy of ${\ell }_1$.

Moreover, an operator $T\in \mathcal{L}(X,Y)$ is said to be completely continuous if every w-convergent sequence $\left(x_n\right)$ is mapped into a norm convergent sequence $\left(T\left(x_n\right)\right)$. Let $\mathcal{V}(X,Y)$ denote the space of all completely continuous operators from X into Y. By 1.6.2 and 4.2.5 of [2], $\mathcal{V}$ is a closed operator ideal.

Next, we characterize the ideal of Bloch mappings having a Rosenthal range, denoted by ${\widehat{\mathcal{B}}}_{\mathcal{R}}$ and introduced in [11], as a normalized Bloch left-hand quotient ideal generated by the linear operator ideal $\mathcal{V}$ and the normalized Bloch ideal ${\widehat{\mathcal{B}}}_{\mathcal{K}}$.

Theorem 7. 

$[{\widehat{\mathcal{B}}}_{\mathcal{R}},p_{\mathcal{B}}]=[{\mathcal{V}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{K}},{∥·∥}_{{\mathcal{V}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{K}}}]$.

Proof. 

Given $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ and applying Theorem 3.3 of [11], 3.2.4 of [2], Definition 2, and Theorem 4 with Theorem 5.4 of [10], respectively, we obtain

$$\begin{array}{cc}\hfill f\in {\widehat{\mathcal{B}}}_{\mathcal{R}}(\mathbb{D},X)& \iff S_f\in \mathcal{R}(\mathcal{G}\left(\mathbb{D}\right),X)\hfill \\ \hfill & \iff T\circ S_f\in \mathcal{K}(\mathcal{G}\left(\mathbb{D}\right),Y),(T\in \mathcal{V}(X,Y))\hfill \\ \hfill & \iff S_f\in {\mathcal{V}}^{-1}\circ \mathcal{K}(\mathcal{G}\left(\mathbb{D}\right),X)\hfill \\ \hfill & \iff f\in {\mathcal{V}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{K}}(\mathbb{D},X).\hfill \end{array}$$

In fact, $p_{\mathcal{B}}\left(f\right)=∥S_f∥={∥S_f∥}_{{\mathcal{V}}^{-1}\circ \mathcal{K}}={∥f∥}_{{\mathcal{V}}^{-1}\circ {\widehat{\mathcal{B}}}_{\mathcal{K}}}$. □

4.2. An Open Problem Related to p-Summing Bloch Mappings

To close this paper, we present an open problem. We do not know what happens with Proposition 4 for the case in which the ideal of normalized Bloch maps ${\mathcal{I}}^{\widehat{\mathcal{B}}}$ does not have the LP in an operator ideal $\mathcal{J}$. The natural idea is to think that a normalized Bloch quotient ${\mathcal{A}}^{-1}\circ {\mathcal{I}}^{\widehat{\mathcal{B}}}$ may not coincide with a composition ideal of the form $({\mathcal{A}}^{-1}\circ \mathcal{J})\circ \widehat{\mathcal{B}}$.

Let $1\le p<\infty$. Let us recall that a bounded linear operator $T:X\to Y$ is p-summing if there is a constant $k\ge 0$ such that, for any $n\in \mathbb{N}$ and ${\left\{x_i\right\}}_{i=1}^n$ in X, it is satisfied that

$${\left(\sum _{i=1}^n{\parallel T\left(x_i\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}\le k\underset{x^{*}\in B_{X^{*}}}{sup}{\left(\sum _{i=1}^n{\left|x^{*}\left(x_i\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

The space of all p-summing linear operators from X into Y, denoted ${\Pi }_p(X,Y)$, is a Banach space endowed with the norm

$${\pi }_p\left(T\right)=inf\left\{k:{\left(\sum _{i=1}^n{\parallel T\left(x_i\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}\le k\underset{x^{*}\in B_{X^{*}}}{sup}{\left(\sum _{i=1}^n{\left|x^{*}\left(x_i\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}\right\}.$$

In the same way (see [17]), a holomorphic function $f:\mathbb{D}\to X$ with $f\left(0\right)=0$ is said to be p-summing Bloch if there exists a constant $k_{\mathcal{B}}\ge 0$ such that

$${\left(\sum _{j=1}^m|{\eta }_j{|}^p{\parallel f^{\prime }\left(z_j\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}\le k_{\mathcal{B}}\underset{h\in B_{\widehat{\mathcal{B}}\left(\mathbb{D}\right)}}{sup}{\left(\sum _{j=1}^m|{\eta }_j{|}^p{\left|h^{\prime }\left(z_j\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}},$$

for any $m\in \mathbb{N}$ and any finite sets ${\left\{{\eta }_j\right\}}_{j=1}^m$ in $\mathbb{C}$ and ${\left\{z_j\right\}}_{j=1}^m$ in $\mathbb{D}$. We denote this space as ${\Pi }_p^{\widehat{\mathcal{B}}}(\mathbb{D},X)$, and it becomes Banach endowed with the norm

$${\pi }_p^{\mathcal{B}}\left(f\right)=inf\left\{k_{\mathcal{B}}:{\left(\sum _{j=1}^m|{\eta }_j{|}^p{\parallel f^{\prime }\left(z_j\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}\le k_{\mathcal{B}}\underset{h\in B_{\widehat{\mathcal{B}}\left(\mathbb{D}\right)}}{sup}{\left(\sum _{j=1}^m|{\eta }_j{|}^p{\left|h^{\prime }\left(z_j\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}\right\}.$$

For $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$ and $z\in \mathbb{D}$, it is clear that $\parallel f^{\prime }\left(z\right)\parallel =\parallel {\gamma }_z\left(f\right)\parallel =\parallel S_f\left({\gamma }_z\right)\parallel$. Let us assume that $S_f\in {\Pi }_p(\mathcal{G}\left(\mathbb{D}\right),X)$. Then, for any $m\in \mathbb{N}$ and ${\left\{{\gamma }_{z_j}\right\}}_{j=1}^m$ in $\mathcal{G}\left(\mathbb{D}\right)$, we have

$${\left(\sum _{j=1}^m{\parallel S_f\left({\gamma }_{z_j}\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}\le {\pi }_p\left(S_f\right)\underset{{\varphi }^{*}\in B_{\mathcal{G}{\left(\mathbb{D}\right)}^{*}}}{sup}{\left(\sum _{j=1}^m{\left|{\varphi }^{*}\left({\gamma }_{z_j}\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

It follows from the above comment that

$$\begin{array}{cc}\hfill {\left(\sum _{j=1}^m{\parallel f^{\prime }\left(z_j\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}& \le {\pi }_p\left(S_f\right)\underset{{\varphi }^{*}\in B_{\mathcal{G}{\left(\mathbb{D}\right)}^{*}}}{sup}{\left(\sum _{j=1}^m{\left|{\varphi }^{*}\left({\gamma }_{z_j}\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={\pi }_p\left(S_f\right)\underset{h\in B_{\widehat{\mathcal{B}}\left(\mathbb{D}\right)}}{sup}{\left(\sum _{j=1}^m{\left|h^{\prime }\left(z_j\right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

Hence, $f\in {\Pi }_p^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ with ${\pi }_p^{\mathcal{B}}\left(f\right)\le {\pi }_p\left(S_f\right)$ and we have shown the following.

Proposition 5. 

Let $1\le p<\infty$ and $f\in \widehat{\mathcal{B}}(\mathbb{D},X)$. If $S_f\in {\Pi }_p(\mathcal{G}\left(\mathbb{D}\right),X)$, then $f\in {\Pi }_p^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ and ${\pi }_p^{\mathcal{B}}\left(f\right)\le {\pi }_p\left(S_f\right)$.

Everything leads us to believe that the reciprocal implication in the above result is not true. However, finding an explicit counterexample is not an easy task. One could think of using the same reasoning as in the Lipschitz setting (see Remarks 2.8 of [18] and Remark 3.3 of [19]), but in the Bloch context, the map $\Gamma :\mathbb{D}\to \mathcal{G}\left(\mathbb{D}\right)$ is not Bloch. Assuming that we find a function $g\in {\Pi }_p^{\widehat{\mathcal{B}}}(\mathbb{D},X)$ such that $S_g\notin {\Pi }_p(\mathcal{G}\left(\mathbb{D}\right),X)$, we would have shown that ${\Pi }_p^{\widehat{\mathcal{B}}}$ does not have LP in ${\Pi }_p$ and, moreover, by Theorem 2.2 of [11], we follow ${\Pi }_p^{\widehat{\mathcal{B}}}\ne {\Pi }_p\circ \widehat{\mathcal{B}}$. Apparently, whatever the operator ideal $\mathcal{A}$ is, then ${\mathcal{A}}^{-1}\circ {\Pi }_p^{\widehat{\mathcal{B}}}$ cannot be a composition ideal generated by the operator ideal ${\Pi }_p$, that is, ${\mathcal{A}}^{-1}\circ {\Pi }_p^{\widehat{\mathcal{B}}}\ne ({\mathcal{A}}^{-1}\circ {\Pi }_p)\circ \widehat{\mathcal{B}}$. Nevertheless, ${\mathcal{A}}^{-1}\circ {\Pi }_p^{\widehat{\mathcal{B}}}$ is a normalized Bloch left-hand quotient ideal since ${\Pi }_p^{\widehat{\mathcal{B}}}$ is a normalized Bloch ideal by Proposition 1.2 of [17].