# Holomorphic mappings with relatively $p$-compact range

A. Jiménez-Vargas  Departamento de Matemáticas, Universidad de Almería, 04120,
Almería, Spain<EMAIL_ADDRESS>
###### Abstract.

Related to the concept of $p$-compact operator with $p\in[1,\infty]$
introduced by Sinha and Karn [17], this paper deals with the space
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ of all Banach-valued holomorphic
mappings on an open subset $U$ of a complex Banach space whose ranges are
relatively $p$-compact sets. We show that this space is a Banach ideal
generated by composition with the ideal of $p$-compact operators. For such
holomorphic mappings, some results on both linearisation from the canonical
predual of $\mathcal{H}^{\infty}(U)$ and factorization through a quotient
space of $\ell_{p^{*}}$ are stated. Further, we characterize such bounded
holomorphic mappings as those whose transposes are quasi $p$-nuclear operators
(respectively, factor through a subspace of $\ell_{p}$). We also prove that
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ is the smallest surjective
bounded-holomorphic ideal which contains the space of all $p$-nuclear
holomorphic mappings from $U$ into $F$.

###### Key words and phrases:

Vector-valued holomorphic mapping, $p$-compact set, $p$-compact operator,
locally $p$-compact holomorphic mapping.

###### 2020 Mathematics Subject Classification:

47B07, 47B10, 46E15, 46E40

## Introduction

Inspired by classical Grothendieck’s characterization of a relatively compact
set of a Banach space as a subset of the convex hull of a norm null sequence
of vectors [9], Sinha and Karn [17] introduced and studied $p$-compact sets
and $p$-compact operators for $p\in[1,\infty]$.

Let $E$ be a complex Banach space with closed unit ball $B_{E}$. Let
$p\in(1,\infty)$ and let $p^{*}$ denote the conjugate index of $p$ given by
$p^{*}=p/(p-1)$.

As usual, given $p\in[1,\infty)$, $\ell_{p}(E)$ denotes the Banach space of
all absolutely $p$-summable sequences $(x_{n})$ in $E$, equipped with the norm

$\left\|(x_{n})\right\|_{p}=\left(\sum_{n=1}^{\infty}\left\|x_{n}\right\|^{p}\right)^{1/p},$

and $c_{0}(E)$ the Banach space of all norm null sequences $(x_{n})$ in $E$,
endowed with the norm

$\left\|(x_{n})\right\|_{\infty}=\sup\left\\{\left\|x_{n}\right\|\colon
n\in\mathbb{N}\right\\}.$

In the case $E=\mathbb{C}$, we will simply write $\ell_{p}$ and $c_{0}$.

For $p\in(1,\infty)$, the $p$-convex hull of a sequence
$(x_{n})\in\ell_{p}(E)$ is defined by

$p\text{-}\mathrm{conv}(x_{n})=\left\\{\sum_{n=1}^{\infty}a_{n}x_{n}\colon(a_{n})\in
B_{\ell_{p^{*}}}\right\\}.$

Similarly, set

$\displaystyle 1\text{-}\mathrm{conv}(x_{n})$
$\displaystyle=\left\\{\sum_{n=1}^{\infty}a_{n}x_{n}\colon(a_{n})\in
B_{c_{0}}\right\\}\qquad\left((x_{n})\in\ell_{1}(E)\right),$
$\displaystyle\infty\text{-}\mathrm{conv}(x_{n})$
$\displaystyle=\left\\{\sum_{n=1}^{\infty}a_{n}x_{n}\colon(a_{n})\in
B_{\ell_{1}}\right\\}\qquad\left((x_{n})\in c_{0}(E)\right).$

Note that
$\infty\text{-}\mathrm{conv}(x_{n})=\overline{\mathrm{abco}}(\\{x_{n}\colon
n\in\mathbb{N}\\})$ which is the norm-closed absolutely convex hull of the set
$\\{x_{n}\colon n\in\mathbb{N}\\}$ in $E$.

Following [17], given $p\in[1,\infty]$, a set $K\subseteq E$ is said to be
relatively $p$-compact if there is a sequence $(x_{n})\in\ell_{p}(E)$
($(x_{n})\in c_{0}(E)$ if $p=\infty$) such that $K\subseteq
p\text{-}\mathrm{conv}(x_{n})$. Such a sequence is not unique but Delgado,
Piñeiro and Serrano [7, p. 297] (see also [12] by Lassalle and Turco) provided
a measure of the size of the $p$-compact set $K$, defining

$m_{p}(K,E)=\begin{cases}\inf\\{\left\|(x_{n})\right\|_{p}\colon(x_{n})\in\ell_{p}(E),\,K\subseteq
p\text{-}\mathrm{conv}(x_{n})\\}&\text{if }1\leq p<\infty,\\\
\inf\\{\left\|(x_{n})\right\|_{p}\colon(x_{n})\in c_{0}(E),\,K\subseteq
p\text{-}\mathrm{conv}(x_{n})\\}&\text{if }p=\infty,\end{cases}$

and $m_{p}(K,E)=\infty$ if $K$ is not $p$-compact. We frequently will write
$m_{p}(K)$ in place of $m_{p}(K,E)$.

A linear operator $T\colon E\to F$ is said to be $p$-compact if $T(B_{E})$ is
a relatively $p$-compact subset of $F$. The space of all $p$-compact linear
operators from $E$ to $F$ is denoted by $\mathcal{K}_{p}(E,F)$ and it is a
Banach operator ideal endowed with the norm $k_{p}(T)=m_{p}(T(B_{E}))$ (see
[17, Theorem 4.2] and [7, Proposition 3.15]).

From a Grothendieck’s result [9], a set $K\subseteq E$ is relatively compact
if and only if for every $\varepsilon>0$, there is a sequence $(x_{n})\in
c_{0}(E)$ with $\left\|(x_{n})\right\|_{\infty}\leq\sup_{x\in
K}\left\|x\right\|+\varepsilon$ such that
$K\subseteq\infty\text{-}\mathrm{conv}(x_{n})$. Hence we can consider compact
sets as $\infty$-compact sets. In this form, compact operators can be viewed
as $\infty$-compact operators and $k_{\infty}$ is the usual operator norm.

The work of Sinha and Karn [17] provided motivation for many papers on
$p$-compactness in operator spaces. We can cite [1, 5, 6, 7, 8, 12, 16].

The extension of the theory of $p$-compact operators to the polynomial and
holomorphic settings was addressed by Aron, Caliskan, García and Maestre [2]
and Aron, Maestre and Rueda [3]. In these settings, the property of
$p$-compactness was dealed from the following local point of view: a mapping
$f\colon U\to F$ is said to be locally $p$-compact if every point $x\in U$ has
a neighborhood $U_{x}\subseteq U$ such that $f(U_{x})$ is relatively
$p$-compact in $F$.

The aim of this note is to study a subclass of locally $p$-compact holomorphic
mappings, namely, holomorphic mappings with relatively $p$-compact range.
Notice that every such mapping is locally $p$-compact but the converse is not
true. For instance, if $\mathbb{D}$ denotes the open complex unit disc, the
holomorphic mapping $f\colon\mathbb{D}\to c_{0}$ defined by
$f(z)=\sum_{n=1}^{\infty}z^{n}e_{n}$, where $(e_{n})$ is the canonical basis
of $\ell_{1}$, is locally $1$-compact but it has not relatively compact range
(see [13, Example 3.2]) and, consequently, neither relatively $p$-compact
range for any $p\geq 1$.

Our motivation to deal with this class of mappings also arises from the study
(initiated by Mujica [13] and completed in [11]) on the Banach space
$\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$ formed by all holomorphic mappings
from $U$ to $F$ with relatively compact range, equipped with the supremum
norm.

We now briefly describe the content of this paper. Let $E$ and $F$ be complex
Banach spaces, $U$ an open subset of $E$ and $p\in[1,\infty]$. Let
$\mathcal{H}^{\infty}(U,F)$ denote the Banach space of all bounded holomorphic
mappings from $U$ into $F$, endowed with the supremum norm. In particular,
$\mathcal{H}^{\infty}(U)$ stands for $\mathcal{H}^{\infty}(U,\mathbb{C})$. In
[13], Mujica provided a linearization theorem of the members of
$\mathcal{H}^{\infty}(U,F)$, which will be an essential tool in our analysis
of the subject.

We prove that the space of all holomorphic mappings $f\colon U\to F$ with
relatively $p$-compact range, denoted
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ and equipped with the natural
norm $k^{\mathcal{H}^{\infty}}_{p}(f)=m_{p}(f(U))$, is a surjective Banach
ideal which becomes regular whenever $F$ is reflexive. Furthermore, we show
that $\mathcal{H}^{\infty}_{\mathcal{K}_{\infty}}(U,F)$ coincides with
$\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$, and the norm
$k^{\mathcal{H}^{\infty}}_{\infty}$ with the supremum norm.

If $\mathcal{G}^{\infty}(U)$ is the canonical predual of
$\mathcal{H}^{\infty}(U)$ obtained by Mujica [13] via an identification
denoted $g_{U}$, we establish that a bounded holomorphic mapping $f\colon U\to
F$ has relatively $p$-compact range if and only if Mujica’s linearization
$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ is a $p$-compact operator.

This fact has some interesting applications. For example, we show that the
ideal $\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ is generated by
composition with the ideal $\mathcal{K}_{p}$ of $p$-compact operators. This
means that each mapping $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$
admits a factorization $f=T\circ g$, where $G$ is a complex Banach space,
$g\in\mathcal{H}^{\infty}(U,G)$ and $T\in\mathcal{K}_{p}(G,F)$. Moreover,
$k^{\mathcal{H}^{\infty}}_{p}(f)$ coincides with
$\inf\\{k_{p}(T)\left\|g\right\|_{\infty}\\}$, where the infimum is extended
over all such factorizations of $f$ and, curiously, this infimum is attained
at the factorization $f=T_{f}\circ g_{U}$ due to Mujica [13].

Among other results, we obtain some inclusion relations of
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}$-spaces, factorize their members
through a quotient space of $\ell_{p^{*}}$, characterize holomorphic mappings
with relatively $p$-compact range as those bounded holomorphic mappings whose
transposes are quasi $p$-nuclear operators (respectively, factor through a
subspace of $\ell_{p}$), and show that
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ is the smallest surjective
bounded-holomorphic ideal which contains the space of all $p$-nuclear
holomorphic mappings from $U$ into $F$.

##  1\. The results

From now on, unless otherwise stated, $E$ and $F$ will denote complex Banach
spaces, $U$ an open subset of $E$ and $p\in[1,\infty]$.

This section is devoted to the study of holomorphic mappings $f\colon U\to F$
so that $f(U)$ is a relatively $p$-compact subset of $F$. We denote by
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ the set formed by all these
mappings and, for each $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$,
define $k_{p}^{\mathcal{H}^{\infty}}(f)=m_{p}(f(U))$.

By [11, Corollary 2.11], the space of all holomorphic mappings with relatively
compact range from $U$ to $F$, denoted
$\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$, is a Banach space equipped with the
supremum norm.

On account of the following result, we will only study in this paper the case
$1\leq p<\infty$.

######  Proposition 1.1.

$\mathcal{H}^{\infty}_{\mathcal{K}_{\infty}}(U,F)=\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$
and $k_{\infty}^{\mathcal{H}^{\infty}}(f)=\left\|f\right\|_{\infty}$ for all
$\mathcal{H}^{\infty}_{\mathcal{K}_{\infty}}(U,F)$.

###### Proof.

Let $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{\infty}}(U,F)$ and let $(y_{n})\in
c_{0}(F)$ be such that $f(U)\subseteq\infty\text{-}\mathrm{conv}(y_{n})$.
Since $\infty\text{-}\mathrm{conv}(y_{n})$ is relatively compact in $F$, it
follows that $f\in\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$ with
$\left\|f\right\|_{\infty}\leq\left\|(y_{n})\right\|_{\infty}$, and taking
infimum over all such sequences $(y_{n})$, we have
$\left\|f\right\|_{\infty}\leq k_{\infty}^{\mathcal{H}^{\infty}}(f)$.

Conversely, let $f\in\mathcal{H}^{\infty}_{\mathcal{K}}(U,F)$. By classical
Grothendieck’s result, for every $\varepsilon>0$, there exists $(y_{n})\in
c_{0}(F)$ with
$\left\|y_{n}\right\|_{\infty}\leq\left\|f\right\|_{\infty}+\varepsilon$ such
that $f(U)\subseteq\infty\text{-}\mathrm{conv}(y_{n})$. Hence
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{\infty}}(U,F)$ and
$k_{\infty}^{\mathcal{H}^{\infty}}(f)=\left\|f\right\|_{\infty}$. ∎

As usual, $\mathcal{L}(E,F)$ denotes the Banach space of all bounded linear
operators from $E$ to $F$ endowed with the operator canonical norm, and
$E^{*}$ stands for the dual space of $E$. The subspaces of $\mathcal{L}(E,F)$
consisting of all compact operators and finite-rank bounded operators from $E$
to $F$ will be denoted by $\mathcal{K}(E,F)$ and $\mathcal{F}(E,F)$,
respectively. The symbol $\kappa_{F}$ represents the isometric linear
embedding from $F$ into $F^{**}$.

Inspired by the notion of operator ideal between Banach spaces [15], the
following type of ideals was considered in [4].

An ideal of bounded holomorphic mappings (or simply, a bounded-holomorphic
ideal) is a subclass $\mathcal{I}^{\mathcal{H}^{\infty}}$ of
$\mathcal{H}^{\infty}$ such that for each complex Banach space $E$, each open
subset $U$ of $E$ and each complex Banach space $F$, the components

$\mathcal{I}^{\mathcal{H}^{\infty}}(U,F):=\mathcal{I}^{\mathcal{H}^{\infty}}\cap\mathcal{H}^{\infty}(U,F)$

satisfy the following three conditions:

  1. (I1)

$\mathcal{I}^{\mathcal{H}^{\infty}}(U,F)$ is a linear subspace of
$\mathcal{H}^{\infty}(U,F)$,

  2. (I2)

For any $g\in\mathcal{H}^{\infty}(U)$ and $y\in F$, the mapping $g\cdot
y\colon x\mapsto g(x)y$ from $U$ to $F$ is in
$\mathcal{I}^{\mathcal{H}^{\infty}}(U,F)$,

  3. (I3)

The ideal property: if $H,G$ are complex Banach spaces, $V$ is an open subset
of $H$, $h\in\mathcal{H}(V,U)$, $f\in\mathcal{I}^{\mathcal{H}^{\infty}}(U,F)$
and $S\in\mathcal{L}(F,G)$, then $S\circ f\circ
h\in\mathcal{I}^{\mathcal{H}^{\infty}}(V,G)$.

Suppose that a function
$\left\|\cdot\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}\colon\mathcal{I}^{\mathcal{H}^{\infty}}\to\mathbb{R}_{0}^{+}$
satisfies the following three properties:

  1. (N1)

$(\mathcal{I}^{\mathcal{H}^{\infty}}(U,F),\left\|\cdot\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}})$
is a normed (Banach) space with
$\left\|f\right\|_{\infty}\leq\left\|f\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}$
for all $f\in\mathcal{I}^{\mathcal{H}^{\infty}}(U,F)$,

  2. (N2)

$\left\|g\cdot
y\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}=\left\|g\right\|_{\infty}\left\|y\right\|$
for every $g\in\mathcal{H}^{\infty}(U)$ and $y\in F$,

  3. (N3)

If $H,G$ are complex Banach spaces, $V$ is an open subset of $H$,
$h\in\mathcal{H}(V,U)$, $f\in\mathcal{I}^{\mathcal{H}^{\infty}}(U,F)$ and
$S\in\mathcal{L}(F,G)$, then $\left\|S\circ f\circ
h\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}\leq\left\|S\right\|\left\|f\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}$.

Then
$(\mathcal{I}^{\mathcal{H}^{\infty}},\left\|\cdot\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}})$
is called a normed (Banach) bounded-holomorphic ideal.

A normed bounded-holomorphic ideal $\mathcal{I}^{\mathcal{H}^{\infty}}$ is
said to be:

  1. (R)

regular if for any $f\in\mathcal{H}^{\infty}(U,F)$, we have that
$f\in\mathcal{I}^{\mathcal{H}^{\infty}}(U,F)$ with
$\left\|f\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}=\left\|\kappa_{F}\circ
f\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}$ whenever $\kappa_{F}\circ
f\in\mathcal{I}^{\mathcal{H}^{\infty}}(U,F^{**})$.

  2. (S)

surjective if for any mapping $f\in\mathcal{H}^{\infty}(U,F)$, any open subset
$V$ of a complex Banach space $G$ and any surjective mapping
$\pi\in\mathcal{H}(V,U)$, we have that
$f\in\mathcal{I}^{\mathcal{H}^{\infty}}(U,F)$ with
$\left\|f\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}=\left\|f\circ\pi\right\|_{\mathcal{I}^{\mathcal{H}^{\infty}}}$
whenever $f\circ\pi\in\mathcal{I}^{\mathcal{H}^{\infty}}(V,F)$.

The following result gathers the properties of the ideal
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$.

######  Theorem 1.2.

For each $p\in[1,\infty)$,
$(\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F),k_{p}^{\mathcal{H}^{\infty}})$
is a surjective Banach bounded-holomorphic ideal. Besides, it is regular
whenever $F$ is reflexive.

###### Proof.

We will check that
$(\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F),k_{p}^{\mathcal{H}^{\infty}})$
satisfies the required conditions.

(N1) We first see that
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)\subseteq\mathcal{H}^{\infty}(U,F)$
with $\left\|f\right\|_{\infty}\leq k_{p}^{\mathcal{H}^{\infty}}(f)$ for all
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$. Indeed, let
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ and let $(y_{n})$ be in
$\ell_{p}(F)$ (in $c_{0}(F)$ if $p=1$) such that $f(U)\subseteq
p\text{-}\mathrm{conv}(y_{n})$. Clearly,
$\left\|f(x)\right\|\leq\left\|(y_{n})\right\|_{p}$ for all $x\in U$, and thus
$f\in\mathcal{H}^{\infty}(U,F)$ with
$\left\|f\right\|_{\infty}\leq\left\|(y_{n})\right\|_{p}$. Extending this
infimum over all such sequences $(y_{n})$, we deduce that
$\left\|f\right\|_{\infty}\leq k_{p}^{\mathcal{H}^{\infty}}(f)$.

We now show that
$(\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F),k_{p}^{\mathcal{H}^{\infty}})$
is a normed space. Plainly, $k_{p}^{\mathcal{H}^{\infty}}(f)\geq 0$ for all
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$. Let
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ be so that
$k_{p}^{\mathcal{H}^{\infty}}(f)=0$. Then $\left\|f\right\|_{\infty}\leq
k_{p}^{\mathcal{H}^{\infty}}(f)$ implies that $f=0$.

Let $f_{1},f_{2}\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ and
$\lambda_{1},\lambda_{2}\in\mathbb{C}$. Then, for $i=1,2$, there is a sequence
$(y^{(i)}_{n})$ in $\ell_{p}(F)$ (in $c_{0}(F)$ if $p=1$) such that
$f_{i}(U)\subseteq p\text{-}\mathrm{conv}(y^{(i)}_{n})$. Clearly, the sequence
$(y_{n})$, defined by

$y_{2n-1}=\lambda_{1}y^{(1)}_{2n-1},\quad
y_{2n}=\lambda_{2}y^{(2)}_{2n}\qquad(n\in\mathbb{N}),$

is in $\ell_{p}(F)$ (in $c_{0}(F)$ if $p=1$) and satisfies that
$(\lambda_{1}f_{1}+\lambda_{2}f_{2})(U)\subseteq
p\text{-}\mathrm{conv}(y_{n})$. Hence
$\lambda_{1}f_{1}+\lambda_{2}f_{2}\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$
with
$k_{p}^{\mathcal{H}^{\infty}}(\lambda_{1}f_{1}+\lambda_{2}f_{2})\leq\left\|y_{n}\right\|_{p}$.
Since
$\left\|y_{n}\right\|_{p}\leq\lambda_{1}||y^{(1)}_{n}||_{p}+\lambda_{2}||y^{(2)}_{n}||_{p}$,
it easily follows that
$k_{p}^{\mathcal{H}^{\infty}}(\lambda_{1}f_{1}+\lambda_{2}f_{2})\leq\left|\lambda_{1}\right|k_{p}^{\mathcal{H}^{\infty}}(f_{1})+\left|\lambda_{2}\right|k_{p}^{\mathcal{H}^{\infty}}(f_{2})$.

To show that the norm $k_{p}^{\mathcal{H}^{\infty}}$ is complete, we will
prove that if $(f_{n})$ is a sequence in
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ such that
$\sum_{n=1}^{\infty}k_{p}^{\mathcal{H}^{\infty}}(f_{n})<\infty$, then
$\sum_{n=1}^{\infty}f_{n}<\infty$ in
$(\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F),k_{p}^{\mathcal{H}^{\infty}})$.
Since $\left\|f_{n}\right\|_{\infty}\leq k_{p}^{\mathcal{H}^{\infty}}(f_{n})$
for all $n\in\mathbb{N}$ and
$(\mathcal{H}^{\infty}(U,F),\left\|\cdot\right\|_{\infty})$ is a Banach space,
there exists $f\in\mathcal{H}^{\infty}(U,F)$ such that
$\left\|\sum_{k=1}^{n}f_{k}-f\right\|_{\infty}\to 0$ as $n\to\infty$. We claim
that $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ and
$k_{p}^{\mathcal{H}^{\infty}}(f)\leq\sum_{n=1}^{\infty}k_{p}^{\mathcal{H}^{\infty}}(f_{n})$.
Indeed, since $(f_{n}(U))$ is a sequence of relatively $p$-compact sets in $F$
such that
$\sum_{n=1}^{\infty}m_{p}(f_{n}(U))=\sum_{n=1}^{\infty}k_{p}^{\mathcal{H}^{\infty}}(f_{n})<\infty$,
Lemma 3.1 in [12] guarantees that $\sum_{n=1}^{\infty}||f_{n}(x)||<\infty$ for
any choice of points $x\in U$ and the set

$K=\left\\{\sum_{n=1}^{\infty}f_{n}(x)\colon x\in U\right\\}$

is relatively $p$-compact in $F$ with
$m_{p}(K)\leq\sum_{n=1}^{\infty}m_{p}(f_{n}(U))$. Clearly, $K=f(U)$ and this
proves our claim. The above proof can be applied to show that for every
$n\in\mathbb{N}$,
$\sum_{k=n+1}^{\infty}f_{k}\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$
with
$k^{\mathcal{H}^{\infty}}_{p}\left(\sum_{k=n+1}^{\infty}f_{k}\right)\leq\sum_{k=n+1}^{\infty}k^{\mathcal{H}^{\infty}}_{p}(f_{k})$.
Hence

$k^{\mathcal{H}^{\infty}}_{p}\left(f-\sum_{k=1}^{n}f_{k}\right)\leq\sum_{k=n+1}^{\infty}k^{\mathcal{H}^{\infty}}_{p}(f_{k})$

for every $n\in\mathbb{N}$, and thus
$k^{\mathcal{H}^{\infty}}_{p}(f-\sum_{k=1}^{n}f_{k})\to 0$ as $n\to\infty$.

(N2) Let $g\in\mathcal{H}^{\infty}(U)$ and $y\in F$. Clearly, the sequence
$(y_{n})$ given by

$y_{1}=\left\|g\right\|_{\infty}y,\quad y_{n}=0\qquad(n\geq 2),$

is in $\ell_{p}(F)$ (in $c_{0}(F)$ if $p=1$) and $(g\cdot y)(U)\subseteq
p\text{-}\mathrm{conv}(y_{n})$. Hence $g\cdot
y\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with
$k_{p}^{\mathcal{H}^{\infty}}(g\cdot
y)\leq\left\|(y_{n})\right\|_{p}=\left\|g\right\|_{\infty}\left\|y\right\|$.
Conversely, $\left\|g\right\|_{\infty}\left\|y\right\|=\left\|g\cdot
y\right\|_{\infty}\leq k_{p}^{\mathcal{H}^{\infty}}(g\cdot y)$ by (N1).

(N3) Let $H,G$ be complex Banach spaces, $V$ be an open subset of $H$,
$h\in\mathcal{H}(V,U)$, $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ and
$S\in\mathcal{L}(F,G)$. Clearly, $S\circ f\circ h\in\mathcal{H}^{\infty}(V,G)$
with $\left\|S\circ f\circ
h\right\|_{\infty}\leq\left\|S\right\|\left\|f\right\|_{\infty}$. Let
$(y_{n})$ be a sequence in $\ell_{p}(F)$ (in $c_{0}(F)$ if $p=1$) such that
$f(U)\subseteq p\text{-}\mathrm{conv}(y_{n})$. Plainly, $(S\circ f\circ
h)(V)\subseteq p\text{-}\mathrm{conv}(S(y_{n}))$. Hence $S\circ f\circ
h\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(V,G)$ with
$k_{p}^{\mathcal{H}^{\infty}}(S\circ f\circ
h)\leq\left\|(S(y_{n}))\right\|_{p}\leq\left\|S\right\|\left\|(y_{n})\right\|_{p}$,
and taking infimum over all such sequences $(y_{n})$, we arrive at
$k_{p}^{\mathcal{H}^{\infty}}(S\circ f\circ
h)\leq\left\|S\right\|k_{p}^{\mathcal{H}^{\infty}}(f)$.

(R) Suppose now that $F$ is reflexive. Let $f\in\mathcal{H}^{\infty}(U,F)$ and
assume that $\kappa_{F}\circ
f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F^{**})$. We can take a sequence
$(y_{n})$ in $\ell_{p}(F)$ (in $c_{0}(F)$ if $p=1$) such that
$(\kappa_{F}\circ f)(U)\subseteq p\text{-}\mathrm{conv}(\kappa_{F}(y_{n}))$,
that is, $\kappa_{F}(f(U))\subseteq\kappa_{F}(p\text{-}\mathrm{conv}(y_{n}))$
which yields $f(U)\subseteq p\text{-}\mathrm{conv}(y_{n})$ by the injectivity
of $\kappa_{F}$. Hence $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with
$k_{p}^{\mathcal{H}^{\infty}}(f)\leq\left\|(y_{n})\right\|_{p}=\left\|(\kappa_{F}(y_{n}))\right\|_{p}$,
and so $k_{p}^{\mathcal{H}^{\infty}}(f)\leq
k^{\mathcal{H}^{\infty}}_{p}(\kappa_{F}\circ f)$ by extending this infimum
over all such sequences $(\kappa_{F}(y_{n}))$. The converse inequality follows
from (N3). This proves that
$(\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F),k_{p}^{\mathcal{H}^{\infty}})$
is regular.

(S) Let $f\in\mathcal{H}^{\infty}(U,F)$ and assume that
$f\circ\pi\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(V,F)$, where $V$ is an
open subset of a complex Banach space $G$ and $\pi\in\mathcal{H}(V,U)$ is
surjective. Since $f(U)=(f\circ\pi)(V)$, it is immediate that
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with
$k_{p}^{\mathcal{H}^{\infty}}(f)=k_{p}^{\mathcal{H}^{\infty}}(f\circ\pi)$.
Hence $\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ is surjective. ∎

Our following objective is a characterization of holomorphic mappings with
relatively $p$-compact range in terms of the $p$-compactness of their
linearizations on the canonical predual of $\mathcal{H}^{\infty}(U)$.

Towards this end, we first recall a result due to Mujica [13] on linearization
of holomophic mappings on Banach spaces.

Given an open subset $U$ of a complex Banach space $E$, let
$\mathcal{G}^{\infty}(U)$ denote the norm-closed linear subspace of
$\mathcal{H}^{\infty}(U)^{*}$ generated by the functionals
$\delta(x)\in\mathcal{H}^{\infty}(U)^{*}$ with $x\in U$, defined by

$\left\langle\delta(x),f\right\rangle=f(x)\qquad\left(f\in\mathcal{H}^{\infty}(U)\right).$

######  Theorem 1.3.

[13, Theorem 2.1] Let $E$ be a complex Banach space and let $U$ be an open set
in $E$.

  1. (i)

The mapping $g_{U}\colon U\to\mathcal{G}^{\infty}(U)$ defined by
$g_{U}(x)=\delta(x)$ is holomorphic with $\left\|\delta(x)\right\|=1$ for all
$x\in U$.

  2. (ii)

For every complex Banach space $F$ and every mapping
$f\in\mathcal{H}^{\infty}(U,F)$, there exists a unique operator
$T_{f}\in\mathcal{L}(\mathcal{G}^{\infty}(U),F)$ such that $T_{f}\circ
g_{U}=f$. Furthermore, $\left\|T_{f}\right\|=\left\|f\right\|_{\infty}$.

  3. (iii)

For every complex Banach space $F$, the mapping $f\mapsto T_{f}$ is an
isometric isomorphism from $\mathcal{H}^{\infty}(U,F)$ onto
$\mathcal{L}(\mathcal{G}^{\infty}(U),F)$.

  4. (iv)

$\mathcal{H}^{\infty}(U)$ is isometrically isomorphic to
$\mathcal{G}^{\infty}(U)^{*}$, via the mapping
$J_{U}\colon\mathcal{H}^{\infty}(U)\to\mathcal{G}^{\infty}(U)^{*}$ given by

$\left\langle
J_{U}(f),g_{U}(x)\right\rangle=f(x)\qquad(f\in\mathcal{H}^{\infty}(U),\;x\in
U).$

  5. (v)

$B_{\mathcal{G}^{\infty}(U)}$ coincides with
$\overline{\mathrm{abco}}(g_{U}(U))$. $\hfill\Box$

We now proceed to prove the promised characterization of holomorphic mappings
with relatively $p$-compact range.

######  Theorem 1.4.

Let $p\in[1,\infty)$ and $f\in\mathcal{H}^{\infty}(U,F)$. The following
conditions are equivalent:

  1. (i)

$f$ has relatively $p$-compact range.

  2. (ii)

$T_{f}\colon\mathcal{G}^{\infty}(U)\to F$ is a $p$-compact linear operator.

In this case, $k_{p}^{\mathcal{H}^{\infty}}(f)=k_{p}(T_{f})$. Furthermore, the
mapping $f\mapsto T_{f}$ is an isometric isomorphism from
$(\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F),k_{p}^{\mathcal{H}^{\infty}})$
onto $(\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F),k_{p})$.

###### Proof.

Using Theorem 1.3, we have the following inclusions:

$f(U)=T_{f}\circ g_{U}(U)\subseteq
T_{f}(\overline{\mathrm{abco}}(g_{U}(U)))\subseteq\overline{\mathrm{abco}}(T_{f}\circ
g_{U}(U))=\overline{\mathrm{abco}}(f(U)),$

and also $\overline{\mathrm{abco}}(g_{U}(U))=B_{\mathcal{G}^{\infty}(U)}$.

$(ii)\Rightarrow(i)$: If $T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$,
then $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with

$k_{p}^{\mathcal{H}^{\infty}}(f)=m_{p}(f(U))\leq
m_{p}(T_{f}(B_{\mathcal{G}^{\infty}(U)}))=k_{p}(T_{f})$

by the first inclusion.

$(i)\Rightarrow(ii)$: If $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$,
then $T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ with

$k_{p}(T_{f})=m_{p}(T_{f}(B_{\mathcal{G}^{\infty}(U)}))\leq
m_{p}(\overline{\mathrm{abco}}(f(U)))=m_{p}(f(U))=k_{p}^{\mathcal{H}^{\infty}}(f)$

by the second inclusion and the known fact that a set $K\subseteq F$ is
$p$-compact if and only if $\overline{\mathrm{abco}}(K)$ is $p$-compact, in
whose case $m_{p}(K)=m_{p}(\overline{\mathrm{abco}}(K))$.

For the last assertion of the statement, it suffices to show that $f\mapsto
T_{f}$ from $\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ to
$\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ is surjective. Take
$T\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ and then $T=T_{f}$ for some
$f\in\mathcal{H}^{\infty}(U,F)$ by Theorem 1.3. Hence
$T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ and thus
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ by proved above. ∎

As an easy consequence of the theorem, we get that the ideal
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}$ is generated by composition with the
operator ideal $\mathcal{K}_{p}$.

######  Corollary 1.5.

Let $p\in[1,\infty)$ and $f\in\mathcal{H}^{\infty}(U,F)$. The following
conditions are equivalent:

  1. (i)

$f\colon U\to F$ has relatively $p$-compact range.

  2. (ii)

$f=T\circ g$ for some complex Banach space $G$,
$g\in\mathcal{H}^{\infty}(U,G)$ and $T\in\mathcal{K}_{p}(G,F)$.

In this case,
$k_{p}^{\mathcal{H}^{\infty}}(f)=\inf\\{k_{p}(T)\left\|g\right\|_{\infty}\\}$,
where the infimum runs over all factorizations of $f$ as in $(ii)$, and this
infimum is attained at $T_{f}\circ g_{U}$ (Mujica’s factorization of $f$
[13]).

###### Proof.

$(i)\Rightarrow(ii)$: If $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$, we
have $f=T_{f}\circ g_{U}$, where $\mathcal{G}^{\infty}(U)$ is a complex Banach
space, $T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ and
$g_{U}\in\mathcal{H}^{\infty}(U,\mathcal{G}^{\infty}(U))$ by Theorems 1.3 and
1.4. Moreover,

$\inf\left\\{k_{p}(T)\left\|g\right\|_{\infty}\right\\}\leq
k_{p}(T_{f})\left\|g_{U}\right\|_{\infty}=k_{p}^{\mathcal{H}^{\infty}}(f).$

$(ii)\Rightarrow(i)$: Assume $f=T\circ g$ with $g$ and $T$ being as in $(ii)$.
Since $g=T_{g}\circ g_{U}$ by Theorem 1.3, it follows that $f=T\circ
T_{g}\circ g_{U}$ which yields $T_{f}=T\circ T_{g}$ by the uniqueness of the
factorization of $f$ (Theorem 1.3), and thus
$T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ by the ideal property of
$\mathcal{K}_{p}$. From Theorem 1.4, we infer that
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with

$k_{p}^{\mathcal{H}^{\infty}}(f)=k_{p}(T_{f})=k_{p}(T\circ T_{g})\leq
k_{p}(T)\left\|T_{g}\right\|=k_{p}(T)\left\|g\right\|_{\infty},$

and so
$k_{p}^{\mathcal{H}^{\infty}}(f)\leq\inf\\{k_{p}(T)\left\|g\right\|_{\infty}\\}$
by taking the infimum over all such factorizations of $f$. ∎

Applying Theorem 1.4 and the fact that
$\mathcal{K}_{p}\subseteq\mathcal{K}_{q}$ whenever $1\leq p\leq q\leq\infty$
[17, Proposition 4.3], we get the following inclusion relation:

######  Corollary 1.6.

If $1\leq p\leq q<\infty$ and
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$, then
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{q}}(U,F)$ and
$k_{q}^{\mathcal{H}^{\infty}}(f)\leq
k_{p}^{\mathcal{H}^{\infty}}(f)$.$\hfill\qed$

We now present a factorization result for $p$-compact holomorphic mappings
which should be compared with [8, Proposition 2.9].

######  Corollary 1.7.

Let $p\in[1,\infty)$ and $f\in\mathcal{H}^{\infty}(U,F)$. The following
conditions are equivalent:

  1. (i)

$f\colon U\to F$ has relatively $p$-compact range.

  2. (ii)

There exist a closed subspace $M$ in $\ell_{p^{*}}$ ($c_{0}$ instead of
$\ell_{p^{*}}$ if $p=1$), a separable Banach space $G$, an operator $T$ in
$\mathcal{K}_{p}(\ell_{p^{*}}/M,G)$, a mapping $g$ in
$\mathcal{H}^{\infty}_{\mathcal{K}}(U,\ell_{p^{*}}/M)$ and an operator $S$ in
$\mathcal{K}(G,F)$ such that $f=S\circ T\circ g$.

In this case,
$k_{p}^{\mathcal{H}^{\infty}}(f)=\inf\\{\left\|S\right\|k_{p}(T)\left\|g\right\|_{\infty}\\}$,
where the infimum is extended over all factorizations of $f$ as in $(ii)$.

###### Proof.

We will only prove it for $p\in(1,\infty)$. The case $p=1$ is similarly
obtained.

$(i)\Rightarrow(ii)$: Suppose that
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$. By Theorem 1.4,
$T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ with
$k_{p}(T_{f})=k_{p}^{\mathcal{H}^{\infty}}(f)$. Applying [8, Proposition 2.9],
for each $\varepsilon>0$, there exist a closed subspace
$M\subseteq\ell_{p^{*}}$, a separable Banach space $G$, an operator
$T\in\mathcal{K}_{p}(\ell_{p^{*}}/M,G)$, an operator $S\in\mathcal{K}(G,F)$
and an operator $R\in\mathcal{K}(\mathcal{G}^{\infty}(U),\ell_{p^{*}}/M)$ such
that $T_{f}=S\circ T\circ R$ with
$\left\|S\right\|k_{p}(T)\left\|R\right\|\leq k_{p}(T_{f})+\varepsilon$.
Moreover, $R=T_{g}$ with $\left\|g\right\|_{\infty}=\left\|R\right\|$ for some
$g\in\mathcal{H}^{\infty}_{\mathcal{K}}(U,\ell_{p^{*}}/M)$ by [11, Corollary
2.11]. Thus we obtain

$f=T_{f}\circ g_{U}=S\circ T\circ R\circ g_{U}=S\circ T\circ T_{g}\circ
g_{U}=S\circ T\circ g,$

with

$\left\|S\right\|k_{p}(T)\left\|g\right\|_{\infty}=\left\|S\right\|k_{p}(T)\left\|R\right\|\leq
k_{p}(T_{f})+\varepsilon=k_{p}^{\mathcal{H}^{\infty}}(f)+\varepsilon.$

Since $\varepsilon$ was arbitrary, we deduce that
$\left\|S\right\|k_{p}(T)\left\|g\right\|_{\infty}\leq
k_{p}^{\mathcal{H}^{\infty}}(f)$.

$(ii)\Rightarrow(i)$: Assume that $f=S\circ T\circ g$ is a factorization as in
$(ii)$. Since $S\circ T\in\mathcal{K}_{p}(\ell_{p^{*}}/M,F)$ by the ideal
property of $\mathcal{K}_{p}$, by applying Corollary 1.5 yields that
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with

$k_{p}^{\mathcal{H}^{\infty}}(f)\leq k_{p}(S\circ
T)\left\|g\right\|_{\infty}\leq\left\|S\right\|k_{p}(T)\left\|g\right\|_{\infty},$

and taking infimum over all such factorizations of $f$, we have
$k_{p}^{\mathcal{H}^{\infty}}(f)\leq\inf\\{\left\|S\right\|k_{p}(T)\left\|g\right\|_{\infty}\\}$.
∎

Let us recall that a mapping $f\in\mathcal{H}^{\infty}(U,F)$ has finite
dimensional rank if the linear hull of its range is a finite dimensional
subspace of $F$. We denote by $\mathcal{H}^{\infty}_{\mathcal{F}}(U,F)$ the
set of all finite-rank bounded holomorphic mappings from $U$ to $F$. In the
light of Theorem 1.2, it is clear that
$\mathcal{H}^{\infty}_{\mathcal{F}}(U,F)$ is a linear subspace of
$\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$.

In similarity with the linear case, it seems natural to introduce the
following class of holomorphic mappings.

######  Definition 1.8.

Let $p\in[1,\infty)$. A mapping $f\in\mathcal{H}^{\infty}(U,F)$ is said to be
$p$-approximable if there exists a sequence $(f_{n})$ in
$\mathcal{H}^{\infty}_{\mathcal{F}}(U,F)$ such that
$k_{p}^{\mathcal{H}^{\infty}}(f_{n}-f)\to 0$ as $n\to\infty$. We denote by
$\mathcal{H}^{\infty}_{\overline{\mathcal{F}}_{p}}(U,F)$ the space of all
$p$-approximable holomorphic mappings from $U$ to $F$.

######  Proposition 1.9.

For $p\in[1,\infty)$, every $p$-approximable holomorphic mapping from $U$ to
$F$ has relatively $p$-compact range.

###### Proof.

Let $f\in\mathcal{H}^{\infty}_{\overline{\mathcal{F}}_{p}}(U,F)$. Hence there
is a sequence $(f_{n})$ in $\mathcal{H}^{\infty}_{\mathcal{F}}(U,F)$ such that
$k_{p}^{\mathcal{H}^{\infty}}(f_{n}-f)\to 0$ as $n\to\infty$. Since
$T_{f_{n}}\in\mathcal{F}(\mathcal{G}^{\infty}(U),F)$ by [13, Proposition 3.1],
$\mathcal{F}(\mathcal{G}^{\infty}(U),F)\subseteq\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$
by [17, Theorem 4.2] and
$k_{p}(T_{f_{n}}-T_{f})=k_{p}(T_{f_{n}-f})=k_{p}^{\mathcal{H}^{\infty}}(f_{n}-f)$
for all $n\in\mathbb{N}$ by Theorems 1.3 and 1.4, we deduce that
$T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ by [17, Theorem 4.2], and
so $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ by Theorem 1.4. ∎

Following [11], the transpose of a mapping $f\in\mathcal{H}^{\infty}(U,F)$ is
the bounded linear operator $f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$
defined by

$f^{t}(y^{*})=y^{*}\circ f\qquad(y^{*}\in F^{*}).$

Moreover, $||f^{t}||=\left\|f\right\|_{\infty}$ and
$f^{t}=J_{U}^{-1}\circ(T_{f})^{*}$, where
$J_{U}\colon\mathcal{H}^{\infty}(U)\to\mathcal{G}^{\infty}(U)^{*}$ is the
isometric isomorphism defined in Theorem 1.3.

The Banach ideal $\mathcal{K}_{p}$ is associated by duality with the ideal of
quasi-$p$-nuclear operators, introduced and studied by Persson and Pietsch
[14]. Given $p\in[1,\infty)$, an operator $T\in\mathcal{L}(E,F)$ is quasi
$p$-nuclear if and only if there is a sequence
$(x^{*}_{n})\in\ell_{p^{*}}(E^{*})$ ($(x^{*}_{n})\in c_{0}(E^{*})$ if $p=1$)
such that

$\left\|T(x)\right\|\leq\left(\sum_{n=1}^{\infty}\left|x^{*}_{n}(x)\right|^{p}\right)^{1/p}$

for all $x\in E$. The set of all such operators, denoted
$\mathcal{QN}_{p}(E,F)$, is a Banach operator ideal, endowed with the norm

$\nu^{\mathcal{Q}}_{p}(T)=\inf\left\\{\left\|(x^{*}_{n})\right\|_{p}\colon\left\|T(x)\right\|\leq\left(\sum_{n=1}^{\infty}\left|x^{*}_{n}(x)\right|^{p}\right)^{1/p},\
\forall x\in E\right\\}.$

In [7, Proposition 3.8], it was stated that an operator
$T\in\mathcal{K}_{p}(E,F)$ if and only if its adjoint
$T^{*}\in\mathcal{QN}_{p}(F^{*},E^{*})$. Moreover,
$k_{p}(T)=\nu^{\mathcal{Q}}_{p}(T^{*})$ by [8, Corollary 2.7]. We now present
the holomorphic analogue of this result.

######  Theorem 1.10.

Let $p\in[1,\infty)$ and $f\in\mathcal{H}^{\infty}(U,F)$. The following
conditions are equivalent:

  1. (i)

$f\colon U\to F$ has relatively $p$-compact range.

  2. (ii)

$f^{t}\colon F^{*}\to\mathcal{H}^{\infty}(U)$ is a quasi $p$-nuclear operator.

In this case, $k_{p}^{\mathcal{H}^{\infty}}(f)=\nu_{p}^{\mathcal{Q}}(f^{t})$.

###### Proof.

Applying Theorem 1.4, [8, Corollary 2.7] and [14, p. 32], respectively, we
have

$\displaystyle f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$
$\displaystyle\Leftrightarrow
T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$
$\displaystyle\Leftrightarrow(T_{f})^{*}\in\mathcal{QN}_{p}(F^{*},\mathcal{G}^{\infty}(U)^{*})$
$\displaystyle\Leftrightarrow
f^{t}=J_{U}^{-1}\circ(T_{f})^{*}\in\mathcal{QN}_{p}(F^{*},\mathcal{H}^{\infty}(U)).$

In this case,
$k_{p}^{\mathcal{H}^{\infty}}(f)=k_{p}(T_{f})=\nu^{\mathcal{Q}}_{p}((T_{f})^{*})=\nu^{\mathcal{Q}}_{p}(f^{t})$.
∎

Given $p\in[1,\infty)$, let us recall that an operator $T\in\mathcal{L}(E,F)$
is $p$-summing if there exists a constant $C\geq 0$ such that, regardless of
the natural number $n$ and regardless of the choice of vectors
$x_{1},\ldots,x_{n}$ in $E$, we have

$\left(\sum_{i=1}^{n}\left\|T(x_{i})\right\|^{p}\right)^{\frac{1}{p}}\leq
C\sup_{x^{*}\in
B_{E^{*}}}\left(\sum_{i=1}^{n}\left|x^{*}(x_{i})\right|^{p}\right)^{\frac{1}{p}}.$

The infimum of such constants $C$ is denoted by $\pi_{p}(T)$ and the linear
space of all $p$-summing operators from $E$ into $F$ by $\Pi_{p}(E,F)$. We
refer to [15, 12, Section 17.3.1] for a complete description of this Banach
operator ideal.

The following result could be compared with [7, Proposition 3.13] stated in
the linear setting.

######  Proposition 1.11.

Let $f\in\mathcal{H}^{\infty}(U,F)$ and
$g\in\mathcal{H}_{\mathcal{K}}^{\infty}(U,F^{*})$. Assume that
$T_{f}\in\Pi_{p}(\mathcal{G}^{\infty}(U),F)$ with $p\in[1,\infty)$. Then
$f^{t}\circ
g\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,\mathcal{H}^{\infty}(U))$ with
$k_{p}^{\mathcal{H}^{\infty}}(f^{t}\circ
g)\leq\pi_{p}(T_{f})\left\|g\right\|_{\infty}$.

###### Proof.

By Theorem 1.4, $T_{g}\in\mathcal{K}(\mathcal{G}^{\infty}(U),F^{*})$ with
$||T_{g}||=\left\|g\right\|_{\infty}$. Consequently, by [7, Proposition 3.13],
the linear operator $(T_{f})^{*}\circ
T_{g}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),\mathcal{G}^{\infty}(U)^{*})$
with $k_{p}((T_{f})^{*}\circ T_{g})\leq\pi_{p}(T_{f})||T_{g}||$. From the
equality $f^{t}\circ T_{g}=J_{U}^{-1}\circ(T_{f})^{*}\circ T_{g}$, we infer
that $f^{t}\circ
T_{g}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),\mathcal{H}^{\infty}(U))$ with
$k_{p}(f^{t}\circ T_{g})=k_{p}((T_{f})^{*}\circ T_{g})$ by the ideal property
of $\mathcal{K}_{p}$. Applying Theorem 1.4, there exists
$h\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,\mathcal{H}^{\infty}(U))$ with
$k_{p}^{\mathcal{H}^{\infty}}(h)=k_{p}(T_{h})$ such that $f^{t}\circ
T_{g}=T_{h}$. Hence $f^{t}\circ g=h$ and thus $f^{t}\circ
g\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,\mathcal{H}^{\infty}(U))$ with

$k_{p}^{\mathcal{H}^{\infty}}(f^{t}\circ
g)=k_{p}(T_{h})=k_{p}((T_{f})^{*}\circ
T_{g})\leq\pi_{p}(T_{f})\left\|T_{g}\right\|=\pi_{p}(T_{f})\left\|g\right\|_{\infty}.$

∎

$p$-Compact operators were characterized as those operators whose adjoints
factor through a subspace of $\ell_{p}$ [17, Theorem 3.2]. We now obtain a
similar factorization for the transpose of a holomorphic mapping with
relatively $p$-compact range (compare also to [7, Proposition 3.10]).

######  Corollary 1.12.

Let $p\in[1,\infty)$ and $f\in\mathcal{H}^{\infty}(U,F)$. The following
conditions are equivalent:

  1. (i)

$f\colon U\to F$ has relatively $p$-compact range.

  2. (ii)

There exist a closed subspace $M\subseteq\ell_{p}$ and
$R\in\mathcal{QN}_{p}(F^{*},M)$, $S\in\mathcal{L}(M,\mathcal{H}^{\infty}(U))$
such that $f^{t}=S\circ R$.

###### Proof.

$(i)\Rightarrow(ii)$: If $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$,
then $T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ by Theorem 1.4. By
[7, Proposition 3.10], there exist a closed subspace $M\subseteq\ell_{p}$ and
operators $R\in\mathcal{QN}_{p}(F^{*},M)$ and
$S_{0}\in\mathcal{L}(M,\mathcal{G}^{\infty}(U)^{*})$ such that
$(T_{f})^{*}=S_{0}\circ R$. Taking $S=J_{U}^{-1}\circ
S_{0}\in\mathcal{L}(M,\mathcal{H}^{\infty}(U))$, we have $f^{t}=S\circ R$.

$(ii)\Rightarrow(i)$: Assume that $f^{t}=S\circ R$ being $S$ and $R$ as in the
statement. It follows that $(T_{f})^{*}=J_{U}\circ f^{t}=J_{U}\circ S\circ R$,
and thus $T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ by [7,
Proposition 3.10]. Hence $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ by
Theorem 1.4 ∎

The following holomorphic variant of the notion of $p$-nuclear operator was
introduced and analysed in [4].

Given $p\in[1,\infty)$, a holomorphic mapping $f\colon U\to F$ is said to be
$p$-nuclear if there exist an operator $T\in\mathcal{L}(\ell_{p},F)$, a
mapping $g\in\mathcal{H}^{\infty}(U,\ell_{\infty})$ and a diagonal operator
$M_{\lambda}\in\mathcal{L}(\ell_{\infty},\ell_{p})$ induced by a sequence
$\lambda\in\ell_{p}$ such that $f=T\circ M_{\lambda}\circ g$. We set

$\nu^{\mathcal{H}^{\infty}}_{p}(f)=\inf\left\\{\left\|T\right\|\left\|M_{\lambda}\right\|\left\|g\right\|_{\infty}\right\\},$

where the infimum is taken over all such factorizations of $f$. Let
$\mathcal{N}^{\mathcal{H}^{\infty}}_{p}(U,F)$ denote the set of all
$p$-nuclear holomorphic mappings from $U$ into $F$.

In Proposition 3.11 of [4], it was shown that
$(\mathcal{N}^{\mathcal{H}^{\infty}}_{p}(U,F),\nu^{\mathcal{H}^{\infty}}_{p})$
is a Banach ideal of bounded holomorphic mappings.

The following inclusion relation is readily obtained.

######  Corollary 1.13.

Let $p\in[1,\infty)$ and $f\in\mathcal{N}^{\mathcal{H}^{\infty}}_{p}(U,F)$.
Then $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with
$k_{p}^{\mathcal{H}^{\infty}}(f)\leq\nu_{p}^{\mathcal{H}^{\infty}}(f)$.

###### Proof.

By Proposition 3.12 in [4], we have
$T_{f}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U),F)$ with
$\nu_{p}(T_{f})=\nu_{p}^{\mathcal{H}^{\infty}}(f)$. It follows that
$T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ with
$k_{p}(T_{f})\leq\nu_{p}(T_{f})$ (see [7, p. 295]). Hence
$f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$ with
$k_{p}^{\mathcal{H}^{\infty}}(f)\leq\nu_{p}^{\mathcal{H}^{\infty}}(f)$ by
Theorem 1.4. ∎

We have just seen that $\mathcal{H}^{\infty}_{\mathcal{K}_{p}}$ is a
surjective bounded-holomorphic ideal which contains
$\mathcal{N}^{\mathcal{H}^{\infty}}_{p}$. Our last purpose in this paper is to
show that $\mathcal{H}^{\infty}_{\mathcal{K}_{p}}$ is the smallest one of such
ideals. Borrowing from the terminology of operator ideals [15, Section 4.7],
we introduce the following concept.

######  Definition 1.14.

Let $\mathcal{I}^{\mathcal{H}^{\infty}}$ be a bounded-holomorphic ideal. The
smallest surjective ideal which contains $\mathcal{I}^{\mathcal{H}^{\infty}}$
is called the surjective hull of $\mathcal{I}^{\mathcal{H}^{\infty}}$ and
denoted by $(\mathcal{I}^{\mathcal{H}^{\infty}})^{\mathrm{sur}}$.

We refer to [10, p. 421] for the following definition and results. Given a
Banach space $E$, set

$E^{1}:=\left\\{f\colon B_{E}\to\mathbb{C}\;|\;\sum_{x\in
B_{E}}\left|f(x)\right|<\infty\right\\}.$

The space $E^{1}$ has the lifting property (or it is a projective Banach
space): if $F$ and $G$ are Banach spaces such that there is a surjection
$S\in\mathcal{L}(F,G)$, then for any given $\varepsilon>0$ and each
$T\in\mathcal{L}(E^{1},G)$, there is a $\widetilde{T}\in\mathcal{L}(E^{1},F)$
such that $S\circ\widetilde{T}=T$ and
$\left\|\widetilde{T}\right\|\leq(1+\varepsilon)\left\|S\right\|\left\|T\right\|$.

There is a surjective bounded linear operator $Q_{E}\colon E^{1}\to E$ with
$\left\|Q_{E}\right\|=1$ defined by

$Q_{E}((\lambda_{x})_{x\in B_{E}})=\sum_{x\in
B_{E}}\lambda_{x}x\qquad((\lambda_{x})_{x\in B_{E}}\in E^{1}).$

Given Banach spaces $E,F,G$, let us recall that a normed operator ideal
$\mathcal{I}$ is surjective if for every surjection $Q\in\mathcal{L}(G,E)$ and
every $T\in\mathcal{L}(E,F)$, it follows from $T\circ Q\in\mathcal{I}(G,F)$
that $T\in\mathcal{I}(E,F)$ with $\left\|T\right\|_{\mathcal{I}}=\left\|T\circ
Q\right\|_{\mathcal{I}}$. The smallest surjective ideal which contains
$\mathcal{I}$, denoted by $\mathcal{I}^{\mathrm{sur}}$, is called the
surjective hull of $\mathcal{I}$ and it admits the following description:

$\mathcal{I}^{\mathrm{sur}}(E,F)=\left\\{T\in\mathcal{L}(E,F)\colon T\circ
Q_{E}\in\mathcal{I}(E^{1},F)\right\\},$

equipped with the norm
$\left\|T\right\|_{\mathcal{I}^{\mathrm{sur}}}=\left\|T\circ
Q_{E}\right\|_{\mathcal{I}}$ for all $T\in\mathcal{I}^{\mathrm{sur}}(E,F)$.

We will need the following description of the surjective hull of the ideal
$(\mathcal{N}^{\mathcal{H}^{\infty}}_{p}(U,F),\nu^{\mathcal{H}^{\infty}}_{p})$.

######  Proposition 1.15.

We have

$(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}(U,F)=\left\\{f\in\mathcal{H}^{\infty}(U,F)\colon
T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U)^{1},F)\right\\},$

with
$\left\|f\right\|_{(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}}=\nu_{p}(T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)})$.

###### Proof.

If $\mathcal{A}(U,F)$ denotes the second set of the equality above, we first
show that $\mathcal{A}(U,F)$ is a surjective bounded-holomorphic ideal:

  1. (I1)

Since $f\mapsto T_{f}$ is a linear mapping from $\mathcal{H}^{\infty}(U,F)$ to
$\mathcal{L}(\mathcal{G}^{\infty}(U),F)$ by Theorem 1.3 and $\mathcal{N}_{p}$
is a linear space, it readily follows that $\mathcal{A}(U,F)$ is a linear
subspace of $\mathcal{H}^{\infty}(U,F)$.

  2. (I2)

Given $g\in\mathcal{H}^{\infty}(U)$ and $y\in F$, we have $g\cdot
y\in\mathcal{H}^{\infty}(U,F)$ and

$T_{g\cdot y}\circ Q_{\mathcal{G}^{\infty}(U)}=y(T_{g}\circ
Q_{\mathcal{G}^{\infty}(U)})\in\mathcal{F}(\mathcal{G}^{\infty}(U)^{1},F)\subseteq\mathcal{N}_{p}(\mathcal{G}^{\infty}(U)^{1},F).$

Hence $g\cdot y\in\mathcal{A}(U,F)$.

  3. (I3)

If $V$ is an open subset of a complex Banach space $H$,
$h\in\mathcal{H}(V,U)$, $f\in\mathcal{A}(U,F)$ and $S\in\mathcal{L}(F,G)$, we
have $S\circ f\circ h\in\mathcal{H}^{\infty}(V,G)$. By [11, Corollary 1.4],
there exists a unique operator
$\widehat{h}\in\mathcal{L}(\mathcal{G}^{\infty}(V),\mathcal{G}^{\infty}(U))$
such that $\widehat{h}\circ g_{V}=g_{U}\circ h$. Since

$S\circ f\circ h=S\circ(T_{f}\circ g_{U})\circ h=(S\circ
T_{f}\circ\widehat{h})\circ g_{V},$

we have $T_{S\circ f\circ h}=S\circ T_{f}\circ\widehat{h}$ by Theorem 1.3.
Hence, by the ideal property of $\mathcal{N}_{p}$,

$T_{S\circ f\circ h}\circ Q_{\mathcal{G}^{\infty}(V)}=S\circ
T_{f}\circ\widehat{h}\circ
Q_{\mathcal{G}^{\infty}(V)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(V)^{1},G).$

Hence $S\circ f\circ h\in\mathcal{A}(V,G)$.

  4. (S)

Let $V$ be an open subset of a complex Banach space $G$, let $\pi\colon V\to
U$ be a surjective holomorphic mapping and $f\in\mathcal{H}^{\infty}(U,F)$.
Suppose that $f\circ\pi\in\mathcal{A}(V,F)$, that is,
$f\circ\pi\in\mathcal{H}^{\infty}(V,F)$ and $T_{f\circ\pi}\circ
Q_{\mathcal{G}^{\infty}(V)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(V)^{1},F)$.
As in the proof of (I3), we deduce that
$T_{f\circ\pi}=T_{f}\circ\widehat{\pi}$, where $\widehat{\pi}$ is the unique
operator in $\mathcal{L}(\mathcal{G}^{\infty}(V),\mathcal{G}^{\infty}(U))$
such that $g_{U}\circ\pi=\widehat{\pi}\circ g_{V}$. Since
$\mathcal{G}^{\infty}(U)^{1}$ has the lifting property and $\widehat{\pi}\circ
Q_{\mathcal{G}^{\infty}(V)}\in\mathcal{L}(\mathcal{G}^{\infty}(V)^{1},\mathcal{G}^{\infty}(U))$
is surjective, given $\varepsilon>0$ there is a
$L_{\varepsilon}\in\mathcal{L}(\mathcal{G}^{\infty}(U)^{1},\mathcal{G}^{\infty}(V)^{1})$
such that $\widehat{\pi}\circ Q_{\mathcal{G}^{\infty}(V)}\circ
L_{\varepsilon}=Q_{\mathcal{G}^{\infty}(U)}$ and
$\left\|L_{\varepsilon}\right\|\leq(1+\varepsilon)$. By the ideal property of
$\mathcal{N}_{p}$, we have

$T_{f}\circ Q_{\mathcal{G}^{\infty}(U)}=T_{f}\circ\widehat{\pi}\circ
Q_{\mathcal{G}^{\infty}(V)}\circ
L_{\varepsilon}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U)^{1},F)$

since $T_{f}\circ\widehat{\pi}\circ
Q_{\mathcal{G}^{\infty}(V)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(V)^{1},F)$.
Then $f\in\mathcal{A}(U,F)$, as required.

In addition, $\mathcal{A}(U,F)$ contains
$\mathcal{N}^{\mathcal{H}^{\infty}}_{p}(U,F)$. Indeed, if
$f\in\mathcal{N}^{\mathcal{H}^{\infty}}_{p}(U,F)$, then
$T_{f}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U),F)$ by [4, Proposition 3.12],
thus $T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U)^{1},F)$
by the ideal property of $\mathcal{N}_{p}$, and hence $f\in\mathcal{A}(U,F)$.

From above, we deduce the inclusion
$(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}(U,F)\subseteq\mathcal{A}(U,F)$
by minimality. For the converse inclusion, note that if
$f\in\mathcal{A}(U,F)$, we have
$T_{f}\in\mathcal{L}(\mathcal{G}^{\infty}(U),F)$ by Theorem 1.3 with
$T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U)^{1},F)$,
that is,
$T_{f}\in(\mathcal{N}_{p})^{\mathrm{sur}}(\mathcal{G}^{\infty}(U),F)$, and
thus $f\in(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}(U,F)$ by [4,
Proposition 3.12]. ∎

We are now ready to prove the announced result.

######  Proposition 1.16.

$(\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F),k_{p}^{\mathcal{H}^{\infty}})$
is the surjective hull of
$(\mathcal{N}^{\mathcal{H}^{\infty}}_{p}(U,F),\nu^{\mathcal{H}^{\infty}}_{p})$
for $p\in[1,\infty)$.

###### Proof.

Let $f\in\mathcal{H}^{\infty}_{\mathcal{K}_{p}}(U,F)$. Then
$T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U)^{*},F)$ with
$k_{p}(T_{f})=k_{p}^{\mathcal{H}^{\infty}}(f)$ by Theorem 1.4. Hence
$T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U)^{1},F)$
by the ideal property of $\mathcal{K}_{p}$. Since
$\mathcal{G}^{\infty}(U)^{1}$ is a projective Banach space, it follows that
$T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U)^{1},F)$
with $\nu_{p}(T_{f}\circ Q_{\mathcal{G}^{\infty}(U)})=k_{p}(T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)})$. So,
$f\in\mathcal{A}(U,F)=(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}(U,F)$
by Proposition 1.15, and we have
$\left\|f\right\|_{(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}}\leq
k_{p}^{\mathcal{H}^{\infty}}(f)$.

Conversely, let
$f\in(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}(U,F)$, that is,
$f\in\mathcal{H}^{\infty}(U,F)$ with $T_{f}\circ
Q_{\mathcal{G}^{\infty}(U)}\in\mathcal{N}_{p}(\mathcal{G}^{\infty}(U)^{1},F)$
by Proposition 1.15. Since $T_{f}\in\mathcal{L}(\mathcal{G}^{\infty}(U),F)$ by
Theorem 1.3, it follows that
$T_{f}\in(\mathcal{N}_{p})^{\mathrm{sur}}(\mathcal{G}^{\infty}(U),F)$. Hence
$T_{f}\in\mathcal{K}_{p}(\mathcal{G}^{\infty}(U),F)$ with
$k_{p}(T_{f})=||T_{f}||_{(\mathcal{N}_{p})^{\mathrm{sur}}}$ by [7, Proposition
3.11]. We conclude that $f\in\mathcal{K}^{\mathcal{H}^{\infty}}_{p}(U,F)$ with
$k_{p}^{\mathcal{H}^{\infty}}(f)=k_{p}(T_{f})$ by Theorem 1.4, and now we have
$k_{p}^{\mathcal{H}^{\infty}}(f)\leq\left\|f\right\|_{(\mathcal{N}^{\mathcal{H}^{\infty}}_{p})^{\mathrm{sur}}}$.
∎

Acknowledgements. This research was partially supported by project UAL-FEDER
grant UAL2020-FQM-B1858, by Junta de Andalucía grants P20$\\_$00255 and
FQM194, and by Ministerio de Ciencia e Innovación grant PID2021-122126NB-C31.

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