1. Introduction
Let
be the open complex unit disc in the complex plane
,
X be a complex Banach space, and
be the space of all holomorphic mappings from
into
X. A mapping
is called
Bloch if
We denote to the space of all Bloch mappings from into X, and to the closed subspace formed by all zero-preserving Bloch mappings, that is, such that , endowed with the Bloch norm . In addition, we become used to the abbreviation .
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
is a subclass of the class of all continuous linear operators
such that for all Banach spaces
X and
Y, the components
satisfy the following properties:
- (i)
, where is the one-dimensional Banach space.
- (ii)
is a subspace of .
- (iii)
The ideal property: if , and , then , with W and Z being Banach spaces.
In addition, consider that for all Banach spaces X and Y, we endow with a (complete) norm so that the following hold:
- (iv)
is a Banach space.
- (v)
If
W and
Z are Banach spaces and
,
and
such that
then we will say that is a (Banach) normed operator ideal.
Let
be operator ideals and let
be Banach spaces. Following [
3] (p. 132), a bounded linear operator
is said to belong to the
left-hand quotient , and we write
, if
for all
, with
Z being an arbitrary Banach space. The
right-hand quotient is defined in a similar way. Of course, the symbols
and
have no meaning. It is well known that
and
are operator ideals (see Section 3.2.2 of [
2]).
Moreover, if
and
are Banach operator ideals and
, we define
with
Z covering all Banach spaces. In such a case,
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
-completely continuous operators can be seen as a right-hand quotient generated by the classes of compact operators and
-
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
-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
-mixing operator and completely
-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
, where
is an operator ideal and
is a normalized Bloch ideal. If both ideals are endowed with Banach norms, then we will prove that
endowed with the norm
is a Banach normalized Bloch ideal which becomes surjective whenever
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
has the linearization property in an operator ideal
, then
, while if
has the linearization property in another operator ideal
, then a map
belongs to the normalized Bloch left-hand quotient
if and only if its linearization
belongs to the operator left-hand quotient
, where the Bloch-free space
is defined by
with
being the functional given as
for all
. In this case, we also prove that
is a composition ideal of the form
.
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 to denote the open unit disc of the complex plane . 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 to denote the dual space of X, that is, . As usual, denotes the closed unit ball of X. On the other hand, for a set , we denote the linear hull and the norm-closed absolutely convex hull of A in X by and , respectively.
2. Preliminaries
Firstly, recall by Definition 5.11 of [
10] that a (Banach) normed normalized Bloch ideal, denoted as
, is a subset of the class of all zero-preserving Bloch maps
which is characterized by the condition that the components
verify the properties that follow:
- (P1)
is a (Banach) normed space and for .
- (P2)
The function
, defined as
is in
with
for any
and
.
- (P3)
The ideal property: Given , and , where Y is a complex Banach space, then the map is in and .
A normed normalized Bloch ideal is
- (C)
Closed if every component is a closed subspace of endowed with the Bloch norm topology.
- (S)
Surjective if for any complex Banach space X, with , whenever , is a metric surjection, and .
Our main tool in this paper is the method of linearization of Bloch mappings gathered in the following result.
Theorem 1 - (i)
The map , given byis in and for any . - (ii)
, where .
- (iii)
For each complex Banach space X and each map , there exists a unique operator such that . Moreover, . In addition, the map is an isometric isomorphism from onto .
Our research will mainly be based on a certain linearization property for the functions of the normalized Bloch ideal in a suitable operator ideal .
Definition 1. For a normed operator ideal and a normed normalized Bloch ideal , is said to have the linearization property (hereafter, we simply write LP for short) in if for any , it is satisfied that if and only if . In such a case, .
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 be a normalized Bloch ideal and let be an operator ideal. A map belongs to the normalized Bloch left-hand quotient , and shall be written as , if for all , with Y being a complex Banach space.
If the operator ideal is equipped with a Banach norm and with a norm , we define the Bloch left-hand quotient norm as follows: The first goal is to demonstrate the existence of
. Toward this end, let us recall Cartesian
-products of Banach spaces (see C.4 of [
2]). Let
with
be a family of Banach spaces. For
, the
Cartesian -product is defined as the set of all families
, where
for
, such that
, being that
is the Banach space comprising all scalar families
that are absolutely
p-summable, with
. It is well known by C.4.1 of [
2] that
is a Banach space equipped with the following norm:
If the underlying index set is , we just write .
Proposition 1. Let be a Banach operator ideal and let be a normed normalized Bloch ideal. If , then Proof. Let us assume that the preceding supremum is not finite. Then, for each
, we are in a position to find a complex Banach space
and an operator
satisfying
Consider the sequence of Banach spaces
with
, and the Cartesian
-product
. For each
, let us define the continuous linear operators
and
by
with
being the Kronecker delta. It is easy to check
and
. Notice that
is a sequence of elements of the Banach space
, and
for all
. Thus, the sequence of partial sums
forms a
-Cauchy sequence. Hence,
and we have
which is a contradiction, where we use that
□
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 and are normed operator ideals, we write in case and for all .
Proposition 2. Let and be normed normalized Bloch ideals satisfyingThen,for any Banach operator ideal . It is well known that is a normalized Bloch ideal which becomes Banach if we endow it with the Bloch norm . The following result is an immediate consequence of the previous proposition. In this, we ensure that is the largest normalized Bloch left-hand quotient for any Banach operator ideal in the following sense.
Corollary 1. Let be a normed normalized Bloch ideal. Thenfor any Banach operator ideal . Under a suitable assumption over the normalized Bloch ideal , and in relation to Corollary 1, we have the following useful result.
Proposition 3. Let be a Banach operator ideal and be a normed normalized Bloch ideal. Then,Furthermore, if has the LP in , then Proof. Firstly, for
, we claim that
. Indeed, notice that
and
for all
. Let
. Following [
12] (p. 131), we may consider a functional
so that
. Taking into account that
is a Banach operator ideal, we have that the function
, given by
is in the component
with
. Note that, for
, we have
Taking the supremum over all
, we have
. Furthermore,
and then
with
. Finally,
and taking the supremum over all
, we conclude that
.
Let us now assume that
has the LP in the operator ideal
. Let
and let
, for a complex Banach space
Y. It is not difficult to see that
. By Theorem 1, we can ensure the existence and uniqueness of the linearizations
and
with
and
satisfying
Hence,
by Remarks 2.8 (2) of [
10]. By the ideal property of
, we have
with
By hypothesis, has the LP in , and then we have with . Therefore, we are in a position to conclude that with 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 be a Banach operator ideal and let be a (Banach) normed normalized Bloch ideal. Then, is a (Banach) normed normalized Bloch ideal.
Proof. (P1): Let and for . Let , where Y is a complex Banach space and note that by the linearity of T. Since for and is a linear space, we obtain , and thus .
We will now demonstrate that is a norm on . Let and assume that . Consequently, for all , with and Y being a complex Banach space. In particular, for all , and then since separates the points of X.
For
and
, we claim that
and
. Indeed, note that
and
Now, we are going to study the structure, as the Banach space of the pair
when
is a complete norm on
. Let
be a
-Cauchy sequence. For a complex Banach space
Y, let
. By Proposition 3, we know
on
, and then there exists a map
such that
as
. As a consequence,
as
because
for all
. On the other hand, for all
, the inequality
shows that
is a Cauchy sequence in
. Hence, we could take a mapping
so that
as
. Bearing in mind the fact that
on
, we obtain that
, and then
with
as
.
Next, we prove that
converges to
f in
. For this purpose, let
. Then, there exists
such that, for all
,
. Thus,
for all
,
and
with
. Taking limits with
, it follows that
for all
and
with
. Taking the supremum over all such
T, we obtain that the sequence
is
-convergent to
f, as we wanted.
(P2): Let
and
. Since
is a normed normalized Bloch ideal, we have that
with
. Let
and do not lose sight of the fact that
. Hence,
with
. Therefore,
with
To obtain the reciprocal inequality, see that
for all
with
, and then
.
(P3): Let Y be a complex Banach space, , and . Let , with Z being a complex Banach space. Then, and by the ideal property of . By the definitions of and , we can ensure that with . Hence, with by the ideal property of . As a result, and since for all with , we deduce that .
Consequently, 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 be a surjective normed normalized Bloch ideal and be a normed operator ideal. Then, is surjective.
Proof. Let
and assume that
, where and
is a metric surjection. Then,
for all
, being
Y a complex Banach space. Since the normed normalized Bloch ideal
is surjective and
, it follows that
with
. By the arbitrariness of
, we can ensure that
. Moreover, notice that
Hence, 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 be a Banach operator ideal, be a normed operator ideal, and be a normed normalized Bloch ideal with the LP in . For every , we have if and only if . In such a case, . Moreover, the correspondence function verifies that it is an isometric isomorphism from onto .
Proof. (i) ⇒ (ii): Let . Then, for all we have that . Using an analogous reasoning to that of the proof of Proposition 3, Theorem 1 allows us to guarantee the existence of two operators and such that . Since has the LP in , we deduce that simply taking into account the ideal property of . The arbitrariness of leads us to conclude that .
(ii) ⇒ (i): Let us suppose that . Then, for all , and thus due to . Since has the LP in , it follows that and then .
We now show the equality between the associated norms
where the second equality is because of the LP of
in
.
Finally, to guarantee the map
is an isometric isomorphism, it remains to prove the surjectivity of such a map from
into
. With such a goal in mind, let
. We have
for all
, with
Y being a complex Banach space. Since
is concretely a bounded linear operator, we could apply Theorem 1 to ensure that
for some
. Hence,
by the LP of
in
. Let us define the mapping
given by
Clearly,
with
and
. Moreover, note that
for all
. Thus
and
. Now for all
, we have
Then, for all due to with . Finally, taking into account that separates points of X, we obtain . Hence, and . □
Let us now recall the composition method for generating normalized Bloch ideals which was introduced in [
11]. A Bloch map
with
is in the composition ideal
, with
being an operator ideal, and is denoted as
, if there are a complex Banach space
Y, an operator
, and a map
such that it is verified that
. Furthermore, if
is endowed with a norm
and
, we define
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 be a Banach operator ideal, be a normed operator ideal, and be a normed normalized Bloch ideal with the LP in . Then, Proof. In order to prove this result, let
. Note that
if and only if
by Theorem 2.2 of [
11]. At the same time, applying Theorem 4, we have
if and only if
due to the fact that
has the LP in
. Furthermore, by the above theorems, we have
for all
. □
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
,
,
,
, and
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
is said to belong to any of the above classes if
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]:
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 is a Grothendieck Bloch ifis a Grothendieck subset of X. Let denote the space of all zero-preserving Grothendieck Bloch maps from into X. According to [
10],
and
represent the spaces of all zero-preserving Bloch maps from
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],
and
are normed normalized Bloch ideals with the LP in
and
, respectively. Next, we show that
has the LP in
.
Theorem 5. The following statements are equivalent for a Bloch map :
- (i)
is a Grothendieck Bloch.
- (ii)
is a Grothendieck operator.
- (iii)
, with Y being a complex Banach space, and .
In such a case, and the correspondence is an isometric isomorphism from onto and from onto .
Proof. (i) ⇒ (ii): If
, then we have that
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,
is Grothendieck in
X. Do not lose sight of the fact that
and then
is a Grothendieck subset of
X. Hence,
by Proposition 6.1.1 of [
13].
(ii) ⇒ (i): Let us suppose that
. Then,
is a Grothendieck subset of
X. Since
, it follows that
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 be a closed operator ideal. Then, is a closed normalized Bloch ideal.
Proof. Let
and consider a sequence in
, namely,
so that
as
. A simple application of Theorem 2.2 of [
11] allows us to guarantee that
with
Hence,
and again by Theorem 2.2 of [
11], we conclude that
. □
Let us not forget that is a closed operator ideal (the norm-limit of a convergent sequence of Grothendieck operators is Grothendieck) and by Theorem 5. Thus, Lemma 4.1 yields the following straightforward result.
Corollary 2. 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. .
Proof. Let
. Just using Theorem 5, [
2] (3.2.6), Definition 2, Theorem 4, and Theorem 5.6 of [
10], respectively, we obtain
Furthermore, notice that . □
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
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
.
Moreover, an operator
is said to be
completely continuous if every
w-convergent sequence
is mapped into a norm convergent sequence
. Let
denote the space of all completely continuous operators from
X into
Y. By 1.6.2 and 4.2.5 of [
2],
is a closed operator ideal.
Next, we characterize the ideal of Bloch mappings having a Rosenthal range, denoted by
and introduced in [
11], as a normalized Bloch left-hand quotient ideal generated by the linear operator ideal
and the normalized Bloch ideal
.
Theorem 7. .
Proof. Given
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
In fact,
. □
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 does not have the LP in an operator ideal . The natural idea is to think that a normalized Bloch quotient may not coincide with a composition ideal of the form .
Let
. Let us recall that a bounded linear operator
is
p-summing if there is a constant
such that, for any
and
in
X, it is satisfied that
The space of all
p-summing linear operators from
X into
Y, denoted
, is a Banach space endowed with the norm
In the same way (see [
17]), a holomorphic function
with
is said to be
p-summing Bloch if there exists a constant
such that
for any
and any finite sets
in
and
in
. We denote this space as
, and it becomes Banach endowed with the norm
For
and
, it is clear that
. Let us assume that
. Then, for any
and
in
, we have
It follows from the above comment that
Hence, with and we have shown the following.
Proposition 5. Let and . If , then and .
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
is not Bloch. Assuming that we find a function
such that
, we would have shown that
does not have LP in
and, moreover, by Theorem 2.2 of [
11], we follow
. Apparently, whatever the operator ideal
is, then
cannot be a composition ideal generated by the operator ideal
, that is,
. Nevertheless,
is a normalized Bloch left-hand quotient ideal since
is a normalized Bloch ideal by Proposition 1.2 of [
17].