1. Introduction
In this paper, we introduce vector-valued growth spaces taking values on a Banach space. The definition of these spaces makes use of the notion of the so-called
Gamma order which is nothing but a linear transformation of a Banach space
into a conditionally complete vector lattice
. If
is a domain in
and
is a continuous function from
into the positive cone
in
, then
is the class of all holomorphic maps
f from
into
for which
for some
. The reader is referred to
Section 2.2 for the unexplained notation.
The usual approach to define the
-norm or “sup”-norm of a vector-valued function
f is to take the corresponding norm of the scalar function
. Our definition differs in (
1) in that we make use of the order structure, which is assumed to hold on the given Banach space via the transformation
. A natural example is when the Banach space
has an unconditional Schauder basis (
Section 2.3). It turns out that in this case there exists a natural conditionally complete vector lattice
and a bijective linear transformation
from
onto
; hence, one can consider the order structure on
induced by
.
The central focus of this paper is on the uniform embedding of the growth space
into the
space of a family
of vector-valued measures; that is, there is the existence of a constant
independent of
f so that
for every
. Now, we consider only a bounded, circular and strictly convex domain
with at least a
boundary in
. When
consists of one element only, such a measure is called
p-
Carleson for the space
. The earliest examples of such measures appear in the seminal work of L. Carleson in [
1,
2], where
p-Carleson measures for the Hardy space
were characterized. They were used in connection with interpolation and corona problems. There is a vast list of different results about
p-Carleson measures for (scalar-valued) Hardy, Bergman-type, Dirichlet-type and Besov spaces.
Scalar-valued growth spaces and their embedding properties were investigated in the work of E. Abakumov and E. Doubtsov ([
3,
4,
5]). In [
5], the author completely characterizes the
q-Carleson measures for the scalar-valued
and provides a characterization for the bounded or compact composition operators on these spaces when the weight
is of the form
for some
, or
for
. The paper [
3] is mainly focused on the case when the domain
is the open unit ball in
. The authors prove a characterization of a bounded or compact composition and Volterra operators on growth spaces in the setting of the open unit ball. The general case when
is a bounded circular domain is treated in [
4]. The key point in all these works is a two-sided estimate for functions in the growth space. This estimate together with a characterization of composition and multiplication operators, Volterra-type operators and extended Cesàro operators are given in [
4] for when the domain is bounded, circular and strictly convex.
Our definitions and results generalize the previous work on (scalar-valued) growth spaces in two different ways: We consider vector-valued holomorphic functions. Secondly, the problem is to characterize the embedding of
into a family of
spaces as explained above. Our compactness characterization is new even in the settings of scalar-valued growth spaces. We also provide an essential norm estimate for the embedding operator (see
Section 4).
Let
denote the class of all holomorphic maps from
into
. For a scalar holomorphic function
on
with
and a holomorphic map
on
into the sequence space
of bounded sequences, the weighted composition operator
is the operator that maps
into the function
for all
. Hence,
can be thought of as the composition followed by a multiplication. When
is the constant sequence, we have
: that is, the usual composition operator. Composition operators on function spaces are a vast subject. We refer the reader to [
6,
7] for the general theory.
The structure of the paper is as follows. In
Section 2.1, we introduce the Gamma order and show that the
transform induces a lattice structure on
. The growth space
is defined in
Section 2.2, and we prove that this is a Banach space. The specific case when
has a Schauder basis is treated in
Section 2.3. In
Section 3, we treat (vector-valued) Carleson measures. We provide essential norm estimates on the embedding operator, and we completely characterize bounded and compact Carleson embeddings of growth spaces in
Section 3 and
Section 4. As an application, we provide the necessary and sufficient criteria for the bounded or compact generalized composition operators (
Section 5).
5. Applications: Weighted Composition Operators
Let
be a Banach space with an unconditional basis
. For vectors
one can define the product
to be the element of
obtained by multiplying coordinate-wise the coefficients of
a and
b. For
and
, let
.
Let
, with
and
. The weighted composition operator
is an operator that maps
into the function
for all
. Hence,
is the composition followed by a multiplication operator. When
, the constant sequence
is the usual composition operator. Composition operators on function spaces are a vast subject. We refer the reader to [
6,
7] for the general theory.
For
and a positive measure
, we define a measure
in
by setting
for any non-negative, continuous
u on
that takes values in
. Now, if
and
denote the measures obtained by applying (
4) to
and
, respectively, as a result of Proposition 2, then
We assume in the rest of this section that
is a family of measures from the class
. Hence,
We assume that
and
are Banach spaces of holomorphic functions on
with the norm given by
and
X satisfies all the properties in
Section 4. An application of Theorems 6 and 7 gives norm and essential norm estimates for composition operators on
.
Theorem 8. Suppose that is a Banach lattice with an unconditional Schauder basis, and ω is an admissible weight that is equivalent to a log-convex function on Ω and takes values in . Let , with and . Then the following are equivalent:
- (i)
is bounded from into X.
- (ii)
.
For the essential norm, we haveTherefore, the following are equivalent: - (i)
is compact.
- (ii)
.
6. Discussion
This paper introduces the growth spaces: spaces of holomorphic functions with certain growth conditions and that take values in a Banach space . We believe that this is the first paper focusing on Banach-valued growth spaces in the literature. Our definition makes use of the abstract notion of the Gamma order, which is also introduced for the first time in this paper. A natural example fitting in this notion is when has an unconditional basis, in which case, the Gamma order on can be explicitly defined. The class of Banach spaces that have an unconditional basis contains most of the important spaces, such as the classical Lebesgue spaces. Any separable Hilbert space has an unconditional basis.
The growth space itself is also a Banach space. In the paper, bounded or compact embeddings of the growth spaces are completely characterized. Our results extend the results related to scalar-valued growth spaces. The compactness criteria are new even for the scalar-valued settings. We provide essential norm estimates for the embedding operator, which are also new information and are interesting on their own. As an application, a characterization of the bounded or compact generalized composition operators defined on these spaces is explicitly given.
For future research, versions of classical Banach spaces, such as Bergman, Hardy, etc., consisting of holomorphic Banach-valued functions can be defined analogously. We are planning to expand the work in this direction and the study of operators on such spaces.
The author is grateful for the comments of the reviewers, which improved the exposition of this paper.