Composition Operators between -rearrangement Invariant -
Module Spaces
İLKER ERYILMAZ
Department of Mathematics,
Ondokuz Mayıs University,
Faculty of Sciences, Kurupelit campus, 55139 Atakum-Samsun,
TURKEY
Abstract: - This paper investigates the behavior and structural properties of composition operators within the
framework of -rearrangement -module spaces. Building upon the foundational concepts of -modules
and rearrangement-invariant spaces, we explore the intricate interplay between these spaces under the action of
composition operators. Our study delves into the algebraic and topological aspects of composition operators,
elucidating their impact on the underlying space structures. After establishing the necessary background on
-modules and -rearrangement-invariant spaces and laying the groundwork for our subsequent analysis, a
rigorous examination of composition operators, we uncover fundamental properties such as -continuity, -
boundedness, and -compactness, shedding light on the intrinsic characteristics of these operators within -
module spaces.
Key-Words: - Bicomplex numbers, -valued function, Hyperbolic norm, -Distribution Function, -
Rearrangement, -Banach function space, multiplication operator, Composition operator.
1 Introduction and Preliminaries on
Numerous mathematical disciplines, such as
probability theory, mathematical and functional
analysis, naturally use bicomplex (-valued
functions. While traditional functional analysis
operates within vector spaces over real or complex
numbers, considering modules with bicomplex
scalars extends the framework, leading to
exploration of new mathematical structures and
properties. Important contributions included in the
book [1], presenting pioneer opinions into modules
with bicomplex scalars. Besides, several articles are
written about studying topological bicomplex
modules and fundamental theorems related to them.
These papers cover fundamental topics such as
Hahn-Banach theorem, bounded linear operators,
topological properties and functional analysis.
Moreover, a comprehensive review of bicomplex
analysis and geometry is presented in [2]. The other
references such as [3], [4], [5], [6], [7], [8], [9], [10]
and [11] guide to the understanding of bicomplex
modules, functional analysis, and related areas,
giving insights, theorems, and applications for
researchers in these fields.
The set bicomplex numbers which is a four-
dimensional extension of the real numbers is
defined as
where and are imaginary units satisfying
and . The set of bicomplex numbers
forms a commutative ring under the usual addition
and usual multiplication operations. The production
of the imaginary units and find out a new
hyperbolic unit , where . According to this
is a square root of and is distinct from and .
The product operation of all units and in the
bicomplex numbers is commutative and satisfies
Furthermore, is a normed space with the norm
for any in
. In light of this,
for every , and finally is a
modified Banach algebra, [12]. Hyperbolic numbers
are two-dimensional extension of the real
numbers that form a number system known as the
hyperbolic plane or hyperbolic plane algebra. They
can be represented in the form , where
Received: September 15, 2023. Revised: April 16, 2024. Accepted: June 16, 2024. Published: July 19, 2024.
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and are real numbers, and is the hyperbolic
unit. If the hyperbolic numbers and are
defined as:
then it is easy to see that
and
. By using this linearly
independent set , any
can be written as a linear combination of and
uniquely. That is,
(1)
where and [1]. The
formula in (1) is called the idempotent
representation of the bicomplex number . Besides
the Euclidean-type norm , another norm named
with (-valued) hyperbolic-valued norm of
any bicomplex number is defined
as
For any hyperbolic number , the
idempotent representation can also be written as
where and are real
numbers. If and for any
, then we say that is a
positive hyperbolic number and denote by .
Now, let and be any two elements of . In [1]
and [2], a relation is defined on by
It is shown in [1] that this relation "" defines a
partial order on . If idempotent representations of
the hyperbolic numbers and are written as
and , then
and . By , we mean
and .
Any function defined on is called -increasing
if , -decreasing if , -
nonincreasing if and -nondecreasing
if whenever . For more details
on hyperbolic numbers and partial order "", one
can refer to [1] and [2].
Definition 1 Let be a subset of . is called a -
bounded above set if there is a hyperbolic number
such that for all . If is -
bounded from above, then the -supremum of is
defined as the smallest member of the set of all
upper bounds of [1], [7].
Remark 2 [1] Let be a -bounded above subset
of , and
. Then the sup is given by
Similarly, for any -bounded below set , -
infimum of is defined as
Remark 3 A -module space or -module space
can be decomposed as
(2)
where and are -vector or
vector spaces. The spelling in (2) is called the
idempotent decomposition of the space [2], [7].
Definition 4 Let be a -algebra on a set . A
bicomplex-valued function defined
on is called
(i) -measure on if are complex
measures on ,
(ii) -measure on if are positive measures
on ,
(iii) -measure on if are real measures
on ,[13], [14].
Assume that is a -finite
complete measure space and are complex-
valued (real-valued) measurable functions on . The
function having idempotent decomposition
is called a -measurable function and
is called a -valued
measurable function on [13], [14].
For any -valued measurable function
, it is easy to see that
is -valued measurable. Also for any two
-valued measurable functions and it can be
easily seen that their sum and multiplication
functions are also -measurable functions [13],
[14].
More results on -topology such as -limit, -
continuity, -Cauchy and -convergence etc. can
be found in [2], [3], [4], [5], [7], [13], [14] and the
references therein.
Definition 5 Let be a -measure
and be a -measure on . Then
is said to be absolutely -continuous with
respect to , and denoted by , if is
absolutely continuous with respect to for
[14].
If for any , is concentrated on for
, then is said to be -concentrated on .
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Any two -measures
on are called mutually -singular,
and denoted by if and are mutually
singular for [14].
Theorem 6 (Lebesgue-Radon-Nikodym Theorem)
Let be a -algebra on . Let be a -finite -
measure on , and let be -measure on .
(a) There is a unique pair of -measures on
such that
where and . If is -finite
measure on then are also so.
(b) There exists a unique
such that
for all [14].
2 Main Results
Let be a -finite complete -measure
space with and indicate
the set of all -measurable, -valued functions on
.
Definition 7 [8] Let and
for any . If the set is
defined as
,
then essential supremum of , denoted by
essup or
is defined by
Definition 8 Let be an element of
. Then
, -
distribution function of , is given by
(3)
for all .
Definition 9 Let and .
Then -rearrangement of , is the function
defined by
(4)
where inf.
According to [13], since
and
, one can write
and so
(5)
2.1 -rearrangement Invariant -module
Spaces
The -Banach function space is defined as
where the norm on has the following
properties:
1. if and only if a.e. on
2. for all
3. for every with , we have
4. if is a increasing convergent
sequence and
(a.e.) on , then
5. if and (a.e.)
on then
6. for every with , there is a
constant such that
for all .
Let , be two finite
complete -measure spaces and ,
denote the linear space of all bicomplex
-measurable functions on and bicomplex
measurable functions on , respectively.
Any two functions and
are said to be -equimeasurable if they
have the same distribution function, that is, if
A function in a -Banach function space
is said to have an absolutely continuous norm if
for each sequence
converging to (a.e). We say that is a -
Banach function space with absolutely continuous
norm if each function in has absolutely
continuous norm. A -rearrangement invariant
space is a -Banach function space such that
whenever and is a -equimeasurable
function with , then and .
For details on Banach function spaces, an
interested reader can, [15].
Proposition 10 Let be a -
rearrangement invariant -Banach function space
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on a resonant measure space . Then the
associate space is also a -rearrangement
invariant -module space (under the norm )
and these norms are given by
and
where .
One can see [15] and [16] for detailed study on
rearrangement invariant spaces.
2.2 -Boundedness
Let be a -measurable transformation,
that is, for any . If
for all with , then
is said to be nonsingular. This situation says that
the measure , defined by
for is absolutely -continuous
with respect to (). Then Theorem 6
ensures the existence of a function
on such that
for all . Therefore any measurable
nonsingular transformation induces a linear
operator (which is called composition operator)
from into defined by:
The non-singularity of guarantees that the
operator is well defined as a map from
into since (-a.e.)
implies (-a.e). The study of these
operators on Lebesgue spaces has been made in
[17], [18], [19], [20], [21], [22] and references
therein. Composition operators on the Lorentz
spaces, weighted Lorentz spaces, Lorentz-Karamata
spaces were studied in [23], [24] and [25].
Theorem 11 Let and be two -rearrangement
invariant -module spaces on the resonant
measure spaces and with
the fundamental functions and , respectively.
Also, let be a non–singular measurable
transformation. Then is a -bounded
composition operator from into if and only if
(6)
for all , for some .
Proof. Suppose that the condition (6) holds. Then
Therefore, we get
and consequently
For and , by using the -decreasing
property of , we see that
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Similarly, we have:
for all . Thus is a -bounded composition
operator on and .
Conversely, let with
. Then by definition of -Banach function
space, we have and . Besides
for some and this implies that
(7)
for some Similarly, we have
(8)
for some If we multiply the inequalities (7)
and (8), we get
by [15]. Therefore, , for some
.
Consider the vector space comprising all
-valued functions on a nonempty set . Let
be a -measurable function on such
that whenever , where
and . This gives rise to
a linear transformation defined
as
where the product of functions is pointwise. If
is a topological -vector space and is -
continuous, then it is referred to as a multiplication
operator induced by . Multiplication operators
have been scrutinized on various function spaces by
[22], [24] and [26]. In line with their arguments, we
investigate multiplication operators on the -
rearrangement invariant -module space.
Proposition 12 For any -measurable function
, is a -linear operator on .
Theorem 13 The linear transformation
on the -rearrangement invariant -module
space is bounded if and only if is essentially -
bounded. Moreover,
Proof. Firstly, assume that is essentially -
bounded and
. Since
for any , we have:
for any . Then
implies
and ,
for any .
Therefore
(9)
can be written. This means is -bounded.
Conversely, suppose that is -bounded on
the -rearrangement invariant -module space.
If is not essentially -bounded, then for each
, the set
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has a -positive measure. It means there exists ,
with such that
and for all with
. Since the decreasing -rearrangement of
is
one can get that
by [10]. Now, to calculate the norm of , if
we use the following inequality
then we get
(10)
However, (10) contradicts the boundedness of .
From (9), it can be seen that
. On the
other hand, for any , let
Then
can be written for . Therefore,
for all and
for all . As a result,
and
with (9).
By this result and [24], a condition
sufficient for the -compactness of the composition
operator on can be inferred using [26].
Theorem 14 Let be a non-singular -
measurable transformation such that the Lebesgue-
Radon-Nikodym derivative
is in
and be the set of all
atoms of with
for each . Then is compact on if , are
purely atomic measures and
for .
3 Conclusion
We examined deeply the behavior and structural
features of composition operators in the setting of
-rearrangement -module spaces in the present
work. We provided information on the algebraic,
topological, and functional aspects of the underlying
space structures by providing an in-depth
understanding of the relationships between
composition operators and those.
We have proved basic conclusions about the
compactness, boundedness, and continuity of
composition operators in -module spaces with a
careful investigation. These results explain the
fundamental qualities of composition operators and
how they determine the behavior of functions in
rearrangement-invariant spaces.
Furthermore, our study of the structural features
produced by synthesis operators shows that they
preserve fundamental spatial features such as
separability, reflexivity, and completeness. This
illustrates how crucial composition operators are to
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preserving - rearrangement-invariant BC moduli
spaces' stability across a variety of operations.
Finally, our work increases the understanding of
operator theory and function spaces broadly,
especially about BC-rearrangement BC-module
spaces. Having potential applications in a wide
range of fields, such as signal processing, image
reconstruction, and mathematical physics, the
information obtained from this research offers new
opportunities for investigation and development in
this interesting field of mathematical analysis.
Acknowledgement:
The author would like to thank the referees for their
helpful comments and valuable suggestions for
improving the manuscript.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
İLKER ERYILMAZ contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solutions.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare.
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