A Mean Ergodic Theorem in Bicomplex Lebesgue Spaces
İLKER ERYILMAZ
Department of Mathematics
Ondokuz Mayıs University
Faculty of Science, 55200 Atakum-Samsun
TURKEY
Abstract: - The main result of this paper is a mean ergodic theorem, in the Von Neumann sense, for some operator
acting on the bicomplex Lebesgue space.
Key-Words: - Mean ergodic theorem, bicomplex Lebesgue space, iterates of an operator, bicomplex modules,
hyperbolic norm, bicomplex measure, bicomplex functional analysis
Received: March 9, 2023. Revised: October 7, 2023. Accepted: November 8, 2023. Published: January 26, 2024.
1 Introduction
The research on Ergodic theory began in the 1930s,
initiated in [1] and [2], and originated from applied
physics and statistical mechanics. The fundamental
problem in Ergodic theory is to study and find the nec-
essary conditions for when the sequences of Cesàro
averages n
P
j=1
Tn(·)are convergent where Twas a
mapping defined on a suitable space. The theorem
of mean ergodicity was extended for bounded linear
operators on Banach spaces in [3] and implemented
to Markoff processes in [4]. Moreover, Lorentz-
invariant Markoff processes in relativistic phase space
is studied in [5]. Thenceforward, ergodic theory
and its applications have certainly evolved in various
mathematical and statistical problems and has been
studied by many researchers. For a systematic prepa-
ration and development of ergodic theorems, we can
refer to the classic book [6], which contains rich liter-
ature in this area. In [7, Section VIII.5], the averages
of iterates of a linear operator Tis examined and dis-
cussed and then tried to throw some light upon the
problems which are occured in probability and statis-
tical mechanics. The conditions of an operator Tin an
arbitrary complex Banach space Ywere given which
are necessary and sufficient for the convergence in Y
of the averages
A(n) = 1
n
n−1
X
j=0
Tj
of the iterates of T. These general conditions have
been interpreted for operators in Lebesgue spaces
which occur in statistical mechanics and probability.
BC-valued functions arise naturally in various
mathematical fields, including probability theory,
mathematical analysis, and functional analysis, and
understanding their properties is crucial for advanc-
ing these areas of study. Indeed, the study of mod-
ules with bicomplex scalars in the context of func-
tional analysis has gained significant attention in re-
cent years. One influential work that has contributed
to this area is the book [8]. The book likely presents
groundbreaking results and insights related to this
topic. Functional analysis traditionally deals with
vector spaces over a field, such as the complex num-
bers or the real numbers. However, by considering
modules with bicomplex scalars, where the scalars are
elements of the bicomplex numbers, a broader frame-
work is introduced. This extension allows for the
exploration of new mathematical structures and the
investigation of properties beyond the classical set-
ting. The book [8] is likely a valuable resource for
researchers and enthusiasts interested in this area. It
likely presents notable results, techniques, and appli-
cations pertaining to the study of modules with bi-
complex scalars in the context of functional analy-
sis. These results may encompass various aspects of
functional analysis, such as operator theory, function
spaces, and spectral theory, among others. They may
shed light on the behavior of modules with bicomplex
scalars, reveal connections to other areas of mathe-
matics, and potentially find applications in physics,
engineering, or other disciplines.
The series of articles mentioned in the references
highlight the systematic study of topological bicom-
plex modules and various fundamental theorems re-
lated to them. Here is a breakdown of the articles and
their contributions:
In [9], the authors studied of topological bicom-
plex modules, likely exploring their topological prop-
erties and investigating concepts such as conver-
gence, continuity, and compactness in this context.
Fundamental theorems, including the principle of
uniform boundedness, open mapping theorem, inte-
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rior mapping theorem for bicomplex modules and
closed graph theorem are presented in [10].
In [11], collaboration with [10], the study of fun-
damental theorems are extended to the setting of topo-
logical bicomplex modules. The focus may be on
generalizing classical results from functional analy-
sis to the bicomplex module framework, providing a
deeper understanding of their properties. Also the au-
thors likely delve further into the study of topological
hyperbolic modules, topological bicomplex modules,
exploring the properties of linear operators, continu-
ity, and related topological concepts specific to these
settings.
The Hahn-Banach theorem for bicomplex mod-
ules and hyperbolic modules are examined in [12].
The book [13] likely provides an in-depth explo-
ration of bicomplex analysis and geometry. It may
cover a wide range of topics, including holomorphic
functions, integration, differential equations, and ge-
ometric properties specific to the bicomplex domain.
In [14], the authors focused on BC bounded linear
operators and bicomplex functional calculus. It may
provide a detailed study of operators acting on bicom-
plex modules and explore the construction and prop-
erties of functional calculi specific to the bicomplex
framework.
These references collectively represent significant
contributions to the study of bicomplex modules,
functional analysis, and related areas. They showcase
the exploration of properties, the development of new
theorems, and the application of functional analysis
techniques in the context of bicomplex numbers. Re-
searchers and readers interested in these topics can re-
fer to these articles and the books for detailed insights
into the respective areas of study.
2 Preliminaries on BC and BC
Lebesgue spaces
Now, we will give a small summary of bicomplex
numbers with some basic properties .The set bicom-
plex numbers BC which is a four-dimensional exten-
sion of the complex numbers is defined as
BC := {W=w1+jw2|w1, w2∈C(i)}
where iand jare imaginary units satisfying ij =ji,
i2=j2=−1. Here C(i)is the field of complex
numbers with the imaginary unit i. According to ring
structure: For any Z=z1+jz2, W =w1+jw2in
BC usual addition and multiplication are defined as
Z+W= (z1+w1) + j(z2+w2)
ZW = (z1w1−z2w2) + j(z2w1+z1w2).
The set BC forms a commutative ring under the usual
addition and multiplication of bicomplex numbers.
The bicomplex numbers have a unit element denoted
as 1BC := 1 and this acts as the identity for mul-
tiplication, such that for any bicomplex number W,
1W=W1 = W. In the sense of module structure,
the set BC is a module over itself. This means that BC
satisfies the properties of a module, including scalar
multiplication and distributivity. The product of the
imaginary units iand jbring out a hyperbolic unit k,
such that k2= 1. This implies that kis a square root
of 1 and is distinct from iand j. The product opera-
tion of all units i, j, and kin the bicomplex numbers
is commutative. Specifically, the following relations
hold:
ij =k, jk =−iand ik =−j.
These properties summarize the basic characteristics
of bicomplex numbers and their algebraic structure.
Hyperbolic numbers Dare a two-dimensional ex-
tension 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 α=
β1+kβ2, where β1and β2are real numbers, and kis
the hyperbolic unit. In the hyperbolic number system,
for any two hyperbolic numbers α=β1+kβ2and
γ=δ1+kδ2, addition and multiplication are defined
as follows:
α+γ= (β1+δ1) + k(β2+δ2)
αγ = (β1δ1+β2δ2) + k(β1δ2+β2δ1).
The hyperbolic numbers form a ring, however, un-
like the complex numbers, the hyperbolic numbers do
not have a multiplicative inverse for all nonzero el-
ements. The nonzero hyperbolic numbers that have
multiplicative inverses are called units. The bicom-
plex numbers contain two imaginary units iand j, and
the hyperbolic numbers can be taken as a subset of the
bicomplex numbers by restricting the imaginary part
of jto be zero.
Let W=w1+jw2∈BC where w1, w2∈C(i).
By the notation of Wwith imaginary units iand j,
the conjugations are formed for bicomplex numbers
in [8], [13] as W1=w1+jw2,W2=w1−jw2
and W3=w1−jw2where w1and w2are the
usual complex conjugates of w1, w2∈C(i). For
any bicomplex number W, they also wrote the three
moduli of Win [8], [13] and [15]. Furthermore,
BC is a normed space with the norm kWkBC =
p|w1|2+|w2|2for any W=w1+jw2in BC. Ac-
cording to this, kW1W2kBC ≤√2kW1kBC kW2kBC
for every W1, W2∈BC, and finally BC is a quasi-
Banach algebra [8].
If the hyperbolic numbers e1and e2defined as
e1=1 + k
2and e2=1−k
2,
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then it is easy to see that the set {e1, e2}is a funda-
mental set in C(i)-vector space BC and linearly inde-
pendent. The set {e1, e2}also satisfies the following
properties:
e2
1=e1, e2
2=e2,(e1)3=e1,(e2)3=e2,
e1+e2= 1, e1·e2= 0
with ke1kBC =ke2kBC =√2
2. By using this linearly
independent set {e1, e2}, any W=w1+jw2∈BC
can be written as a linear combination of e1and e2
uniquely. That is, W=w1+jw2can be written as
W=w1+jw2=e1z1+e2z2(1)
where z1=w1−iw2and z2=w1+iw2[8]. Here z1
and z2are elements of C(i)and the formula in (1) is
called the idempotent representation of the bicomplex
number W.
Besides the Euclidean-type norm k·kBC, another
norm named with (D-valued) hyperbolic-valued norm
|W|kof any bicomplex number W=e1z1+e2z2is
defined as
|W|k=e1|z1|+e2|z2|.
For any hyperbolic number α=β1+kβ2∈D, an
idempotent representation can also be written as
α=e1α1+e2α2
where α1=β1+β2and α2=β1−β2are real
numbers. If α1>0and α2>0for any α=
β1+kβ2∈D, then we say that αis called a posi-
tive hyperbolic number. Thus, the set of non-negative
hyperbolic numbers D+∪ {0}can be defined by
D+∪ {0}=β1+kβ2:β2
1−β2
2≥0, β1≥0
={e1α1+e2α2:α1, α2≥0}.
Now, let αand γbe any two elements of D. In [8],
[12] and [13], a relation is defined on Dby
αγ⇔γ−α∈D+∪ {0}.
It is showed in [8] that this relation ”” has reflexive,
anti-symmetric and transitive properties. Therefore
”” defines a partial order on D. If idempotent rep-
resentations of the hyperbolic numbers αand γare
written as α=e1α1+e2α2and γ=e1γ1+e2γ2,
then αγimplies that α1≤γ1and α2≤γ2. By
α≺γ, we mean α1< γ1and α2< γ2. For more de-
tails on hyperbolic numbers Dand partial order ””,
one can refer to [8, Section 1.5], [13] and [15].
Definition 1 Let Abe a subset of D.Ais called a D-
bounded above set if there is a hyperbolic number δ
such that δαfor all α∈A. If A⊂Dis D-bounded
from above, then the D-supremum of Ais defined as
the smallest member of the set of all upper bounds of
A[13].
In other words, the hyperbolic number λ=e1λ1+
e2λ2, where λ1and λ2are real numbers, is the D-
supremum of Aif
(1) e1α1+e2α2e1λ1+e2λ2for each α=e1α1+
e2α2∈A
(2) For any ε=e1ε1+e2ε20, there exists
θ=e1θ1+e2θ2∈Asuch that e1θ1+e2θ2
e1(λ1−ε1) + e2(λ2−ε2)are satisfied.
Remark 2 Let Abe a D-bounded above subset of
Dand A1:= {λ1:e1λ1+e2λ2∈A},A2:=
{λ2:e1λ1+e2λ2∈A}. Then the supDAis given by
supDA:= e1sup A1+e2sup A2.
Similarly, for any D-bounded below set A,D-infimum
of Ais defined as
infDA=e1inf A1+e2inf A2
where A1and A2are as above [8, Remark 1.5.2].
Definition 3 ABC-module (X, +,·), where (X, +)
is an abelian group, is called a topological BC-
module, if there is a topology τXin X, so that the
operations + : X×X→Xand ·:BC ×X→X
are continuous.
The following result is known from [11].
Remark 4 ABC-module space or D-module space
Ycan be decomposed as
Y=e1Y1+e2Y2(2)
where Y1=e1Yand Y2=e2Yare R-vector or
C(i)−vector spaces. The spelling in (2) is called as
the idempotent decomposition of the space Y. There-
fore, any element yin Ycan be uniquely inscribed as
y=e1y1+e2y2with y1∈Y1and y2∈Y2.
Definition 5 Let Mbe a σ-algebra on a set Ω. A
bicomplex-valued function µ=µ1e1+µ2e2de-
fined on Ωis called a BC-measure on Mif µ1, µ2
are complex measures on M. In particular if µ1, µ2
are positive measures on Mi.e range of both µ1, µ2
are [0,∞]then µis called a D-measure on Mand
if µ1, µ2are real measures on Mi.e range of both
µ1, µ2are [0,∞)then µis called a D+-measure on
M[16].
Assume that Ω = (Ω,M, µ)is a σ-finite complete
measure space and f1, f2are complex-valued (real-
valued) measurable functions on Ω. The function hav-
ing idempotent decomposition f=f1e1+f2e2
is called as a BC-measurable function and |f|k=
|f1|e1+|f2|e2is called a D-valued measurable func-
tion on Ω[17]. Thus for any given complex valued
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function space (F(Ω) ,k·kΩ), one can create a BC-
valued function space (F(Ω,BC),k·kBC)by com-
bining all f1,f2and bringing out functions of the type
f=f1e1+f2e2where f1and f2are in (F(Ω) ,k·kΩ)
with kfk2
BC =1
2kf1k2
Ω+kf2k2
Ω. Similar defini-
tion can be given for any hyperbolic measurable func-
tion.
For any BC-valued measurable function f=
f1e1+f2e2, it is easy to see that |f|k=|f1|e1+
|f2|e2is D-valued measurable. Because if f=
f1e1+f2e2is a BC-valued measurable function, then
f1and f2are C-measurable functions. Therefore
real and imaginary parts of f1and f2are R-valued
measurable and so does |f1|and |f2|. As a result,
|f|kis D-measurable. Also for any two BC-valued
measurable functions fand g, it can be easily seen
that their sum and multiplication functions are also
BC-measurable functions [16], [17]. More results on
D-topology such as D-limit, D-continuity, D-Cauchy
and D-convergence etc. can be found in [16], [18],
[19] and the references therein.
Definition 6 Let Mbe a σ-algebra on a set Ωand
µ=e1µ1+e2µ2be a BC-measure on M. Then
two bicomplex valued BC-measurable functions f=
e1f1+e2f2and g=e1g1+e2g2on Ωare called to
be equal (µ-a.e.) if f1=g1(µ1-a.e.) and f2=g2
(µ2-a.e.).
Definition 7 Let µ=e1µ1+e2µ2be a D-measure on
an arbitrary measure space (Ω,M)and 1≤p < ∞.
Suppose Lp(Ω, µ1)and Lp(Ω, µ2)denote the lin-
ear space of all equivalence classes of complex val-
ued,measurable functions f1and f2on Ωwith
Z
Ω
|f1(x)|pdµ1<∞and Z
Ω
|f2(x)|pdµ2<∞.
Then Lp
BC (Ω,M, µ) = Lp
BC (µ)consists of all bicom-
plex valued, bicomplex measurable functions (equiv-
alence classes) f=e1f1+e2f2on Ωsuch that
f1∈Lp(Ω, µ1)and f2∈Lp(Ω, µ2)[19].
Proposition 8 For 1≤p < ∞,Lp
BC (µ)is a BC-
module under usual addition operation in functions
and bicomplex scalar multiplication [19].
Let 1≤p < ∞. By using Definition 3 and Re-
mark 4, we may write an idempotent decomposition
Lp
BC (µ) = e1Lp(µ1) + e2Lp(µ2)
for Lp
BC (µ)where Lp(µ1)and Lp(µ2)are usual
Lebesgue spaces [19]. Therefore a hyperbolic (D-
valued) norm can be defined on the BC-module
Lp
BC (µ)with
kfkp,D=e1kf1kp+e2kf2kp
for any e1f1+e2f2=f∈Lp
BC (µ).
Proposition 9 Let 1≤p < ∞. The space
Lp
BC (µ),k·kp,Dis a bicomplex Banach module
[19].
3 Mean Ergodic Theorem
In [20], the mean ergodic theorem in bicomplex Ba-
nach modules is studied. Also, a result on ergodic-
ity is given for bounded bicomplex strongly contin-
uous semigroups in bicomplex Banach modules. In
this section, we will prove a mean ergodic theorem, in
the Von Neumann sense, which can be written for av-
erages of iterates of an operator Tacting on Lp
BC (µ)
where 1< p < ∞.
Proposition 10 Let 1≤p < ∞. The set
S={s=s1e1+s2e2|s1, s2∈S}
is D−dense in Lp
BC (µ)where Sis the set of simple
functions.
Proof. Let ε=e1ε1+e2ε20and f=e1f1+
e2f2be any element of Lp
BC (µ). By the definition of
Lp
BC (µ), the functions f1and f2belong to Lp(µ1)
and Lp(µ2). Since the set of simple functions Sis
dense in Lp(µ1)and Lp(µ2), then there exist simple
functions h1and h2such that
kf1−h1kp< ε1and kf2−h2kp< ε2.
If one call e1h1+e2h2as h, then h∈Sand
kf−hkp,D≺ε. This means that Sis D−dense in
Lp
BC (µ).
Lemma 11 Let (X, M, ϑ)be a finite positive mea-
sure space, ℵ 6= 0 be a complex Banach space and
ϕbe a map of Xinto itself which satisfies the follow-
ing conditions:
(i)ϕ−1(E)∈Mfor all E∈M(3)
(ii)If ϑ(E)=0then ϑϕ−1(E)= 0.
Then for every function ufrom Xto ℵthe following
operator Tdefined as
T(u) (·) = u(ϕ(·)) (4)
maps measurable functions into measurable functions
and ϑ-equivalent functions into ϑ-equivalent func-
tions. Furthermore Tis a continuous linear map of
the space of all ℵ-valued ϑ−measurable functions
into itself.
Proof. See [7, VIII.5.6, Lemma 6]
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Lemma 12 Let (X, M, µ)be a finite positive mea-
sure space, Y6= 0 be a bicomplex Banach space. As-
sume that ϕis a map of Xinto itself which satisfies
(3). Then for any p > 1, the linear operator Tdefined
in the bicomplex linear space YXof all functions on
Xinto Yby
T u (x) = u(ϕ(x)) , x ∈X, u ∈YX(5)
maps Lp
BC (µ)into itself if and only if there exists
M=M1e1+M2e20such that
M=supD
E∈M
µϕ−1(E)
µ(E)(6)
=supD
E∈M
µ1ϕ−1(E)e1+µ2ϕ−1(E)e2
µ1(E)e1+µ2(E)e2
.
Furthermore, when this condition is satisfied Tis
aD−continuous D−linear map on Lp
BC (µ)and
kTkBC =M
1
p.
Proof. Let µ(·) = µ1(·)e1+µ2(·)e2be a D+−mea-
sure. If µ(E) = µ1(E)e1+µ2(E)e2= 0 for
any E∈M, then Mwill be taken zero. Now as-
sume that µ(E) = µ1(E)e1+µ2(E)e20. If u
is a BC-measurable function and ϕis defined as in
(3), then is easy to see that T u is BC-measurable
by (5). Firstly suppose that Tmaps Lp
BC (µ)into
itself. It will be shown that Tis D−closed and
hence D−continuous by [10]. Since Tis defined on
Lp
BC (µ), then it maps µ-equivalent functions into µ-
equivalent functions and also BC−measurable func-
tions into BC−measurable functions. Now let α6= 0
be a fixed vector in Yand let Ebe a µ-null set in M.
Then µ(E) = µ1(E)e1+µ2(E)e2= 0,χE= 0
(a.e.) and
T(αχE) = αχϕ−1(E)
by the definition of T. Also, linearity of Tim-
plies that µϕ−1(E)= 0. This means that ϕis
a measure-preserving map of Xinto itself and satis-
fies (3). Since Sis D−dense in Lp
BC (µ), for any u=
u1e1+u2e2∈Lp
BC (µ)a sequence of simple functions
(un) = u(1)
ne1+u(2)
ne2⊂Scan be formed
such that kun−ukp,D
D
→0, i.e.
u(1)
n−u1
p→0
and
u(2)
n−u2
p→0. This convergence implies
convergence in µ−measure and so the graph of Tis
closed. Therefore Tis D−bounded and D−continu-
ous on Lp
BC (µ)by Closed graph theorem [11, Theo-
rem 5.5]. On the other hand, for any 06=α∈Yand
E∈M, we have
kT(αχE)kp,D=kT(α1χE)kpe1+kT(α2χE)kpe2=
=
Z
Xα1χϕ−1(E)(x)
pdµ1
1
p
e1
+
Z
Xα2χϕ−1(E)(x)
pdµ2
1
p
e2
=|α1|µ1ϕ−1(E)1
pe1
+|α2|µ2ϕ−1(E)1
pe2
=|α|kµϕ−1(E)1
p
by [18, Definition 2.2]. Therefore, one can get that
|α|kµϕ−1(E)1
p=kT(αχE)kp,D
|α|kkTkBC kχEkp,D
=|α|kkTkBC µ1(E)
1
pe1
+|α|kkTkBC µ2(E)
1
pe2
=|α|kkTkBC µ(E)
1
p
which means M kTkp
BC.
Conversely, let s=s1e1+s2e2be
aµ-integrable function in Shaving values
β(1)
1e1+β(2)
1e2, β(1)
2e1+β(2)
2e2, . . . β(1)
ne1+β(2)
ne2
on the disjoint sets E1, E2, . . . , Enof D−pos-
itive measure. Then T s has the values
β(1)
1e1+β(2)
1e2, β(1)
2e1+β(2)
2e2, . . . β(1)
ne1+β(2)
ne2
on the sets ϕ−1(E1), ϕ−1(E2), . . . , ϕ−1(En).
Since the family {E1, E2, . . . , En}is a decompo-
sition of X, property (3) of ϕimplies that the family
{ϕ−1(E1), ϕ−1(E2), . . . , ϕ−1(En)}is also a de-
composition of X. Therefore, if we use the nota-
tion n
P
i=1 β(1)
ie1+β(2)
ie2χEifor swhere χEi(·) =
e1χEi(·) + e2χEi(·), then
T s (x) = s(ϕ(x))
=
n
X
i=1 β(1)
ie1+β(2)
ie2χEi(ϕ(x))
=
n
X
i=1 β(1)
ie1+β(2)
ie2χϕ−1(Ei)(x)
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and
kT skp,D=
Z
X
|T s1(x)|pdµ1
1
p
e1
+
Z
X
|T s2(x)|pdµ2
1
p
e2
=
Z
X
n
X
i=1
β(1)
iχϕ−1(Ei)
p
dµ1
1
p
e1
+
Z
X
n
X
i=1
β(2)
iχϕ−1(Ei)
p
dµ2
1
p
e2
Z
X
n
X
i=1 β(1)
iχϕ−1(Ei)
pdµ1
1
p
e1
+
Z
X
n
X
i=1 β(2)
iχϕ−1(Ei)
pdµ2
1
p
e2
can be written. Since the elements of the family
ϕ−1(E1), ϕ−1(E2), . . . , ϕ−1(Ek)are disjoint,
kT skp,D
Z
∪k
i=1 ϕ−1(Ei)
n
X
i=1 β(1)
iχϕ−1(Ei)
pdµ1
1
p
e1
+
Z
∪k
i=1 ϕ−1(Ei)
n
X
i=1 β(2)
iχϕ−1(Ei)
pdµ2
1
p
e2
=
n
X
i=1 Z
ϕ−1(Ei)β(1)
i
pdµ1
1
p
e1
+
n
X
i=1 Z
ϕ−1(Ei)β(2)
i
pdµ2
1
p
e2
= n
X
i=1 β(1)
i
pµ1ϕ−1(Ei)!1
p
e1
+ n
X
i=1 β(2)
i
pµ2ϕ−1(Ei)!1
p
e2
n
X
i=1 β(1)
i
pM1·µ1((Ei))!1
p
e1
+ n
X
i=1 β(2)
i
pM2·µ2((Ei))!1
p
e2
=M
1
pkskp,D
is found.
Since the µ-integrable simple functions Sare
dense in Lp
BC (µ)and Tis a D−continuous opera-
tor acting on a dense subset of Lp
BC (µ), we can say
that Tpossesses a unique D−bounded, D−continu-
ous extension e
Tdefined on all of Lp
BC (µ)with norm
e
T
BC M
1
p. Furthermore, by the definition of M,
M=supD
E∈M
µϕ−1(E)
µ(E)
=supD
E∈M
µ1ϕ−1(E)e1+µ2ϕ−1(E)e2
µ1(E)e1+µ2(E)e2
=supD
E∈M
χϕ−1(E)
p
pe1+
χϕ−1(E)
p
pe2
kχEkp
pe1+kχEkp
pe2
=supD
E∈M
kT χEkp
p,D
kχEkp
p,D
kTkp
BC
can be found. Therefore kTkBC =M
1
p.
Proposition 13 Assume that (X, M, µ)be a finite
positive measure space and let ϕbe a mapping of X
into itself with ϕ−1(M)⊂M. Also, suppose that
there is a D−constant Mfor which
1
n
n−1
X
j=0
µϕ−j(E)Mµ (E)(7)
for all n∈Nand E∈M. Then, for every p∈
(1,∞), the operator Tdefined in (5) maps Lp
BC (µ)
into itself. Also the averages A(n) = 1
n
n−1
P
j=0
Tjas op-
erators acting on Lp
BC (µ)are uniformly D−bounded.
Proof. If we write n= 2 in (7), then we get
µϕ−1(E)(2M−1) µ(E)for any E∈M.
Therefore Tis bounded by (6). This inequality also
shows that µϕ−1(E)= 0 whenever µ(E)=0.
Now let E∈Mbe any set, n∈Nand j=
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1,2, . . . , n −1. Since
Tj(χE) (·) = Tj−1(T(χE)) (·) = Tj−1(χE(ϕ)) (·)
=Tj−1χϕ−1(E)(·)
=Tj−2Tχϕ−1(E)(·)
=Tj−2χϕ−2(E)(·) = ···
we can conclude that Tj(χE) (·) = χϕ−j(E)(·).
Thus, for any simple function
s(·) =
m
X
i=1 β(1)
ie1+β(2)
ie2χEi(·),
we have
A(n) (s) (·) = 1
n
n−1
X
j=0
Tj(s) (·)
=1
n
n−1
X
j=0
Tj m
X
i=1 β(1)
ie1+β(2)
ie2χEi!(·)
=e1
m
X
i=1
β(1)
i
1
n
n−1
X
j=0
TjχEi
(·)
+e2
m
X
i=1
β(2)
i
1
n
n−1
X
j=0
TjχEi
(·)
=e1
m
X
i=1
β(1)
i
1
n
n−1
X
j=0
χϕ−j(Ei)
(·)
+e2
m
X
i=1
β(2)
i
1
n
n−1
X
j=0
χϕ−j(Ei)
(·)
=
m
X
i=1 β(1)
ie1+β(2)
ie2
1
n
n−1
X
j=0
χϕ−j(Ei)
(·)
by (8). Hence
kA(n) (s)kp,D=
=
m
X
i=1 β(1)
ie1+β(2)
ie2
1
n
n−1
X
j=0
χϕ−j(Ei)
p,D
=
m
X
i=1
β(1)
i
1
n
n−1
X
j=0
χϕ−j(Ei)
p
e1
+
m
X
i=1
β(2)
i
1
n
n−1
X
j=0
χϕ−j(Ei)
p
e2
m
X
i=1 β(1)
i
1
n
n−1
X
j=0
χϕ−j(Ei)
p
e1
+
m
X
i=1 β(2)
i
1
n
n−1
X
j=0
χϕ−j(Ei)
p
e2
=
m
X
i=1 |βi|k
1
n
n−1
X
j=0
χϕ−j(Ei)
p,D
m
X
i=1 |βi|k
1
n
n−1
X
j=0
χϕ−j(Ei)
p,D
=
m
X
i=1 |βi|k
1
n
n−1
X
j=0
µ1ϕ−j(Ei)e1+µ2ϕ−j(Ei)e2
m
X
i=1 |βi|kM(µ1(Ei)e1+µ2(Ei)e2)
=Mkskp,D
for all s∈S. By using the D−density of Sin Lp
BC (µ)
and Tietze extension theorem, kA(n)kBCMfor
any n∈N. The averages of iterates, namely
A(n), are uniformly bounded as operators acting on
Lp
BC (µ).
Theorem 14 (Mean Ergodic Theorem) Assume that
(X, M, µ)is a finite positive measure space and ϕis
a mapping of Xinto itself which satisfies ϕ−1(M)⊂
M. If the inequality
1
n
n−1
X
j=0
µϕ−j(E)Mµ (E)(8)
is satisfied for all n∈Nand E∈M, then for every
p∈(1,∞), the operator Tdefined by the equation
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(5) is a D−continuous linear map on Lp
BC (µ)and the
sequence of averages A(n), as operators acting on
Lp
BC (µ), is strongly D−convergent. Here Mis inde-
pendent of E,nand ϕ0(E) = E.
Proof. With (8), it can be written that
1
2µϕ0(E)+µϕ−1(E)Mµ (E)
for any E∈M. Therefore the linear operator Tde-
fined by the equation (5) is a D−bounded and D−con-
tinuous map on Lp
BC (µ)by Lemma 12. If we denote
the space of all D−linear and D−continuous opera-
tors on Lp
BC (µ)by B(Lp
BC (µ)), then it can be eas-
ily seen that A(n), the averages, are in this complete
space. Since the averages A(n)are D−uniformly
bounded while operating on Lp
BC (µ), we can write
that the sequence {A(n)f} ⊂ Lp
BC (µ)D−converges
for all f∈Lp
BC (µ)by Riesz–Thorin convexity theo-
rem. By the way, when the averages A(n)are operat-
ing on Lp
BC (µ), we obtained that A(n)f∈Lp
BC (µ)
for all n∈Nand for each f∈Lp
BC (µ). It is
known that the characteristic functions of elements of
Mform a fundamental set for Lp
BC (µ). Then, for any
E∈Mand x∈X, since we have |χE|k1and
Tn(χE) (x)
nk
=1
n|χE(ϕn) (x)|k
=1
n(e1|χE(ϕn) (x)|+e2|χE(ϕn) (x)|)
1
n→0,
we can say that the sequence {A(n)}D−converges
in strongly operator topology by [7, VIII.5.1].
Remark 15 it should be observed that a measure
preserving transformation ϕ(i.e. one for which
µϕ−1(E)=µ(E), for every Ein σ−algebra ) sat-
isfies the hypothesis of the preceding theorem. This
type of maps arise in the study of conservative me-
chanical systems. Also if the map ϕis metrically tran-
sitive (i.e. µE∆ϕ−1(E)= 0 implies µ(E)=0
or µ(X−E) = 0) then ϕis completely dissipative
as in [21]. So by [21], ϕadmits a σ−finite invari-
ant measure υ≈µ. Then ϕbecame a υ−measure
preserving transformation.
Acknowledgment:
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.
Conflicts of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this
article.
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