Product of Composition and Multiplication Operator

(1)

PRODUCT OF COMPOSITION AND MULTIPLICATION

OPERATOR

A Thesis Submitted by

Sharata Charan Gardia 413MA2059

under the supervision of

Prof. Seshadev Pradhan Department of Mathematics

A thesis presented for the degree of Master of Science

Department of Mathematics National Institute of Technology

Rourkela, India

May, 2015

(2)

Department of Mathematics

NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA

DECLARATION

I hereby declare that the project report entitled“Product of Multiplication and Composition operator” submission is of my own work on the concerned topics and to the best of my knowledge and belief. It includes no materials which is previously published or written by another person nor material which to be substantial extent has been excepted for the award of any other degree or other institute of higher learning.

place: Rourkela Sharata Charan Gardia

Date: 10.05.2015 Roll No.413MA2059

(3)

Department of Mathematics

ACKNOWLEDGEMENT

It is my great pleasure to express my heart-felt gratitude to all those who helped and encourage me at various stage during the running of the project work entitled“Product of Composition and Multiplication operator”.

I am indebted to my guide Professor Shesadev Pradhan for his invaluable guidance and constant support with explaining the mistakes with great patience.

His encouragement has always comforted me throughout. I would like to thanks my friends in NIT, Rourkela and its outside to whom I am in contact for their love and friendship.

Finally, to my family I will be always feel proud for their care.

NIT, Rourkela May, 2015

Sharata Charan Gardia

(4)

Department of Mathematics

CERTIFICATE

This is to certify that the project report entitled, “Product of Multiplication and Composition operator” is the bonafied work carried out by Sharata Cha- ran Gardia, student of M.Sc Mathematics at National Institute of Technology, Rourkela, during the year 2014-15, in partial fulfillment of the requirements for the award of degree of Master of Science under the guidance of Prof. Shesadev Pradhan, Professor, National Institute of Technology, Rourkela and that the project is a review work by the student by the collection of papers by various sources.

(Shesadev Pradhan) Professor DEPARTMENT OF MATHEMATICS NIT, ROURKELA

(5)

ABSTRACT

The composition operator CT and the multiplication operator Mu together multiplied to give a new operator named as Weighted composition operator Wu,T = CTMu given by f 7→ u◦T.f ◦T, on the Orlicz space. The thesis discusses various properties of the operatorW_u,T starting with basic necessary preliminaries.

(7)

1 Linear Operators-Definition and Examples

LetX andY be an arbitrary set then a mapping i.e. f:X →Y is a rule which send each element of to an unique element ofY i.e.

f:X →Y :x7→y=f(x)

Now instead of choosing simply a set, takeX andY to be vector spaces. X and Y both are the algebraic structure with suitable addition(+) and multiplication(.) Then the correspondingf we denote byT and we call this an operator i.e.

T:X →Y

Note : operators are basically extension of the function.

Definition 1.1 (linear operator). A linear operator T is an operator such that

(i) the domainD(T) ofT is a vector space and the rangeR(T) lies in a vector space over the same field.

(ii) for allx, y∈D(T) and the scalarsα, β

T(x+y) =T x+T y T(αx) =αT x

and combining above two, T(αx+βy) =αT x+βT y.

•Basically the operator T which preserve the operation addition as well as scalar multiplication is said to be a linear operator whereD(T) is always a vector space andR(T) lies in a vector space. In factR(T) is also a vector space.

Notation Definition D(T) Domain ofT R(T) Range ofT N(T) Null space ofT

1.1 Bijective property of Linear operator

A linear operatorT: D(T)→R(T)⊂Y

(8)

is said to be one-one if

T(x1) =T(x2)

⇒x1 =x2 x1, x2∈D(T) T: D(T)⊂X →R(T)⊂Y is said to be onto if

∀y∈R(T)∃x∈D(T) s.t. y=T x

1.2 Examples

1.2.1 Identity operator

Ix:X →X s.t Ix(x) =x,∀x∈X Ix(αx+βy) =αx+βy=αIx(x) +βIx(y) so it is a linear operator.

1.2.2 Zero Operator

O:X →Y :x7→0 Ox= 0 So, zero operator is a linear operator.

1.2.3 Differentiation or Differential Operator LetX be the vector space of all polynomial defined on [a,b].

x(t) =a₀+a₁t+a₂t²+· · ·+a_ntⁿ, n≥1 DefineT: X→X by

T x(t) = d dtx(t) Also we can define

T:C¹[a, b]→C[a, b] by T :f 7→f⁰

Note: As polynomials are well differentiable, so the operator T is well defined.

But if we will replaceX byC[a, b], then the operatorT is not well defined as

(9)

there are function which are continuous but not differentiable. That is why we took the domain as the vector space of all polynomial defined on [a,b].

So,

T x(t) = _dt^dx(t) =x⁰(t) is again a polynomial.

Again

T(αx+βy) = (αx(t) +βy(t))⁰=α(x⁰(t)) +β(y⁰(t)) =αT x(t) +βT y(t) So, T becomes linear.

1.2.4 Integral operator

LetX =C[a, b] be set of all continuous function defined on the interval [a, b].

Then the operator

T:X →X defined by

T(x(t)) = Z b

a

x(τ)dτ is well defined.

considering

T(x(t)) =tx(t), t∈[a, b]

T is linear as

T(αx+βy) =t[αx(t) +βy(t)] =αtx(t) +βty(t) =αT x+βT y Hence,T is linear.

2 Bounded Linear Operator

Definition 2.1 (Norm). A norm on a field i.e. real or complex vector space X is a real valued function on X whose value at anx∈X is denoted bykxk which satisfy the following properties:

1. kxk ≥0

2. kxk= 0⇔x= 0 3. kαxk=|α|kxk

(10)

4. kx+yk ≤ kxk+kyk Then (X,k.k) is a normed space.

Definition 2.2. (Normed space)

A normed space is a vector space with a norm defined on it.

Let (X,k.kx) and (Y,k.ky) be normed spaces andT :D(T)⊂X→Y be a linear operator. Then we say; the operatorT is bounded, if there is a real numberc >0 such that∀x∈D(T),

kT xk ≤c.kxk (1)

LetB(X, Y) be the set of all bounded linear operator fromX to Y. ForT ∈B(X, Y) norm ofT is defined by

kTk=sup{kT(x)k:kxk ≤1}

kT xk ≤ckxk gives the relation between the norm and the operator also.

What should be the minimum value of c here so that (1) holds ?

kT xk

kxk < c, true∀x6= 0∈D(T) so, sup^{kT xk}_kxk is the minimum value for c.

so letkTk= sup^{kT xk}_kxk

This minimum value for c, we call it as Normof the bounded linear operator T. SokT xk ≤ kTkkxkand IfT = 0,thenkTk= 0.

2.1 Examples

1. Identity operator I :X →X.

2. Zero operator 0 :X→X.

3. Differential operator 4. Integral operator

T:C[0,1]→C[0,1]

:x7→y=T x wherey(t) =Rb

ak(t, τ)x(τ)dτ.wherekis a given function which is called the kernel ofT. Assumingkis continuous on the closed regionJ×J int−τ plane.

(11)

Now

T: (C[0,1],k.k_∞)

whereC[0,1] is a normed space with respect to the normedk.k_∞, where kxk= sup

0≤t≤1

|x(t)|

T is bounded as

kT xk=kyk=max|y(t)|, t∈J

=max| Z 1

0

k(t, τ)x(τ)dτ|

≤max Z 1

0

|k(t, τ)|.|x(τ)|dτ

Since k is a continuous on closed regionJ×J, so kis bounded. So,

|k(t, τ)| ≤M ∀(t, τ)∈J×J.

Hence,kT xk ≤Mkxk

⇒T is bounded operator.

(12)

3 Banach Space

Definition 3.1 (Norm). A norm on a field i.e. real or complex vector space X is a real valued function on X whose value at anx∈X is denoted bykxk which satisfy the following properties:

1. kxk ≥0

2. kxk= 0⇔x= 0 3. kαxk=|α|kxk 4. kx+yk ≤ kxk+kyk Then (X,k.k) is a normed space.

Definition 3.2. (Normed space)

A normed space is a vector space with a norm defined on it.

ABanach spaceis a complete normed space(complete in the norm defined by norm).

Theorem 3.1. LetY be a subspace of a Banach spaceX. ThenY is closed if and only ifY is complete(i.e the concept of converging Cauchy sequence).

Proof. LetY be closed subspace in X. Let{xn}be a Cauchy sequence inY, then{xn} ∈X.

SinceX is complete,

∃ somex∈X s.t. {xn} →x Now, every neighborhood ofxwill contain some points inY.

Take{xn} 6=xwith n very large which implies thatxis an accumulation point ofY and sinceY is closed(assumption),x∈Y.

HenceY is complete.

Conversely,

LetY be complete andxbe a limit point ofY.

⇒ ∃ a sequence{xn} ⊂Y, s.t. {xn} →xforn→ ∞

And we know that every convergent sequence is a Cauchy sequence.

NowY is complete. So{xn}converges inY and hencex∈Y. ThusY is closed.

(13)

4 Product of composition and multiplication operators

Definition 4.1. (Lebesgue outer measure)

Lebesgue outer measure or the simply outer measure of a set is given by m^?(A) =infP

l(In),where the infimum is taken over all finite or countable collection of intervals [In] such thatA⊆ ∪In.

Some well-known properties are as follows.

1. m^?(A)≥0, 2. m^?(φ) = 0,

3. m^?(A)≤m^?(B) ifA⊆B Definition 4.2. (measurable sets)

A setX is said to be measurable if for each set A we have m^?(A) =m^?(A∩X) +m^?(A∩X^c).

Definition 4.3. (σ-algebra)

A collection of subsets of any arbitrary spaceX is said to be aσ−algebraif X belongs to the collection and the collection is closed under formation of countable unions and of complements.

Mathematically,

A collectionB of subsets of the space X is called aσ−algebraif 1. φ∈B

2. ifE∈B, then it’s complimentX−E∈B.

3. ifEn∈B, n=1,2,3,... is a countable sequence of sets inB then S∞

n=1En ∈B.

Examples:

1. B={φ, X}(trivialσ-algebra) 2. B = power set ofX (fullσ-algebra) Definition 4.4. (Measure)

A functionµdefined by µ:B →R⁺∪ {∞} is said to be a measure if 1. µ(φ) = 0.

(14)

2. ifEn is countable collection of sets and are pairwise disjoint inB i.e E_n∩E_n=φforn6=m, thenµ(S∞

n=1E_n) =P∞

n=1µ(E_n).

Definition 4.5. (Measurable space)

LetX be any subspace andB is a σ-algebra ofX. Then the pair (X, B) is called a measurable space.

Definition 4.6. (Measure space)

LetX be any subspace andB is a σ-algebra ofX. Then (X, B, µ) is called a measure space, whereµis a measure on measurable space (X, B).

Definition 4.7. (Convex function)

A continuous functionf:R→Ris said to be a convex function if for any two arbitrary pointsx1andx2 inR(on the graph off(x)),

f(tx1+ (1−t)x2)≤tf(x1) + (1−t)f(x2) Examples:

1. f(x) =x² 2. f(x) =e^x

Definition 4.8. (Measurable function)

Letϕbe a function defined byϕ:X →Y, where (X, E) and (Y, F) are the measurable space. Thenϕis said to be measurable if ϕ⁻¹(F)⊂E.

Now we write Ω for the space of consideration.

Definition 4.9. (Orlicz function)

Let (Ω, B, µ) be a finite measure space and letϕbe a continuous convex function defined byϕ: [0,∞)→[0,∞) such that

1. ϕ(x) = 0 if and only if x= 0 2. ϕ(x)→ ∞asx→ ∞

The functionϕis known as an Orlicz function.

Definition 4.10. (Orlicz space)

The Orlicz space denoted byL_ϕ,Ω,B,µor L_ϕconsist of all complex-valued measurable functionf on Ω such that for somek >0,

R

Ωϕ(k|f(ω)|)dµ <∞, ω >0

(15)

We simply denote the Orlicz space byLϕ. An atom of a measure space (Ω, B, µ) is an elementA∈B withµ(A)>0 such that for eachS∈B, if S⊂A, then eitherµ(S) = 0 orµ(S) =µ(A).And the measure space having no atom is called a non-atomic measure space.

Definition 4.11. (Measurable Transformation)

Let (Ω₁, B₁) and (Ω₂, B₂) be two measurable space. A mapping or transformation T defined by

T: Ω₁→Ω₂

is said to be measurable if for any measurable setA∈B2, the inverse image will be inB₂.

i.e. T⁻¹(A) ={ω1: T(ω1)∈A} ⊂B1

Definition 4.12. (Non-singular measurable transformation)

SupposeT: Ω1→Ω2 is a measurable transformation which satisfies whenever µ(A) = 0⇒µ(T−1(A)) = 0 forA∈B. Then T is called a non-singular measurable transformation.

If T is non-singular, then the measureµT⁻¹ which is given by

(µT⁻¹)(B) =µ(T⁻¹(B)) for A∈B,is absolutely continuous with respect to the measureµ.

4.1 Radon-Nikodym Theorem

The following theorem is important which we will use frequently in composition and multiplication operator.

Theorem

Let (X,Σ) be a measurable space. Now if aσ-finite measure ν on the measurable space (X,Σ) is absolutely continuous with respect to another σ-finite measureµon (X,Σ), then there exits a measurable function f:X →[0,∞) such that for any measurable set AinX

ν(A) =R

Af dµ

and the functionf is called the Radon-Nicodym derivative ofν with respect to µ.It is denoted by ^dν_dµ.

So by using Radon-Nikodym Theorem, there exists a non-negative measurable functionfT such that

(16)

(µT⁻¹)(A) =R

AfTdµ, for everyA∈B.

The functionfT is known as Radon-Nikodym derivative of the measureµT⁻¹ with respect to the measureµand hence it is written as ^dµT_dµ⁻¹.

4.2 Measurable function

Letf be a extended real-valued function defined on a measurable set X. Then f is called measurable function if for eachα∈R, the set{x:f(x)> α}is measurable.

4.3 Multiplication and Composition operator

Definition 4.13. (Multiplication operator) LetT be a non-singular transformation defined by

T: Ω→Ω

and u be a complex valued measurable function defined on Ω.The bounded linear transformationM_u:F(Ω)→F(Ω) defined byM_uf =u.f for every f ∈F(Ω) is called amultiplication operatorinduced byu. In fact a continuous multiplication linear transformation is known as multiplication operator.

(HereF(Ω) is the topological vector space of complex valued function defined on Ω.)

The bounded linear transformation or composition transformation

C_T: F(Ω)→F(Ω) byC_Tf =f◦T for every f ∈F(Ω) on a Banach function space is called acomposition operator and is induced byT.

A weighted composition operator denoted byW_u,T is the product of a multiplication operator with a composition operator.

Wu,T(f) =CTMu(f)

Now we shall derive the equation for the weighted composition operatorWu,T.

W_u,Tf =C_TM_uf

=C_T(u.f), ∵M_uf =u.f

= (u.f)◦T ∵C_Tf =f◦T

=u(T)f(T)

(17)

Hence we got the equation forWu,T i.e

W_u,T =u(T)f(T)

We proceed for the weighted composition operatorWu,T given by W_u,Tf(ω) =u(T(ω))f(T(ω))

whereuandf are complex valued measurable function and T is a non-singular measurable transformation on the Orlicz spaceL_ϕ.

4.4 Boundedness of operator W

_u,T

The class of all bounded operator on a Banach spaceX is denoted by B(X) and the kernel and range of an operatorP onX byker(P) andR(P) respectively.

Now, supposeWu,T is bounded in Lϕi.e it’s range is inLϕ. Then this thing is denoted by writingW_u,T ∈B(L_ϕ).

For verifying the boundedness of weighted composition operatorWu,T, it can be easily seen thatC_T andM_u are both bounded linear transformation and therefore the operatorWu,T is bounded, writingWu,T ∈B(Lϕ).

5 Adjoint of W

_u,T

5.1 ∆

₂

- condition

Definition 5.1. (∆2- condition)

We say the Orlicz functionϕsatisfy the ∆2- condition if for somek >0, ϕ(2x)≤kϕ(x) ∀x >0

Now we assumeϕsatisfies ∆2- condition.

uis a complex-valued measurable mapping.

T is a non-singular measurable transformation.

L(µ) is the linear space of all complex-valued measurable functions. Let (Ω, B, µ) be a finite measure space . Then we define

L^φ(µ) ={f:X →C is measurable s.t. R

Xφ(|f|)dµ <∞for some >0}.

Theorem 5.1. If the mappingW_u,T:L_ϕ→L(µ), is such that Wu,T(Lϕ)⊆Lϕ thenWu,T ∈B(Lϕ).

Proof. let us consider a sequence f_n andf, g both are inL_ϕ i.ef ∈L_ϕ, g∈L_ϕ such that

(18)

kfn−fkϕ→0 andkWu,Tfn−gkϕ→0

asn→ ∞. Then we can find a subsequencef_n_k off_n which will satisfy ϕ(|fn_k−f|)→0 almost everywhere(a.e) on the space Ω. Now,

ϕ(|f_n_k−f|)→0 a.e or, ϕ(|u.f_n_k−u.f|)→0 a.e

Now sinceT is non-singular,

ϕ(|u◦T.f_n_k◦T−u◦T.f◦T|)→0 a.e onΩ.

Again we can find a subsequencefn_k0 offn_k such that ϕ(|u◦T.fn_k0 ◦T −g|)→0 a.e onΩ also from above, we have

ϕ(|u◦T.fn_k0 ◦T−u◦T.f◦T|)→0 a.e onΩ.

So, we can write

kWu,Tfn_k0−gkϕ→0 and (2) kWu,Tfn_k0−Wu,Tfkϕ→0 (3) asn_k⁰ → ∞. This implies from (2) and (3),W_u,Tf =g and by closed graph theorem,Wu,T is bounded onLϕ i.e. Wu,T ∈B(Lϕ).

Theorem 5.2. Letuandf_T belongs toL^∞(µ). ThenW_u,T ∈B(L_ϕ).

Proof. LetuandfT belongs toL^∞(µ) i.e. u∈L^∞(µ) andfT ∈L^∞(µ).

(19)

Ifft>1, then Z

Ω

ϕ

|Wu,Tf| kfkϕkuk_∞kfTk_∞

dµ

= Z

Ω

ϕ

|u◦T.f◦T| kfk_ϕkuk_∞kf_Tk_∞

dµ

= Z

Ω

ϕ

|(u.f)◦T| kfkϕkuk∞kfTk∞

dµ

= Z

Ω

ϕ

|(u.f)|

kfkϕkuk∞kfTk∞

dµ◦T⁻¹

≤ Z

Ω

ϕ

|f| kfkϕkfTk∞

f_Tdµ

≤ Z

Ω

ϕ |f|

kfkϕ

fT

kfTk_∞dµ

≤ Z

Ω

ϕ |f|

kfkϕ

dµ

≤1

∴kWu,Tfkϕ≤ kfkϕkuk_∞kfTk_∞ and whenf_T ≤1,

Z

Ω

ϕ

|Wu,Tf| kfk_ϕkuk_∞

dµ

≤ Z

Ω

ϕ |f|

kfk∞

f_Tdµ

≤ Z

Ω

ϕ |f|

kfk_∞

dµ

≤1

∴ kWu,Tfkϕ≤ kfkϕkuk∞

So, we takek=max{1,kf_Tk_∞} which follows to

kW_u,Tfk_ϕ≤k.kfk_ϕkuk_∞

∴Wu,T ∈B(Lϕ)

Using the above results and observation, we extend the results for the weighted composition operator on the Orlicz spaceLϕ as follows.

(20)

Theorem 5.3. The linear transformationWu,T:L7→L(µ), is a bounded operator onL_ϕif and only if u∈L^∞(µT⁻¹).

Proof. Suppose the linear transformation isWu,T:L7→L(µ) is a bounded operator onL_ϕ.

Assume the contrary that, u is not bounded with respect to the measureµT⁻¹ i.e. u /∈L^∞(µT⁻¹),which is equivalently sayingu◦T is not bounded with respect to the measureµ.Then from theorem 5.1 , for each natural number n, χE_n∈L_ϕ and

kWu,T χE_nkϕ

=kMuχE_nk_ϕ,µT⁻¹

=kM_{u◦T χE}_nkϕ

≥nkχEnkϕ,

whereE_n={x∈Ω :|u(T(x))|> n}. This means that the weighted composition operatorWu,T has no bound and hence it contradicts the hypothesis. ∴u∈L^∞(µT⁻¹).

Conversely,

Letu∈L^∞(µT⁻¹) i.e. u◦T ∈L^∞. Then by using theorem 5.1,f 7→u◦T.f is a bounded operator onLϕ. Also sincefT ∈L^∞,f 7→f◦T is bounded. Hence W_u,T is bounded onL_ϕ.

5.2 Adjoint of W

_u,T

Definition 5.2. LetWu,T:Lϕ7→L(µ) is a bounded linear operator onLϕ, then there exists an unique operatorW_u,T^? :Lϕ7→L(µ) such that

(x, W_u,T^? y) = (Wu,Tx, y) for allx, y∈Lϕ

The operatorW_u,T^? is linear and bounded with the following criterion.

1. kW_u,T^? k=kWu,Tk 2. (W_u,T^? )^?=Wu,T

The operatorW_u,T^? is called the adjoint ofW_u,T.

Theorem 5.4. LetWu,T be bounded onLϕ. Then the adjointW_u,T^? is given by

W_u,T^? (g) =fT.u.E(g)◦T⁻¹ for eachg∈Lψ.

(21)

(Hereϕandψare two complementary Orlicz function.)

Proof. LetA∈B be such thatµ(A)<∞, whereB is a measurable space.

Then forg∈Lψ,

(W_u,T^? Fg)(χA)

=Fg(Wu,TχA) (by the def inition of adjoint)

= Z

(Wu,TχA).gdµ

= Z

u◦T.χA◦T gdµ

= Z

E(u◦T.g).χA◦T dµ

= Z

u◦T.E(g).χA◦T dµ

= Z

f_T.u.E(g)◦T⁻¹.χAdµ

= (F_f_T_.u.E(g)◦T−1)(χA) Hence, (W_u,T^? Fg) =F_f_T_.u.E(g)◦T−1 and (W_u,T^? g) =fT.u.E(g)◦T⁻¹, for eachg∈Lψ

6 Bounded Weighted Composition on Orlicz space

We have the transformationT i.e.

T: Ω→Ω

. By using Radon-Nikodym derivative there exists non-negative measurable functionf₀ such thatµT⁻¹(A) =R

Af₀dµ. Heref₀ is called the

Radon-Nikodym derivative of the measureµT⁻¹ with respect to the measure µand it is denoted by

f0=^dµT_dµ⁻¹ On the basis of above we have the following results.

Theorem 6.1. LetT: Ω→Ω andu: Ω→C be two measurable

transformation. Then the operatorMu,T:L^φ(µ)→L^φ(µ) is bounded operator if and only if there exists a constantM >0 such that

f₀(x)φ(|u(x)y|)≤φ(M|y|)

(22)

forµ−almost allx∈Ω,y∈C.

Proof. Suppose that, f₀(x)φ(|u(x)y|)≤φ(M|y|).

Then for everyf ∈L^φ(µ), Z

Ω

φ

M_u,T(f) Mkfkφ

dµ

= Z

Ω

φ

(u◦T).(f ◦T) Mkfkφ

dµ

= Z

Ω

φ

u(T(x)).f(T(x)) Mkfkφ

dµ

= Z

T⁻¹(Ω)

φ

(u◦T).(f ◦T) Mkfk_φ

dµT⁻¹(y) taking T(x) =y

= Z

Ω

φ

|u.f|

Mkfkφ

dµT⁻¹ dµ dµ

≤ Z

Ω

f₀φ

|u.f|

Mkfkφ

dµ ∵ dµT⁻¹

dµ =f₀

≤ Z

Ω

φ |f|

kfkφ

dµ

≤1 Therefore,

kMu,Tkφfor every f ∈L^φ(µ).

Conversely, LetM_u,T:L^φ(µ)→L^φ(µ) is a bounded operator. And Suppose that the conditionf0(x)φ(|u(x)y|)≤φ(M|y|) is not true. Then for every n∈N there exists a measurable set{Fn} of Ω and somey_n ∈Csuch that

Fn={x∈Ω :f0(x)φ(|u(x)yn|)> φ(2ⁿn²|yn|)}

is a measurable set having positive measure. Now we know thatµis

non-atomic and so we can choose a disjoint sequence of measurable sets{En} such that

1. En⊂Fn and 2. µ(E_n) = ₂_n_φ(n^φ(|y₂¹_|y^|)

n|)

Letf is defined as simple function as follows.

f =

∞

X

n=1

bnχEn, where bn=nyn

(23)

. Now, consider Z

Ω

φ(αf)dµ

= Z

Ω

φ α

∞

X

n=1

b_nχE_n

! dµ

=

∞

X

n=1

Z

Ω

φ(αbn)χEndµ

=

n0

X

n=1

φ(αbn)µ(En) +

∞

X

n=n0

φ(αbn)µ(En)

≤

n0

X

n=1

φ(αbn)µ(En) +

∞

X

n=n0

φ(αb_n)φ(|y1|) 2ⁿφ(n²|yn|)

=

n0

X

n=1

φ(αbn)µ(En) +

∞

X

n=n0

φ(n²|yn|)φ(|y1|) 2ⁿφ(n²|yn|)

<∞ wheren₀> αBut

Z

Ω

φ(αM_u,Tf)dµ

= Z

Ω

f₀φ(α|u.f|)dµ

≥

∞

X

n=n0

Z

E_n

f₀φ(1

n|u.bn|)dµ α < n₀and α > 1 n0

≥

∞

X

n=n0

Z

E_n

f₀φ(|u.y_n|)dµ ∵b_n=ny_n

>

∞

X

n=n₀

φ(2ⁿn²|yn|)µ(En)

=

∞

X

n=n₀

2ⁿφ(n²|yn|)µ(En)

≥

∞

X

n=n₀

φ(|y1|)

=∞

which is a contradiction to the fact thatM_u,T is a bounded operator onL^φ(µ).

Hence the inequalityf0(x)φ(|u(x)y|)≤φ(M|y|) holds.

(24)

References

[1] S. C. Arora and Gopal Datt, Multiplication and composition induced operator on Orlicz Lorentz space, J. Adv. Res. Pure Math. 1(1) (2009), 49-64.

[2] S. C. Arora, Gopal Datt and Satish Verma, Weighted composition operators on Lorentz spaces,Korean Math. Soc 44(4) (2007), 701-708.

[3] Gopal Datt, Product of composition and multiplication operators, ACTA MATHEMATICA VIETNAMICA, 37(2) (2012), 293-300.

[4] S. Gupta, B. S. Komal and Nidhi Suri, Weighted composition operator on Orlicz spaces, Int. J. Conntem. Math. Sciences, 5(1), (2010), 11-20.

[5] M. R. Jabbarzadeh, A note on weighted composition operators on measurable function space, J. Korean math. Soc. 41(1)(2004) 95-105.

[6] B. S. Komal and Shally Gupta, Multiplication operators on Orlicz spaces, Integral equation and operator theory, 41(2001) 321-330.

Product of Composition and Multiplication Operator