International Journal of Mathematical Analysis Vol. 9, 2015, no. 33, 1645 - 1659

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijma.2015.53106

Weighted Composition Operators

G. Balaji 1, C. V. Seshaiah 2, B. Vennila2 and K. Meenambika3

1 Thangavelu Engineering College

Chennai, Tamil Nadu, India

2 Sri Ramakrishna Engineering College

Coimbatore-641022, Tamil Nadu, India

3 Sengunthar Engineering College

Namakkal, Tamil Nadu, India

Copyright © 2015 G. Balaji et al. This article is distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Abstract

Weighted composition operators have been related to products of

composition operators and their adjoints and to isometries of Hardy spaces. In this

paper, we identify the Hermitian weighted composition operators on H2 and

compute their spectral measures. Some relevant semigroups are studied. The

resulting ideas can be used to find the polar decomposition, the absolute value,

and the Aluthge transform of some composition operators on H2.

Mathematics Subject Classification: 39B52, 26A33

Keywords: The distributor function; Radon-Nikodym derivative; Non-atomic

measure; Lorentz space

1. Introduction

The purpose of this paper is to study the weighted composition operators:

S.C. Arora, G. Batt and S. Verma studies boundedness, compactness and closed-

ness of ranges of weighted composition operators on ,p qL .

In this paper, we study the weighted compositon operators on weighted

Bergman spaces of bounded symmetric domains. The necessary and sufficient

conditions for a weighted composition operator Wϕ,ψ to be bounded and compact

are studied by using the Carleson measure techniques.

1646 G. Balaji et al.

1.1: (Weighted Composition Operator)

Let , , be a - finite measurable space. Let :T be

measurable non – singular transformation and u be a complex – valued

measurable function defined on . The weighted composition operator induced

by u and T is defined by

,

, , , p q

u TW f x u x f T x for x f L (1.1)

1.1 Boundedness Theorem 1.1.1

Let , , be a - finite measurable space and :u a

measurable function. Let :T be a non – singular measurable

transformation such that the Randon - Nikodym derivative 1

Tf d T d is

in .L Suppose ,u L then the weighted composition operator

, : .u TW f u f T is bounded on , ,1 ,1p qL p q

Proof:

Suppose .Tb f

Then for f in , ,p qL consider the distribution function of

, .u TWf W f u f T

:Wf s x u x f Tx s

:x u f T x s

1 :T x u f x s

(1.2)

:Tf x u f x s

ub s

Therefore, for each 0,t we have

0 : 0 :f

fus s t b s s t

It follows that

*

inf 0 : fWf t s s t

inf 0 :f

s u s t b

inf 0 : :s x u f x s t b

inf 0 : :u s x f x s t b

*u f t b

For 1 ,1p q

Weighted composition operators 1647

*1

,

0

qq p

p q

dtWf t Wf t

t

*1

0

qp dt

t u f t bt

1 *

0

qq p dtu t f t b

t

,

q qq b

p qb u f

That is,

1

, ,

p

p q p qWf b u f

(1.3)

Therefore W is bounded for 1 ,p q

*1

,0

sup p

pt

Wf t Wf t

1 *

0

sup p

t

t u f t b

1

,

p

pb u f

(1.4)

It follows that W is also bounded.

The following example shows that u need not be bounded in order to have

a bounded weighted composite on operator , .u TW

Example 1.1.2

Let 0,1 2 , where 0,1 has the Lebesgue measure and 2 is an

atom with unit mass. Let 1u x x and 2T x for every .x Then u is

not bounded on 0,1 , but u and T define a bounded weighted composition

operator

,u T

f zW f u x f Tx

x on ' .L Notice

that , ,1u T u TW f W f d

1

20

2fd

x

1

1 12 2 2

2 2f f

where 1 denotes the norm of '.L

1648 G. Balaji et al.

Under some conditions on u and ,, u TT W is bounded and implied that

.u L

Theorem 1.1.3

Let , , be a finite measurable space. Let :u be a measurable

function and :T a non – singular measurable transformation such that

r rT E E for each 0,r where : .rE x u x r If ,u T is bounded on

, ,1 ,1 ,p qL p q then u L .

Proof:

Assume that ,u T is bounded i.e.,

, ,,u T p qp qf C f (1.5)

For all ,p qf L and some constant C.

If possible, suppose u is not in .L For each x . Let

:nE x u x x

Using the fact that ,n nT E E we have

1 ,

n nE T E

and

1: :

n nE x T E

x s x u x x ns

Therefore,

1

*

inf 0 : :E nn n

E T EW t s x u x x ns t

inf 0 : :nEn s x u x T x ns t

1inf 0 : :

nT En s x u x x ns t

inf 0 : :nEn s x x s t

* .nEn t

This gives,

,,

,n nE E

p qp qW x

which contradicts the boundedness of .

Therefore .u L

Weighted composition operators 1649

Conclusion

Theorem 1.1.4

Let , , be a finite measurable space and :u a measurable

function. Let :u be a non – singular measurable transformation such that

the Radon – Nikodym derivative 1

Tf d T d is in L and r rT E E

for each 0,r where :rE x u x r .

Then the weighted composition operator , :u TW f u f T is bounded on

, ,1 ,1p qL p q if and only if u L .

Proof:

Based on the previous theorems, all we need to do is to show that

u L implied the boundedness of ,u TW write sup : ,u u x x

and

consider,

:f s x u x f Tx s

:x u f Tx s

: /y f y s u

/f s u

Hence,

*

inf 0 : fWf t s s t

inf 0 : fs s u t

inf : fu s u s u t

*u f t

1

*1

,

0

qq

p

p q

dtWf t Wf t

t

1

1 *

0

q

qp dt

t u f tt

,

.p q

u f

This implies that ,u T is bounded on , .p qL

1650 G. Balaji et al.

1.2 Compactness

Theorem 1.2.1

Let ,u TW W be a weighted composition operator on the Lorentz space

, ,p qL for 1 ,1 ,p q induced by a non – singular measurable

transformation T on and a measurable : .u C Decompose into two

disjoint sets 1 and 2 , where 1 does not possess any atom and 2 is a

countable union of atoms of finite measure. If is compact, then

,

,

0,n p q

n

n p q

W Ab x

A

(1.6)

Where nA are all the atoms of 2 with 0,nA for each x. If more

over, u is bounded below, we also have

1

1 0T

Proof:

To prove 0,nb assume to the contrary that nb 0, then without loss of

generality, there exists some 0 such that nb for all n. For each .n Let

,

,n

An

A

n

p q

fW

Then ,

,

,

n

An

Ap q

n p q

p q

fW

1

The sequence of nf is bounded sequence in ,p qL with an upper bound 1

.

However, ,

,

,

1An

An

p q

n p q

p q

W

WfW

.

Since nWf and mWf have supports with ,n m ,n m n m p q

Wf Wf Wf Wf .

Hence,

,

1 11

2 2n m n m m n p q

Wf Wf Wf Wf Wf Wf

, ,

1

2n m n mp q p q

Wf Wf Wf Wf

, , ,

1

2n m n m n mp q p q p q

Wf Wf Wf Wf Wf Wf

Weighted composition operators 1651

This means nWf does not admit any convergent subsequence with the

bounded sequence .nf It is a contradiction to the compactness of W. Therefore

0 as 0nb n .

Since, 1 ,T the Random – Nikodym derivative 1

Tf d T d satisfies

1 , for .T

E

T E f t d E (1.7)

If 1

1 0,T by (1.7), there exists some 1 such that the set

1 : 1TE x f x

has positive measure. Since 1 is non – atomic we can find a sequence nE of

measurable subsets of E such that 1 ,n nE E

, 0 .n n

aE a E

z

Let

,

n

n

E

n

Ep q

e

,

Then ne is a bounded sequence in , .p qL For , , let andm m N m n

be the lower bound of u , then for 0.t

*

n mWe We t

1inf 0 : : n ms x u x e Tx u x e Tx s t

1inf 0 : : n ms x e Tx e Tx s t

1inf 0 : :n n ms T y E e y e y s t

inf 0 : :n n ms y E e y e y s t

inf 0 : :n n ms y E e y e y s t

inf 0 : :n m n ms y E E e y e y s t

Using the same method in (2.11), we have,

*

*

,

(n m

n

E E

n m

Ep q

tWe We t

which implies

1652 G. Balaji et al.

1

*1

,

0

q

p

n m n mp q

dtWe We t We We t

t

1

1 *

0,

n m

n

q

qp

E E

Ep q

dtt t

t

1

1 *

0,

n m

n

qq

p

E E

Ep q

d ss s

s

1 1

1 1

1 1p p

n m

p p

n

E E

E z

Thus nWe does not admit a convergent subsequence, which contradicts

the compactness of W. Hence 0Tf a e on 1 and

1

1

1 0TT f d

Hence 1

1 0T .

Condition (1.6) gives us a simple criteria for checking the non –

compactness of ,u TW in the following examples.

Example 1.2.2

Let , 1 and the measure on where it is the induced

Lebesgue measure on the interval ,1 and each n is an atom with mass

21 .

n

Define :T by if or 1,T x n x n x n n and :u

by 21u x

x for all .x Then the weighted composition operator

,u TW is

well – defined on ' .L Indeed for each 'f L ,

, 211

1n

u T

n N n

W f f x u x dx u nn

4

1 1

1n N

f nn n n

Now for each n,

, 4 211

1 1 1and

1u T n nW

n n n n

.

we have,

Weighted composition operators 1653

, 1

2

1

11

1

u T n

n

n

n n

So that

,u T is not compact.

Theorem 1.2.3:

Let ,u TW W be a weighted composition operator on the Lorentz space

,p qL u for 1 ,1 ,p q induced by a non – singular measurable

transformation T on and a measurable : :u

Decompose into two disjoint sets 1 2and where 1 does not

possess any atom and 2 is a countable union of atoms of finite measure. Then

is compact is the following statements are true:

(i) .u L

(ii) 1

1 0T

(iii)

1

0,n

n

n

T AC

A

where nA

are all the atoms of 2 with 0nA , for each n.

Proof:

Suppose 1

1 0T and 2 1 nnA

.

Let 1

n

N

n T An N

W f f A u

Then,

:

N

N

W W fx W W f x

1 :n n

n N

x T A f A u x

1 :n n

n N

x T A f A u

1

, n

n

n N f A u

T A

, n

n n

n N f A u

C A

,

sup ,n

n nn N n N n N f A u

C A

0

supn

n nn f A u

C A

sup .n fn N

C u

1654 G. Balaji et al.

Hence,

* sup 0 :N N

fW W f t W W t

sup 0 : sup n fn N

C u t

sup 0 : supf nn N

u u u t C

* sup n

n N

u f t C

and

1

*1

,0

qq

N Np

p q

dtW W f t W W f t

t

1

,sup .

p

n p qn N

u C f

Equivalently,

1

sup 0 as 0.

pN

nn N

W W b N

Moreover, consider

:N

W N fx W f x

1 :n n

n N

x T A f A u x

1 :n n

n N

x T A f A u

1

, n

n

n N f A u

T A

, n

n n

n N f A u

C A

supn

n n

f A u

C A

sup .n fC u

for every N . Hence,

* sup 0 : N

N

fW f t t

sup 0 :sup n fC u t

sup 0 : supf nu u u t C

* sup nu f t C

,

and

Weighted composition operators 1655

1*

1

,0

qq

N Np

p q

dtW f t W f t

t

1

1 *

0

sup

q

qp

n

dtt u f t C

t

1

1 *

0

sup

q

qp

n

dtu t f t C

t

1

,sup

p

n p qu C f

This implied

,p qN

W is bounded for every .N N Since W is the limit of

bounded finite rank operators ,

NW it is compact.

Observations:

1. Conditions (i) and (iii) are not necessary for the weighted composition operator

to be compact. For (i), see example 1.1.2. For (iii), consider the mapping on 'l

defined by .n nx x n It is a weighted composition operator with T the

identity function and u the function 1 .n x Since 1 ,T n n (iii) is clearly not

satisfied.

To see that W is compact, define 1 2 3, 2, 3,..., ,0,...N

n NW x x x x x N

Then

1

NN n

nl

n

xW x

n

1

N

n n ln

x x

.

Hence N

W is bounded. Moreover, consider

1

N N i

nl i

xW W x

N i

1

1 1 N i

i

N x

1

1 1 n

n

N x

1 1 n lN x

we have,

1 1 0.N

W W N

1656 G. Balaji et al.

Hence W is the limit of bounded finite rank operators and is therefore

compact.

2. Conditions (i) and (iii) together imply 0nb , where nb is the sequence in

Theorem 1.2.1

In fact,

,

,

nAp q

n

n p q

Wb

A

1

*1

0

11

n

qq

p

A

pq

n

dtt u T t

t

p q A

11 1

11

pq

n

pq

n

u p q T A

p q A

11

p

n

n

T Au

A

1

0p

nu C

If (i) and (iii) hold.

1.3 Closedness of Ranges

Theorem 1.3.1

Let :T be a non – singular injective transformation such that the

Radon – Nikodym derivative 1

Tf d T d is in L and is bounded

below.

Let :u be a measurable function such that ,u TW is bounded and

injective on the Lorentz space , ,1 ,1 .p qL p q Then ,u TW has a

closed range if and only if there exists a 0 such that u x such that

.u x a e on : 0 ,S x u x the support of u.

Proof:

Suppose that ,u TW W has a closed range. Then there exists and 0

such that

, ,p q p q

Wf f (1.8)

for all ,p qf L

Choose 1

.p

fT

We claim that u x a e otherwise, there is a measurable subset

Weighted composition operators 1657

: 0F x u x such that

0 .F Let .E T F

Since T is non – singular, E too has positive measure. In fact, if 0,E

then by the non – singularity of 1, 0T F T E

Also 1

T

E

T E f d

T

E

f d

Tf E

Therefore,

,,E E T p qp q

W u

1

,T E

p q

u

1

,T E

p q

1

11

qpp

T Eq

1

1q

p

T

pf E

q

1

,

p

T E p qf

,E p q

This contradicts (1.8). Hence u x a e on S. Conversely if

u x a e on S, then for each ,E consider the norm of :E

W

,,E E p qp q

W u T

1

,T E

p q

u

1

,T E

p q

11 pq

T

E

pf d

q

1

1

'

q

ppE

q

1

,'

p

E p q

1658 G. Balaji et al.

Where ' is a lower bound of Tf . For each , ,p qf L using increasing

simple functions to approximate f. Then f satisfy, 1

, ,'

p

p q p qWf f as well.

Therefore ,u TW has a closed range.

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Received: April 1, 2015; Published: June 17, 2015