A Study on Quasi Class Q Operator

ISSN: 2231-5373

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Page 1

A Study on Quasi Class Q Operator

Dr. S.K.Latha,

Professor, Department of Mathematics, Kathir College of Engineering,

Coimbatore District, -Tamilnadu-641 062, INDIA

ABSTRACT: In this paper quasi class

Q

operator is introduced with example and compared with normaloid operator. Necessary and sufficient condition for a composition operator to be quasi class

Q

is given with an example.

Also weighted composition operator

W

and generalized Aluthge transform

C

_r are characterized to be of quasi class

Q

. Keywords: Class

Q

operators, composition Operators

1. INTRODUCTION

CLASS Q OPERATOR : In [2], B.P.Duggal, C.S.Kubrusly and N.Levan introduced a new class ‘Class

Q

’ operator and showed that this class of operators properly induces the paranormal operators and studied various properties of Class

Q

operators.

In [5], S.Panayappan and D.Senthilkumar have given necessary and sufficient condition for a composition operator to be class

Q

and studied various properties of class

Q

composition operators.

An operator

T



L H

(

)

is of class

Q

if and only if

T T

*2 2



2

T T

*

 

I

0.

[2] 2. COMPOSITION OPERATOR

The Radon-Nikodym derivative of



_

T

1

with respect to



is denoted by

h

and that of



_

T

k

with respect to



is denoted by

h

_k where

k



N, the set of natural numbers. Here

T

k is obtained by composing

T

-

k

times. From [4],

w

_k denotes

(

)(

2

)....(

k 1

)

w w T w T

_

_

w T

_



so that

(

),

k k

k

W f



w

f T



*

(

)

,

k _k

k k

W

f



h E w f



T



* 2

(

)

k _{k k}

k k

W W f



h E w



T



f

and

*

(

) (

).

k

k k

k k k

W W

f



w h



T

E w f

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Page 2 More generally in [1], Charles Burnap, Il Bong Jung and Alan Lambert have defined the family of

operators



T

_r

: 0

 

r

1



where

T

_r



T U T

r 1r

.

In [1], for a composition operator

C

the polar decomposition is given by

C



U C

where

,

C f



h f

and

Uf

1

f T

.

h T







This is valid for all composition operators. In [1], Charles Burnap, Il Bong Jung and Alan Lambert have given more generally Aluthge transformation for composition operators as

1

r r

r

C



C U C

 for

0

 

r

1

and

/ 2

.

r

r

h

C f

f T

h T





 











Here we see that

C

_r is a weighted composition operator with weight

/ 2 r

h

h T



_{ }





_







where

0

 

r

1

.

Since

C

_r is a weighted composition operator it is easy to show that

2 1

[ (

)]

r

C f

h E



T



f





* 2 1

[ (

)]

.

r r

C C f

h E



T



f







And

C

r*

f



vE vf

 

where





1 2 ₄

h T

v

E

h T























(ie)

C C f

r r*



vE vf

(

).

If

T

1

  

,

then

E

becomes identity operator and hence

C C f

_{r r}*



v f

4

.

Also we have that

C f

rk





k

(

f T



k

),

where

2 1

(

)(

)....(

k

).

k

T

T

T



_

 

_



_



_



*

(

)

,

k _k

r k k

C f



h E



f



T



* 2 1

(

C C

r r

)

k

f

h E

k

( (



)

T

)

k

f

,







* 2

(

)

k _{k k}

r r k k

C C f



h E





T



f

and

*

(

) (

).

k

k k

r r k k k

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Page 3 3. QUASI CLASS Q OPERATOR

In this paper quasi class

Q

operator is introduced with example and compared with normaloid operator. Necessary

and sufficient condition for a composition operator to be quasi class

Q

is given with example.

Also weighted composition operator

W

and generalized Aluthge transform

C

r are characterized to be of quasi class

Q

.

Definition: An operator

T



L H

(

)

is said to be of quasi class

Q

if

2 2 2

3 2

( )

2

( )

( ) ,

T

x



T

x



T x

 

x

H

with

x



1.

Theorem 1: An operator

T



L H

(

)

is of quasi class

Q

iff

T T

*3 3



2

T T

*2 2



T T

*



0.

Proof:

For all

x



H

, consider

3 2

* 3 * 2 *

2

0

T T



T T



T T



3 2

* 3 * 2 *

(

T T

2

T T

T T x x

) ,

0









3 2

* 3 * 2 *

(

T T

) ,

x x

2 (

T T

) ,

x x

(

T T x x

) ,

0











T x T x

3

,

3



2

T x T x

2

,

2



Tx Tx

,



0

.

2 2 2

3 2

( )

2

( )

( )

0

T

x

T

x

T x









.

2 2 ₂

3 2

( )

2

( )

( ) .

T

x

T

x

T x







Corollary: A weighted shift

T

with decreasing weighted sequence

(



_n

)

is quasi class

Q

if

2 2 2

1 2

2

1

1

0.

n n n



_



_





_

 

Proof:

Since

T

is a weighted shift, its adjoint

T

* is also a weighted shift and defined by

T e

( )

_n





_{n n}

e

_1

,

*

1 1

( )

_{n n n}

,

T e





_

e

_

* 2

(

T T e

)

_n





_n

e

_n

,

*2 2 2 2

1

(

T T

)

e

_n



 

_{n n}

e

_n and

*3 3 2 2 2

1 2

(

T T e

)

n



 

n n



n

e

n

.

ISSN: 2231-5373

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Page 4

3 2

* 3 * 2 *

2

0.

T T



T T



T T





 

_n2 _n12



_n22



2

 

_n2 _n12





_n2



0.





_n12



_n22



2



_n12

 

1

0.

Example: Consider the operator

T l

:

2



l

2 defined by

T x

( )



(0,



1 1

x

,



2

x

2

,...)

where

1 2

1

1

1

,

,

4

2

4

n n n







_





_



for

n



1.

Then

T

is of quasi class

Q

.

Proof:

Given

T x

( )



(0,



_{1 1}

x

,



₂

x

₂

,...).

Then *

1 2 2 3

( )

(

,

,...).

T

x





x



x

T T x

*

( )



(



₁2

x

₁

,



₂2

x

₂

,



₃2

x

₃

...).

2

2 1 1 3 2 2 4 3 3

( )

(0, 0,

,

,

...).

T

x



 

x

 

x

 

x

* 2 2 2 2

2 1 1 3 2 2 4 3 3

( )

(0,

,

,

...).

T T

x



 

x

 

x

 

x

*2 2 2 2 2 2 2 2

2 1 1 3 2 2 4 3 3

( )

(

,

,

...).

T T

x



 

x

 

x

 

x

T

3

( )

x



(0, 0, 0,

  

_{3 2 1 1}

x

,

  

_{4 3 2}

x

₂

,...).

* 3 2 2

3 2 1 1 4 3 2 2

( )

(0, 0,

,

,...).

T T

x



  

x

  

x

T T

*2 3

( )

x



(0,

  

₃2 ₂2 _{1 1}

x

,

  

₄2 ₃2 ₂

x

₂

,...).

*3 3 2 2 2 2 2 2 2 2 2

3 2 1 1 4 3 2 2 5 4 3 3

( ) (

,

,

...).

T T x



  

x

  

x

  

x

Now consider

(

T T

*3 3



2

T T

*2 2



T T x x

*

) ,

2 2 2 2 2 2

3 2 1 1 4 3 2 2

(

  

x

,

  

x

, ....)



2 2 2 2

2 1 1 3 2 2

2(

 

x

,

 

x

,...)





(



₁2

x

₁

,



₂2

x

₂

,...),

1 2

( ,

x x

,...)



(

  

₃2 ₂2 ₁2



2

 

₂2 ₁2





₁2

)

x

₁ 2 2 2 2 2 2

4 3 2 3 2 2 2 1 2

(

  

2

 



)

x

...), ( ,

x x

,...)









2 2 2 2 2 2

3 2 1 2 1 1 1 1

(

  

2

 



) ,

x x









(

  

₄2 ₃2 ₂2



2

 

₃2 ₂2





₂2

) ,

x x

_{2 2}



...

2

2 2 2 2 2 2

3 2 1 2 1 1 1

(

  

2

 



)

x









(

  

₄2 ₃2 ₂2



2

 

₃2 ₂2





₂2

)

x

₂ 2



...



0.

Example: Let

T

be a weighted shift with weighted sequence

1

2

n n





. Then

T

is of quasi class

Q

but it is not normaloid.

Proof:

1

1

( )

.

2

n n n

T e



e

_

*

1 1

1

( )

.

2

n n n

T e



_

e

_

3 2

* 3 * 2 *

2

0,

.

T T



T T



T T



 

n

N

T



is of quasi class

Q

.

ISSN: 2231-5373

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Page 5 4. QUASI CLASS

Q

COMPOSITION OPERATOR.

Theorem 2: Let

C



B L

(

2

( ))



. Then

C

is of quasi class

Q

iff

f

0(3)



2

f

0(2)



f

0



0

a.e. Proof:

From Theorem 1,

C

is of quasi class

Q



C C

*3 3



2

C C

*2 2



C C

*



0

*3 3 *2 2 * 2

(

C C

2

C C

C C f f

) ,

0,

f

L

( )











 

(3) (2) 2

0 0 0

(

2

)

0,

E

f

f

f

f d



E



_







 

(3) (2)

0

2

0 0

0

f

f

f









a.e.

Example 1: Let

X



N

, the set of all natural numbers and



be the counting measure on it. Define

T N

:



N

, by

T

(1)



1,

T

(2) 1,



T

(3)



1,

T

(4) 1,



T

(5)



2,

T

(6

n m



)

 

n

1,

m



0,1, 2,3, 4,5

and

.

n



N

Since

f

₀(3)



2

f

₀(2)



f

₀



0,

 

n

N

,

C

is of quasi class

Q

.

Example 2: Let

X



N

, the set of all natural numbers and



be the counting measure on it. Define

T N

:



N

by

T

(1)



1,

T

(4

n m





2)

 

n

1,

m



0,1, 2,3

and

n



N

.

Since

f

0(3)



2

f

0(2)



f

0



0,

 

n

N

,

C

is of quasi class

Q

.

Theorem 3: The weighted composition operator

W

is of quasi class

Q

iff

2 3 2 2 2 1

3 3 2 2

(

h E w

(

)

T



) 2(

h E w

(

)

T



) (

hE w

( )

T



) 0













a.e.

Proof:

From theorem 1,

W

is of quasi class

Q

*3 3 *2 2 *

2

0

W W

W W

W W









*3 3 *2 2 * 2

(

W W

2

W W

W W f f

) ,

0,

f

L

( )











 

(

*3 3

2

*2 2 *

)

2

0,

E

W W

W W

W W f d



E



_







 

2 3 2 2 2 1

3 3 2 2

(

h E w

(

)

T



) 2(

h E w

(

)

T



) (

hE w

( )

T



) 0















a.e.

Corollary: The generalized Aluthge transformation

C

_r is of quasi class

Q

2 3 2 2 2 1

3 3 2 2

(

h E

(



)

T



) 2(

h E

(



)

T



)

(

hE

(



)

T



)

0















a.e.

Proof:

From theorem 1,

C

_r is of quasi class

Q

*3 3 *2 2 *

2

0

r r r r r r

C

C

C

C

C C

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Page 6

*3 3 *2 2 * 2

(

C C

_{r r}

2

C C

_{r r}

C C

_{r r}

) ,

f f

0,

f

L

( )















2

*3 3 *2 2 *

(

r r

2

r r r r

)

0,

E

C C

C C

C C

f

d



E



_







  

2 3 2 2 2 1

3 3 2 2

(

h E

(



)

T



) 2(

h E

(



)

T



) (

hE

(



)

T



)

0.















5. REFERENCES

[1]: Charles Burnap, Il Bong Jung and Alan Lambert, Separating Partial Normality classes with Composition operators, Journal of operator theory 53:2(2005) 381-397

[2]: B. P. Duggal, C.S. Kubrusly, N. Levan, Contractions of Class Q and invariant subspaces, Bull. Korean Math. Soc., 42, 2005, 169-177

[3]: Mahmoud M. 33, On the classes of Parahyponormal operator, Journ. Math. Sci., 4(2)(1993),73-88

[4]: Panayappan S., Non-hyponormal weighted composition operators, Indian Journal of pure and applied Mathematics, 27(10): 979-983, October 1996