(Alpha, Beta)-Normal and Skew n-Normal Composite Multiplication Operator on Hilbert Spaces

http://www.sciencepublishinggroup.com/j/dmath doi: 10.11648/j.dmath.20190401.17

ISSN: 2578-9244 (Print); ISSN: 2578-9252 (Online)

(Alpha, Beta)-Normal and Skew n-Normal Composite

Multiplication Operator on Hilbert Spaces

Senthil

1, *

, Nithya

2

, Suryadevi

3

, David Chandrakumar

3

1

Department of Economics and Statistics, Government of Tamilnadu, DRDA, Dindigul, India

2_{Department of Mathematics, Mother Teresa Women’s University, Kodaikanal, India} 3

Department of Mathematics, Vickram College of Engineering, Enathi, India

Email address:

*

Corresponding author

To cite this article:

Senthil, Nithya, Suryadevi, David Chandrakumar. (Alpha, Beta)-Normal and Skew n-Normal Composite Multiplication Operator on Hilbert Spaces. International Journal of Discrete Mathematics. Vol. 4, No. 1, 2019, pp. 45-51. doi: 10.11648/j.dmath.20190401.17

Received: February 15, 2019; Accepted: March 19, 2019; Published: May 6, 2019

Abstract:

In this paper, the condition under which composite multiplication operators on Hilbert spaces become skew n-normal operators, (Alpha, Beta)-n-normal, parahypon-normal and quasi-parahypon-normal have been obtained in terms of radon-nikodym derivative.

Keywords: Composite Multiplication Operator, Conditional Expectation, Aluthge Transformation,

Skew n-Normal Operator, Parahyponormal

1. Introduction

Let ( ,X

∑

, )µ be a σ -finite measure space. Then a mapping T from X into X is said to be a measurable transformation if T−1( )E ∈

∑

for every E∈

∑

. A measurable transformation T is said to be non-singular if

1

(T ( ))E 0

µ

− _{= whenever}

µ

_{( )E =0. If T is non-singular then}

the measure

µ

T−1 defined as

µ

T−1( )E =

µ

(T−1( ))E for every E in

∑

, is an absolutely continuous measure on

∑

with respect to

µ

. Since

µ

is a σ -finite measure, then by the Radon-Nikodym theorem, there exists a non-negative

function f0 in L1( )

µ

such that 1

0

( )

E

T E f d

µ − ₌

∫

µ

for

every E∈

∑

. The function f0 is called the

Radon-Nikodym derivative of

µ

T−1 with respect to

µ

.

Every non- singular measurable transformation T from X into itself induces a linear transformation CT on Lp( )

µ

defined as C fT = f T for every f inLp( )

µ

. In case CT is

continuous from Lp( )

µ

into itself, then it is called a

composition operator on Lp( )

µ

induced by T. We restrict our study of the composition operators on L2( )

µ

which has Hilbert space structure. If u is an essentially bounded complex-valued measurable function on X, then the mapping Muon

2

( )

L

µ

defined by M fu = ⋅u f , is a continuous operator with

range in L2( )

µ

. The operator Mu is known as the

multiplication operator induced by u. A composite multiplication operator is linear transformation acting on a set of complex valued

∑

measurable functions f of the form

, ( ) ( ) ( ) ( )

u T T u

M f =C M f = u T f T

where u _{is a complex valued,}

∑

_{measurable function. In}

case u=1 almost everywhere, Mu T, becomes a composition

operator, denoted by CT.

In the study considered is the using conditional expectation of composite multiplication operator on _L2_{-spaces. For eac}

( , , )

p

f ∈L X

∑

µ , 1≤ ≤ ∞p , there exists an unique

1_{( )}

( )

A A

g f dµ= g E f dµ

∫

∫

for every T−1(

∑

)-measurable function g, for which the left integral exists. The function E f( ) is called the

conditional expectation of f with respect to the subalgebra

1

( )

T−

∑

. As an operator of Lp( )

µ

, E is the projection onto the closure of range of T and E is the identity on

( ) p

L

µ

, p≥1 if and only if T−1(

∑

)=

∑

. Detailed discussion of E is found in [1-4].

1.1. (Alpha, Beta)--Normal Operator [13]

An operator T is called

(

α β

,

)

-normal operator if

2 * * 2 *

T T T T T T

α

≤ ≤

β

,

(

0≤ ≤ ≤

α

1

β

)

1.2. Skew n-Normal Operator [12]

An operator T is called skew n-normal operator if

( )

_{T T}n * _{T=T T T}

( )

* n

, for all natural number n.

1.3. p-Hyponormal Operator [15]

An operator T is called p-hyponormal operator if

( ) ( )

* p * p

T T ≥ T T , for

(

0≺p≺∞

)

2. Related Work in the Field

The study of weighted composition operators on 2

L spaces

was initiated by R. K. Singh and D. C. Kumar [5]. During the last thirty years, several authors have studied the properties of various classes of weighted composition operator. Boundedness of the composition operators in Lp(

∑

),

( 1≤ <∞p )spaces, where the measure spaces are σ -finite, appeared already in [6]. Also boundedness of weighted operators on C X E( , ) has been studied in [7]. Recently S. Senthil, P. Thangaraju and D. C. Kumar have proved several theorems on n-normal, n-quasi-normal, k-paranormal, and (n,k) paranormal of composite multiplication operators on

2

L spaces [8-11]. In this paper we investigate composite

multiplication operators of

(

α β

,

)

-normal operator and skew n-normal operator L2( )

µ

-spaces.

3. Hyponormality for Composite

Multiplication Operator

The results in the following proposition were proved in [12], as part of his doctoral dissertation.

3.1. Proposition [3]

Let E=E( \ )⋅ A and let

ϕ

be a non-negative Fmeasurable function.

Define the positive operator P_ϕby P f_ϕ =ϕ ϕE( f).

Let 1

2 ₄

( (E ))

ϕ ϕ

ϕ

∧

=

. Then

1 2 P_ϕ P

ϕ∧ = .

Define the operator R_{ϕ by} R f_ϕ =E(ϕ f) _{. Then}

2

( )

R_ϕ E

ϕ

∞

= .

In [3], has proved the following lemma, as noted for any non-negative function f,

support f⊂support E f( r)for any r>0

3.2. Lemma [14]

Let α and

β

be non-negative functions, with sup

S= port

α

. Then the following are equivalent:

For every f∈L2( )

µ

2 2

( \ )

X X

f d E f A d

α µ≥ β µ

∫

∫

support

β

⊂Sand

2

\ 1

S

E β χ A

α

 

≤

 

 

  almost everywhere.

3.3. Proposition

Let the composite multiplication operator

2 , ( ( ) )

u T

M ∈B L µ . Then for u≥0

2

, , 0

( )i M∗u T Mu T f =u f f

2

, , 0

( )ii M_{u T}M∗_{u T} f =(u T) (f T) E f( )

,

(iii) Mn_{u T}( )f =(C M_{T u}) ( )n f =u_n (f Tn),u_n =(u T) (u T2) (u T3)...(u Tn)

1

, 0

( )iv M∗_{u T} f =(u f )( ( )E f T− )

( 1)

, 0 0

( )v M∗n_{u T} f =u f ( (E u f ) T− −n ) ( ( )E f T−n)

whereE u f( ₀) T− −(n 1) =( (E u f₀) T−1) (E u f( ₀) T−2)...( (E u f₀) T− −(n 1))

, 0

( )vi M_{u T} =u f f

2 2

, 0 0

(vii) M∗_{u T} =( u f T ) ( (E u f T f))

2 0

,

u T _{u f T} v

M∗ =P =P _{, where}

(

)

(

)

2 0

1

2 4

0

( )

( )

u f T v

E u f T

=

Theorem 3.1

Let the composite multiplication operator M_{u T,} ∈B L( 2( ) )µ with weight u≥0 and S be the support of f0:

, u T

M ≥ C if and only if 2

1

u ≥

,

u T

C ≥ M∗ if and only ifsupport v=support f₀

where

(

)

(

)

2 0

1

2 4

0

( )

( )

u f T v

E u f T

=

,

(

)

2 0

1

2 2

0 0

( )

( )

1

S

u f T

E u f T E

f χ

 

 

 

 _≤

 

 

 

 

 

almost everywhere.

Proof:

Since Mu T, and C are multiplication operators, we need only compare their symbols. After squaring, we get ,

u T

M ≥ C if and only if u f0 ≥ f0

, u T

M ≥ C if and only if u f2 ₀ ≥ f₀Because f0 >0almost everywhere,

we obtain (i).

As for this f, C ≥ M∗u T,

f

∀ ,

∫

f0 f 2d

µ

≥ M∗u T, f, f

,

u T v

M∗ =P, where

(

)

(

)

2 0

1

2 4

0

( )

( )

u f T v

E u f T

= ₌

∫

_{v E v f( ) f d}

µ

However,

2

( ) ( ), ( )

v E v f f d

µ

= E v f v f = E v f

∫

2

( )

E v f d

µ

=

∫

Since support v=support f₀ , the desired conclusion follows from lemma 3.2. Theorem 3.5

Let the composite multiplication operator M_{u T,} ∈B L( 2( ) )µ . Then Mu T, is p-hyponormal if and only if uf0 >0,uf0 T>0

and 2 2

0 0

1 1

p p

E

u f u f T

 

≤

 

  .

Proof:

Here

(

)

2

* 2

, , , 0

p

p p

u T u T

X

M M f f =

∫

u f f d

µ

and

(

)

2

* 2

, , , ( 0 )

p p

p

u T u T

X

M M f f E u f T f dµ

 

 

= _{ }

 

∫

This implies

2 2

2 2

0 ( 0 )

p

p p p

X X

u f f dµ E u f T f dµ

 

 

≥ _{ }

 

∫

∫

⇒ ₂

0 0

(( ) ) ( )

p

p p p

u f T u f

σ ⊂σ and

2 0 2

0

( )

1

p p

p p

u f T E

u f

 

≤

 

 

 

⇒

2 2

0 0

1 1

( )

p p p p

E

u f u f T

 

≤

 

 

  if

2 0

(u pf p) T >0andu2pf₀p>0.

4. Parahyponormal for Composite Multiplication Operator

Mahmound M. Kutkut [16] , has proved that an operator A is parahyponormal if and only if (AA∗)2−2C A A C∗ + 2 ≥0for all real C. In an analogous manner, we derive some characterization of parahyponormal and quasi-parahyponormal composite multiplication operator on _L2_-spaces.

Theorem 4.1

Let the composite multiplication operatorM_{u T,} ∈B L( 2( ) )µ . Then Mu T, is parahyponormal if and only if

4 2 2 2

0 ( ) 2 0 0

u T ⋅f T E f − C u f f +C ≥ almost everywhere, for all C≥0 Proof:

Suppose Mu T, is parahyponormal. Then (M_{u T,} M∗_{u T,} )2−2C M∗_{u T,} M_{u T,} +C2 ≥0for all C≥0.

This implies that

2 2

, , , ,

((M_{u T}M∗_{u T}) −2C M∗_{u T}M_{u T} +C ) ,f f ≥0for all f∈L2( )

µ

By the proposition 3.3 we get,

{

2 2 2 2

}

0 0

( ( )) 2 0

E

u T⋅ f T E f⋅ − C u f f +C dµ≥

∫

for everyE∈

∑

.

⇔ 4 2 2 2

0 ( ) 2 0 0

u T ⋅f T E f − C u f f +C ≥ almost everywhere, for all C≥0 Theorem 4.2

Let the composite multiplication operatorM_{u T,} ∈B L( 2( ) )µ . Then Mu T, is M-parahyponormal if and only if

2 4 2 2 2

0 ( ) 2 0 0

m u T f⋅ T E f − C u f f+C ≥ almost everywhere, for all C≥0. Proof:

Suppose Mu T, is M-parahyponormal.

Then m2(M_{u T,} M∗_{u T,} )2−2C M∗_{u T,} M_{u T,} +C2 ≥0for all C≥0. This implies that

2 2 2

, , , ,

(m (M_{u T} M∗_{u T}) −2C M∗_{u T}M_{u T} +C ) ,f f ≥0for all f∈L2( )

µ

By the proposition 3.3 we get,

{

2 2 2 2 2

}

0 0

( ( )) 2 0

E

m u T⋅ f T E f⋅ − C u f f +C dµ≥

∫

for everyE∈

∑

.

⇔ 2 4 2 2 2

0 ( ) 2 0 0

m u T f⋅ T E f − C u f f+C ≥ almost everywhere, for all C≥0 Theorem 4.3

Let the composite multiplication operatorM_{u T,} ∈B L( 2( ) )µ . Then Mu T, is M∗-parahyponormal if and only if

2 2 2 1 2 2

0 ( 0) 2 0 ( ) 0

m u f E u f T− f − C u T f⋅ T E f +C ≥ almost everywhere, for all C≥0. Proof:

Suppose Mu T, is M∗-parahyponormal.

Then 2 2 2 2

, , 2 , , 0

u T u T u T u T

m M∗ M − C M M∗ +C ≥ for all C≥0. This implies that

2

2 2 2

, , , ,

(m M∗ _{u T}M _{u T} −2C M_{u T} M∗_{u T} +C )f, f ≥0for all f∈L2( )

µ

By the proposition 3.3 and M∗2_{u T,} M2_{u T,} f =u f E u f2 ₀ ( 2 ₀) T−1f u≥0 we get,

{

2 2 2 1 2 2

}

0 ( 0) 2 0 ( ) 0

E

m u f E u f T− f − C u T ⋅f T E f +C dµ≥

⇔ 2 2 2 1 2 2

0 ( 0) 2 0 ( ) 0

m u f E u f T− f − C u T f⋅ T E f +C ≥ almost everywhere, for all C≥0

5. Quasi-Parahyponormal for Composite Multiplication Operator

By result of [16] an operators A on H is quasi-parahyponormal if and only if _{(A A}2 ∗2₎2_{−2C AA(} ∗₎2_+C2 _{≥0for all C. In an}

analogous manner, we derive the characterization of quasi-parahyponormal composite multiplication operator on 2

L -spaces.

Theorem 5.1

Let the composite multiplication operatorM_{u T,} ∈B L( 2( ) )µ . Then Mu T, is quasi-parahyponormal if and only if

2 4 2 2 2 2 4 2 2

0 ( ( 0)) ( ) 2 0 ( ) 0

u T u⋅ T ⋅f T ⋅ E uf T E f⋅ − C u T f⋅ T E f⋅ +C ≥ almost everywhere, for all C≥0 Proof:

Suppose Mu T, is quasi-parahyponormal.

Then (M2_{u T,} M∗2_{u T,} )2−2C M( _{u T,} M∗_{u T,} )2+C2 ≥0for all C≥0. This implies that

2

2 2 2 2

, , , ,

((M _{u T}M∗_{u T}) −2C M( _{u T}M∗_{u T}) +C )f, f ≥0for all f ∈L2( )

µ

By the proposition 3.3 and 2 2 2 2 2

, , 0 ( 0) ( )

u T u T

M M∗ f =u T u⋅ T ⋅f T ⋅E uf T E f⋅

{

2 2 2 2 2 2 2

}

0 0 0

( ( ) ( )) 2 ( ( )) 0

E

u T u⋅ T ⋅f T ⋅E uf T E f⋅ − C u T⋅ f T E f⋅ +C dµ≥

∫

for everyE∈

∑

⇔ 2 4 2 2 2 2 4 2 2

0 ( ( 0)) ( ) 2 0 ( ) 0

u T u⋅ T ⋅f T ⋅ E uf T E f⋅ − C u T f⋅ T E f⋅ +C ≥

almost everywhere, for all C≥0 Theorem 5.2

Let the composite multiplication operatorM_{u T,} ∈B L( 2( ) )µ . Then Mu T, is M-quasi-parahyponormal if and only if

2 2 4 2 2 2 2 4 2 2

0 ( ( 0)) ( ) 2 0 ( ) 0

m u T u⋅ T ⋅f T ⋅ E uf T E f⋅ − C u T f⋅ T E f⋅ +C ≥ almost everywhere, for all C≥0 Proof:

SupposeMu T, is M-quasi-parahyponormal.

Then m2(M2_{u T,} M∗2_{u T,} )2−2C M( _{u T,} M∗_{u T,} )2+C2 ≥0for all C≥0. This implies that

2

2 2 2 2 2

, , , ,

(m (M _{u T}M∗ _{u T}) −2C M( _{u T}M∗_{u T}) +C )f, f ≥0for all f∈L2( )

µ

By the proposition 3.3 and M2u T_, M∗2u T_, f =u T u⋅ 2 T2⋅f₀ T2⋅E uf( ₀) T E f⋅ ( )

{

2 2 2 2 2 2 2 2

}

0 0 0

( ( ) ( )) 2 ( ( )) 0

E

m u T u⋅ T ⋅f T ⋅E uf T E f⋅ − C u T⋅ f T E f⋅ +C dµ≥

∫

for everyE∈

∑

.

⇔ 2 2 4 2 2 2 2 4 2 2

0 ( ( 0)) ( ) 2 0 ( ) 0

m u T u⋅ T ⋅f T ⋅ E uf T E f⋅ − C u T f⋅ T E f⋅ +C ≥ almost everywhere, for all C≥0

6. Skew n-normal and (Alpha, Beta)-normal Composite Multiplication Operator

Theorem 6.1

Let the composite multiplication operatorM_{u T,} ∈B L( 2( ) )µ . ThenMu T, is skew n-normal if and only if

2 2

0 0

( n) ( n) ( ) ( ) ( ( ) )

n n

u u T f T = u T f T E u T almost everywhere. Proof:

,

u T

M is skew n-normal. Then

, , , , ,

(Mn_{u T} M∗_{u T})M_{u T} f =(Mn_{u T}M∗_{u T}) (uf) T

1 , ( 0 (( ) ) )

n u T

M u f E uf T T−

2

, 0

n u T

M u f f

=

2 0

( ) n

n

u u f f T =

2

0

( n) ( n) ( n)

n

u u T f T f T

=

Also,

, ( , , ) , , ( ( ))

n n

u T u T u T u T u T n

M M∗ M f =M M∗ u f T

1

, 0 ( ( ))

n

u T n

M u f E u f T T−

=

1

, 0 ( ) ( )

n

u T n

M u f E u f T −

=

1 0

( ( _n) ( n ))

u u f E u f T − T

=

2 0

(u T) (f T) E u( _n) T (f Tn)

=

⇔ 2 2

0 0

( n) ( n) ( ) ( ) ( ( ) )

n n

u u T f T = u T f T E u T almost everywhere. Theorem 6.2

Let the composite multiplication operatorM_{u T,} ∈B L( 2( ) )µ . Then M*_{u T,} is skew n-normal if and only if

2 2 2 ( 1) ( 1)

0 ( ) ( 0 )

n n

u f = u T− − f T− − almost everywhere. Proof:

* ,

u T

M is skew n-normal. Then

(

) (

)

(

( 1)

)

, ( , , ) , , 0 ( 0) ( )

n n n

u T u T u T u T u T

M∗ M M∗ f =M∗ M u f E u f T− − E f T−

(

) (

)

(

)

(

( 1)

)

, 0 ( 0) ( )

n n

u T

M∗ u u f E u f T− − E f T− T

=

(

2 ( 2) ( 1)

)

, ( ) ( 0 ) ( ( 0) ) ( ( ) )

n n

u T

M∗ u T f T E u f T− + E f T− −

=

(

2 ( 2) ( 1)

)

1

0 ( ) ( 0 ) ( ( 0) ) ( ( ) )

n n

u f E u T f T E u f T− + E f T− −  T−

= _{ }

3 2 ( 1)

0 ( ( 0) ) ( ( ) )

n n

u f E u f T− − E f T− =

Also,

(

1

)

, , , , , 0

(M∗n_{u T}M_{u T})M∗_{u T} f =M∗n_{u T}M_{u T} u f E f( ) T−

(

2

)

, ( ) ( 0 ) ( )

n u T

M∗ u T f T E f

=

( 1) 2 ( 1) ( 1)

0 ( 0) ( ) ( 0 ) ( )

n n n n

u f E u f T− − u T− − f T− − E f T− =

⇔ 2 2 2 ( 1) ( 1)

0 ( ) ( 0 )

n n

u f = u T− − f T− − almost everywhere. Theorem 6.2

Let the composite multiplication operator M_{u T,} ∈B L( 2( ) )µ . Then Mu T, is

(

α β

,

)

-normal if and only if

2 2 2 2 2

0 ( 0) ( ) 0

u f f u f T E f u f f

α

≤ ≤

β

almost everywhere.

Proof:

,

u T

2

, , 0

u T u T

M∗ M f =u f f , M_{u T,} M∗_{u T,} f =(u2 T) (f₀ T) E f( )

By definition,

2 2 2 2 2

0 ( 0) ( ) 0

u f f u f T E f u f f

α

≤ ≤

β

almost everywhere.

8. Conclusion

In the study of p-hyponormal operator, the Aluthge transform is a very useful tool. We investigate some basic properties of such operators and study the relation among skew n-normal operators,

(

α β

,

)

-normal operator, parahyponormal and quasi-parahyponormal composite multiplication operators on L2( )

µ

-space. In future try to generalize the composite multiplication operator on Poisson weighted sequence spaces.

Acknowledgements

We would like to thank the reviewers for carefully reading manuscript and for their constructive comments. I want to thank Professor Dr. R. David Chandrakumar, Department of Mathematics, Vickram College of Engineering for his support and encouragement during preparation of the paper.

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