On n-binormal Operators

Gen. Math. Notes, Vol. 10, No. 2, June 2012, pp. 1-8 ISSN 2219-7184; Copyright © ICSRS Publication, 2012 www.i-csrs.org

Available free online at http://www.geman.in

On n-binormal Operators

S. Panayappan¹ and N. Sivamani²

1 Department of Mathematics, Government Arts College, Coimbatore-641018,Tamilnadu, India.

E-mail: [email protected]

2 Department of Mathematics, Tamilnadu College of Engineering, Coimbatore-641659,Tamilnadu, India.

E-mail: [email protected] (Received: 24-4-12 / Accepted: 6-6-12)

Abstract

In this paper we introduce n-binormal operators acting on a Hilbert spaceH An operator . T∈L(H) is n-binormal if T^∗Tⁿ commutes with TⁿT^∗or

[

^{T∗Tn,TnT∗}

]

⁼⁰ and it is denoted by

[

nBN . We investigate some basic

]

properties of such operators. In general a n-binormal operator need not be a normal operator. Further we study n-binormal composite integral operators.

Keywords: Normal, n-normal, binormal, n -isometry and Hilbert space.

1 n-binormal Operators

Let H be a Hilbert space and L(H) be the algebra of all bounded linear operators acting on H An operator . T∈L(H) is called normal if T^∗T=TT^∗, n-normal if T^∗Tⁿ=TⁿT^∗, binormal if T^∗Tcommutes with TT ^∗,

isometry if T^∗T=I, 2-isometry if T^∗2T² −2T^∗T+I=0, 3-isometry if ,

0 3

3 ²

3 ²

3 − ^∗ + ^∗ − =

∗T T T T T I

T n-isometry if

( )

^{1 0}

0

 =







−  ^{∗ −}

=

∑

^{n k} − ^{n k}

k

T k T n ^nk

or

( ) ( )

^{1 0}

1 1 . ...

2 1

2 1

1 ²

1  + − =









− −





 +







− ^{∗ − ∗ − − ∗}

∗ ^{− −}

I T

n T T n

n T T

n T T

T ^{n n n n n n n n} and

n-binormal if T^∗TⁿTⁿT^∗=TⁿT^∗T^∗Tⁿ (refer[1], [2], [3] and [8]). In this section we investigate some basic properties of n-binormal operators.

Theorem 1.1 If ^T∈

[

^nBN

]

then so are (i) kTfor any real number k.

(ii) any S∈L(H)that is unitarily equivalent to .T (iii) the restriction

TM of T to any closed subspace Mof H that reduces .T

Proof . (i) The proof is straightforward.

(ii) Let S∈L(H)be unitarily equivalent to T then there is a unitary operator )

(H L

U∈ such that S =UTU^∗which implies that S^∗ =UT^∗U^∗andSⁿ =UTⁿU^∗ If Tis n-binormal then T^∗TⁿTⁿT^∗=TⁿT^∗T^∗Tⁿ

now S^∗SⁿSⁿS^∗ =UT^∗U^∗UTⁿU^∗UTⁿU^∗UT^∗U^∗ =UT^∗TⁿTⁿT^∗U^∗ and SⁿS^∗S^∗Sⁿ=UTⁿU^∗UT^∗U^∗UT^∗U^∗UTⁿU^∗=UTⁿT^∗T^∗TⁿU^∗ Hence S is unitary equivalent to T (refer [5]).

(iii) If T is n-binormal then T^∗TⁿTⁿT^∗=TⁿT^∗T^∗Tⁿ

Consider

( ) ( ) ( ) ( )

^T_M ^{∗ T}_M ^{n T}_M ^{n T}_M ^∗=

( )( )( )( )

^T∗_M ^Tn_M ^Tn _M ^T∗_M

⁼

(

^T∗TnTnT∗_M

)

⁼

(

^TnT∗T∗Tn_M

)

( )( )( )( )

T M

T M T M

Tⁿ M ^{∗ ∗ n}

=

⁼

( ) ( ) ( ) ( )

^T_M ^{n T}_M ^{∗ T}_M ^{∗ T}_M ⁿ

Hence ^T_M ∈

[

^nBN

]

^.

Theorem 1.2 If T∈L(H) is n-normal then ^T∈

[

^nBN

]

_.

Proof. If Tis n-normal then T^∗Tⁿ=TⁿT^∗

Post multiply by TⁿT^∗on both sides

T^∗TⁿTⁿT^∗ =TⁿT^∗TⁿT^∗ =TⁿT^∗T^∗Tⁿ Hence Tis n-binormal.

The following example shows that the converse need not be true.

Example 1.3 Let 









= −

1 0

1

T 1 be an operator on R , which is²

[

^3BN

]

but neither 3-normal nor normal.

Theorem 1.4 Let ^T∈

[

^nBN

]

^{and S∈}

[

^nBN

]

^{. If T}and S are doubly commuting then TS is n-binormal.

Proof .

( ) ( ) ( ) ( )

^{TS n TS ∗ TS ∗ TS n}

=SⁿTⁿS^∗T^∗S^∗T^∗SⁿTⁿ =SⁿS^∗TⁿT^∗T^∗S^∗TⁿSⁿ =SⁿS^∗TⁿT^∗T^∗TⁿS^∗Sⁿ

=SⁿS^∗T^∗TⁿTⁿT^∗S^∗S^{n, since} Tis

[

^nBN

]

=SⁿT^∗S^∗TⁿTⁿS^∗T^∗Sⁿ =T^∗SⁿTⁿS^∗S^∗TⁿSⁿT^∗ =T^∗TⁿSⁿS^∗S^∗SⁿTⁿT^∗

=T^∗TⁿS^∗SⁿSⁿS^∗TⁿT^∗, since S is

[

^nBN

]

=T^∗S^∗TⁿSⁿSⁿTⁿS^∗T^∗ =S^∗T^∗SⁿTⁿSⁿTⁿS^∗T^{∗ =}

( ) ( ) ( ) ( )

^{TS ∗ TS n TS n TS ∗}

Hence TS is n-binormal.

Example 1.5 Let 









= −

1 0

0

S 1 and 







= 0 1

1

T 1 be not commuting

[

^2BN

]

operators.Then ST is not

[

^2BN

]

^.

The following example shows that the sum and difference of two commuting n- binormal operators need not be n-binormal.

Example 1.6 Let 







= 1 0

0

S 1 and 







= 0 2

0

T 0 on R . Then ² Sand Tare commuting

[

^2BN

]

operators but 



 



= + 2 1

0 T 1

S is not

[

^2BN

]

^and 



 





= −

− 2 1

0 T 1

S is not

[

^2BN

]

^.

In the following theorem, we obtain sufficient condition for the sum of n-binormal operators be n-binormal( refer [7]).

Theorem 1.7 Let Sand T be commuting

[

nBN operators such that

] (

^S+T

)

^∗commutes with ^{n k k}

n

k

T k S n ₋

−

∑

= 



 





1

1

. Then

(

^S+T

)

is n-binormal operator.

Proof. Consider

(

^S+T

) (

^{∗ S+T}

) (

^{n S+T}

) (

^{n S+T}

)

^∗

( ) ( )

_^









  +



 



















 

 + 

=

∑ ∑

=

− ∗

=

∗ − n

k

k k n n

k

k k

n S T S T

k T n

k S T n

S

0 0

( ) ( ) ( ) ( ) ( ) ( )

_^









  + + +



 

 + 

 +











  + +



 

 + 

+ +

=

∑ ∑

⁻

=

∗

− ∗

− ∗

=

− ∗

∗

∗ 1

1 1

1

n

k

n k

k n n

n

k

n k

k n

n S T S T T S T

k T n

S S T T S T k S T n S S T S

( ) ( ) ( ) ( ) ( ) ( )

_^









  + + +



 

 + 

 +











  + +



 

 +  + +

= ^{− − ∗ ∗ ∗}

=

∗

− ∗

=

∗

∗

∗ −

∗

∗

∑ ∑

^{S T S T T S T}

k T n

S S T T S T k S T n S S T

S ^{n k k n}

n

k n

n

k

n k

k n n

1

1 1

1

( ) ( )

_^









  + + +





 + 

 +











  + +





 +  + +

=

∑ ∑

⁻

=

∗

∗ ∗

−

∗

∗

∗

∗

− −

=

∗ ∗

∗ 1

1 1

1

n

k

n n k

k n n

n n n k k n n

k n

n S T S T T S T T

k T n

S S S T T T S T k S T n S S T S S

Since S and T are commuting

[ ] ^nBN

operators such that

(

^S+T

)

^∗ _commutes

with ^{n k k}

n

k

T k S

n ₋

−

=

∑

_



 





1

1

.

( ) ( ) _^









  + +





 + 

+

 +











  + + +





 +  +

=

∑ ∑

⁻

=

∗

∗

∗ −

∗

∗

∗

∗

− ∗

=

−

∗

∗ 1

1 1

1

n

k

n n k k n n

n n

n n

k

k k n n

n S T S T T T

k T n S S T S S T T S T T S T k S T n

S S S

( ) ( ) ( ) ( ) ( ) ( )











  + +





 +  +

 +











  + + +





 +  +

=

∑ ∑

⁻

=

∗

∗

∗ −

∗

− ∗

=

∗

∗ ∗

−

∗

∗ 1

1 1

1

n

k

n k

k n n

n

k

n k

k n

n S T S T T

k T n S S T S T S T T S T k S T n

S S

( ) ( )

_^





















  +





 + 

 +











  +









  +





 + 

=

∑ ∑

⁻

=

−

∗

∗

∗

∗

− −

=

1

1 1

1

n

k

n k k n n

n k k n n

k

n S T T

k S n

T S T S T T k S S n

( ) ( )

^{n n k k}

k k

k n n

k

T k S

T n S T S T k S

n ₋

=

∗

− ∗

=

∑

∑







 + 

 +







= 

0 0

(

^S +^T

) (

^{n S}+^T

) (

^S+^T

) (

^S+^T

)

ⁿ

= ^{∗ ∗} . Hence

( ^{S + T} )

is n-binormal

Theorem 1.8 Let T∈L(H) with the Cartesian decomposition ^{T = A+iB.} Then T is binormal if and only if (i) AB³ +B³A= A³B+BA³ and (ii) A²BA+ABA² =B²AB+BAB².

Proof. Since T is binormal then T^∗TTT^∗=T T^∗T^∗T.

(

^{A iB}

)(

^{A iB}

)(

^{A iB}

)(

^{A iB}

)

TTT

T^{∗ ∗}= − + + −

⁼

(

^{A2+iAB−iBA+B2}

)(

^{A2−iAB+iBA+B2}

)

3 2 2 2 2 3 4

3 2

2 2 2

3 4

B A iB AB iB A B iBAB BABA

BAAB iBA

iAB ABBA ABAB

iABA B

A BA iA B iA A

+ +

− +

− +

−

−

+

− +

+ +

+

−

=

(

^{A iB}

)(

^{A iB}

)(

^{A iB}

)(

^{A iB}

)

T T

TT^{∗ ∗} = + − − +

⁼

(

^{A2−iAB+iBA+B2}

)(

^{A2+iAB−iBA+B2}

)

4 3 2

2 2 2 3

3 2

2 2 2

3 4

B A iB AB iB A B iBAB BABA

BAAB iBA

iAB ABBA ABAB

iABA B

A BA iA B iA A

+

− +

+ +

+

− +

−

− +

− +

− +

=

It is easy to observe that T is binormal if and only if (i) and (ii) are true.

The following examples show that n-binormal and n-isometry operators are independent classes.

Example 1.9 Consider the operator 



 



= 0 1

1

T 1 onR²,which is 3-binormal but not 3-isometry.

Example 1.10 Consider the operator 



 



= 1 0

1

T 1 onR , which is 3-isometry but ² not 3-binormal.

2 n-binormal Compositie Integral Operators

Let

(

^{X, S,}µ

)

^{be a} σ -finite measure space and let φ:X → X be a non-singular measurable transformation

(

^µ

( )

^{E =0⇒µφ−1}

( )

^{E =0}

)

. Then a composition transformation, for 1≤ p≺∞,Cφ :L^p

( )

µ →L^p

( )

µ is defined by C_φ f = f φ for every ^f ∈^Lp

( )

µ . In case C_φ is continuous, we call it a composition operator induced by φ^{. C}_φ is bounded operator if and only if ₀

1

d f d ⁻ =

µ

µφ . This is the

Radon- Nikodym derivative of the measure µφ⁻¹w.r.to the measure µand it is

essentially bounded. For more details about composition operators refer [1]

and[6].

A kernel K∈L^p

(

^µ×^µ

)

always induces a bounded integral operator

( )

µ ^p

( )

µ

p

K L L

T : →

defined by

(

^TK ^f

)( )

^{x =}

∫

^K

( ) ( ) ( )

^{x,y f y d}µ ^{y .}

Given a kernel Kand a non-singular measurable function φ:X → X ^{, the} composite integral operator T_Kφinduced by

(

^K,φ

)

is a bounded linear operator

( )

µ ^p

( )

µ

p

K L L

T : → defined by

( )

^TK_φ ^f

( )

^{x =}

∫

^K

( ) ( )

^{x,y f}

(

φ ^y

) ( )

^dµ ^{y =}

∫

^Kφ

( ) ( ) ( )

^{x,y f y d}µ ^y

We note that

( )

^TKnφ ^f

^{( )}

^{x =}

∫

^Kn

^{( ) ( )}

^{x,y f}

⁽

^{φ y}

^{) ( )}

^{dµ y}

⁼

∫

^Kφⁿ

( ) ( ) ( )

^{x,y f y d}µ ^y where

( )

, ⁼

∫∫∫ ∫

...

( )

, 1

( )

1, 2 ...

(

ⁿ₋2, ⁿ₋1

) (

ⁿ₋1,

)

1 2... ⁿ₋1

n x y K x z K z z K z z K z y dzdz dz

K_{φ φ φ φ φ} .

Theorem 2.1 Let Kφ ∈L²

(

µ×µ

)

. Then T_Kφis n-binormal if and only if

∫∫∫

^K_φ^∗

( ) ( ) ( ) ( ) ( ) ( ) ( )

^{x,y K}_φ^{n y,z K}_φ^{n z,t K}_φ^{∗ t,p d}µ ^{y d}µ ^{z d}µ ^t

⁼

∫∫∫

^Kφⁿ

( ) ( ) ( ) ( ) ( ) ( ) ( )

^x,^{y K}φ^{∗ y},^{z K}φ^{∗ z},^{t K}φ^{n t},^{p d}µ ^{y d}µ ^{z d}µ ^{t .} Proof. Suppose the condition is true . For ^f,g∈^L2

( )

µ , we have ^{T T T T f g}

(

^{T T T T}K ^f

) ( ) ( ) ( )

^{x g x d x}

n K n K K K

n K n K

K^∗_{φ φ φ} ^∗_φ ^{, =}

∫

^∗_{φ φ φ} ^∗_φ µ

⁼

∫∫ [

^Kφ^∗

( )

^x,y

(

^TKnφ^TKnφ^TK∗φ ^f

) ^{( ) ( )}

^{y dµ y}

]

^g

^{( ) ( )}

^{x dµ x}

^K

( )

^{x y}

(

^K

( )

^{y z}

(

^{T T}K ^f

) ^{( ) ( )}

^{z d z}

)

^d

( ) ( ) ( )

^{y g x d x}

n K

n _{φ φ} µ µ µ

φ

φ

∫

∫∫

^{∗ ∗}

= , ,

^K

( ) ( )

^{x y K y z}

(

^K

( )

^{z t T}K ^f

( ) ( )

^{t d t}

)

^d

( ) ( ) ( ) ( )

^{z d y g x d x}

n

n φ φ µ µ µ µ

φ

φ

∫

∫∫∫

^{∗ ∗}

= , , ,

( ) ( ) ( )

^{x y K y z K z t}

(

^K

( ) ( ) ( )

^{t p f p d p}

)

^d

( ) ( ) ( ) ( ) ( )

^{t d z d y g x d x}

Kφ φⁿ φⁿ

∫

φ µ µ µ µ µ

∫∫ ∫∫

^{∗ ∗}

= , , , ,

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

^{x y K y z K z t K t p f p d p d t d z d y g x d x}

K_φ^{∗ ,} _φ^{n ,} _φ^{n ,} _φ^{∗ ,} µ µ µ µ µ

∫∫ ∫∫∫

=

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

^{x,y K y,z K z,t K t,p d y d z d t f p d p g x d x...1}

K_φ^∗ _φⁿ _φⁿ _φ^∗ µ µ µ µ µ

∫∫∫∫∫

=

and ^{T T T T f g}

(

^{T T}K^TK ^TK^{n f}

) ( ) ( ) ( )

^{x g x d x}

n K n

K K K n

K_φ ^∗_φ ^∗_{φ φ} ^{, =}

∫

_φ ^∗_φ ^∗_{φ φ} µ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

^{x y K y z K z t K t p f p d p d t d z d y g x d x}

K_φ^{n ,} _φ^{∗ ,} _φ^{∗ ,} _φ^{n ,} µ µ µ µ µ

∫∫ ∫∫∫

=

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

^{x,y K y,z K z,t K t,p d y d z d t f p d p g x d x...2}

Kφⁿ φ^∗ φ^∗ φⁿ µ µ µ µ µ

∫∫ ∫∫∫

=

It follows from (1) and (2) that T_Kφis n-binormal . Conversely suppose

Kφ

T is n-binormal .Take f =χ_Eand g=χ_Fwe see that from (1) and (2)

^K

( ) ( ) ( ) ( ) ( ) ( ) ( )

^{x y Kn y z Kn z t K t p d y d z d t}

E F

n _{φ φ φ} µ µ µ

φ , , , ^∗ ,

∫∫∫

^K

( ) ( ) ( ) ( ) ( ) ( ) ( )

^{x y K y z K z t Kn t p d y d z d t}

E F

n _{φ φ φ} µ µ µ

φ , ^∗ , ^∗ , ,

∫ ∫ ∫

=

for all E,F∈S×S. Hence the required condition holds.

References

[1] A. Gupta and B.S. Komal, Binormal and idempotent integral operators, Int. J. Contemp. Math. Sciences, 6(34) (2011), 1681-1689.

[2] J. Agler and M. Stankus, m-isometries transformations of Hilbert space, Integral Equations and Operator Theory, 21(1995), 383- 427.

[3] A.A.S. Jibril, On n-power normal operators, The Arabian Journal for Science and Engineering, 33(2A),(2008), 247-25.

[4] A.A.S. Jibril, On 2-normal operators, Dirasat, 23(2) (1996), 190-194.

[5] A.A.S. Jibril, On operators for which ^{T∗2T2 =}

( )

^{T∗T 2,} International Mathematical Forum, 5(46) (2010), 2255-2262.

[6] R.K. Singh and J.S. Manhas, Composition Operators on Function Spaces, North Holland Mathematics Studies 179, Elsevier Sciences Publishers Amsterdam, New York, (1993).

[7] S.A. Alzuraiqi and A.B. Patel, On n-normal operators, Gen. Math. Notes, 1(2) (2010), 61-73.

[8] S.M. Patel, 2-isometric operators, Glasnik Matematicki, 37(57) (2002), 143-147.