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: panayappan@gmail.com
2
Department of Mathematics, Tamilnadu College of Engineering,
Coimbatore-641659,Tamilnadu, India.
E-mail: sivamanitce@gmail.com
(Received: 24-4-12 / Accepted: 6-6-12)
Abstract
In this paper we introduce n-binormal operators acting on a Hilbert
space H . An operator T ∈ L (H ) is n-binormal if T ∗T n commutes with T nT ∗ or
[T T
∗
n
]
, T nT ∗ = 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 ∗T n = T nT ∗ ,
binormal
if
T ∗T commutes
with
TT ∗,
2
S. Panayappan et al.
isometry if
∗3
∗2
T ∗T = I , 2-isometry if
2
2
k
n
∗
T T − 3T T + 3T T − I = 0,
3
T ∗ T 2 − 2T ∗T + I = 0 , 3-isometry if
n-isometry
n
∑ (− 1)  k T
if
 
k =0
∗n − k
T n −k = 0
n
 n  n −1
 n  n−2
n −1  n  ∗
n
T T + (− 1) I = 0 and
or T ∗ T n −  T ∗ T n −1 +  T ∗ T n −2 ...........(− 1) 
1
 2
 n − 1
n-binormal if T ∗T nT nT ∗ = T nT ∗T ∗T n (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) kT for any real number k .
(ii) any S ∈ L (H ) that is unitarily equivalent to T .
(iii) the restriction T of T to any closed subspace M of H that reduces T .
M
Proof . (i) The proof is straightforward.
(ii) Let S ∈ L (H ) be unitarily equivalent to T then there is a unitary operator
U ∈ L (H ) such that S = UTU ∗ which implies that S ∗ = UT ∗U ∗ and S n = UT nU ∗
If T is n-binormal then T ∗T nT nT ∗ = T nT ∗T ∗T n
now S ∗ S n S n S ∗ = UT ∗U ∗UT nU ∗UT nU ∗UT ∗U ∗ = UT ∗T nT nT ∗U ∗
and S n S ∗ S ∗ S n = UT nU ∗UT ∗U ∗UT ∗U ∗UT nU ∗ = UT nT ∗T ∗T nU ∗
Hence S is unitary equivalent to T (refer [5]).
(iii) If T is n-binormal then T ∗T nT nT ∗ = T nT ∗T ∗T n
( M ) (T M ) (T M ) (T M ) = (T M )(T M )(T M )(T M )
Consider T
∗
n
n
∗
∗
n
∗
n
)
)
= (T T T T
M
)(T M )(T M )(T M )
= (T
M
(
∗ n n ∗
=T T T T
n
∗
∗
∈ [nBN ].
n
Theorem 1.2 If T ∈L(H ) is n-normal then T ∈ [nBN ] .
Proof. If T is n-normal then T ∗T n = T nT ∗
∗
n
( M ) (T M ) (T M ) (T M )
= T
M
n
∗
n
Hence T
M
∗
∗
n
On n-binormal Operators
3
Post multiply by T nT ∗ on both sides
T ∗T nT n T ∗ = T nT ∗T nT ∗ = T nT ∗T ∗T n
Hence T is n-binormal.
The following example shows that the converse need not be true.
1 1 
 be an operator on R 2 , which is [3BN ] but neither
Example 1.3 Let T = 
 0 − 1
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 ) (TS ) (TS ) (TS )
n
∗
∗
n
= S nT n S ∗T ∗ S ∗T ∗ S nT n
= S n S ∗T nT ∗T ∗ S ∗T n S n
= S n S ∗T nT ∗T ∗T n S ∗ S n
= S n S ∗T ∗T nT nT ∗ S ∗ S n , since T is [nBN ]
= S nT ∗ S ∗T nT n S ∗T ∗ S n
= T ∗ S nT n S ∗ S ∗T n S nT ∗
= T ∗T n S n S ∗ S ∗ S nT nT ∗
= T ∗T n S ∗ S n S n S ∗T nT ∗ , since S is [nBN ]
= T ∗ S ∗T n S n S nT n S ∗T ∗
= S ∗T ∗ S nT n S nT n S ∗T ∗
= (TS ) (TS ) (TS ) (TS )
∗
n
n
∗
Hence TS is n-binormal.
1 0 
1 1 
 and T = 
 be not commuting
Example 1.5 Let S = 
 0 − 1
1 0 
operators.Then ST is not [2 BN ] .
[2 BN ]
4
S. Panayappan et al.
The following example shows that the sum and difference of two commuting nbinormal operators need not be n-binormal.
1 0
 0 0
 and T = 
 on R 2 . Then S and T are commuting
Example 1.6 Let S = 
 0 1
 2 0
[2 BN ]
[2 BN ] .
 1 0
 is not
operators but S + T = 
2
1


 1 0
 is not
S − T = 
−
2
1


[2 BN ] and
In the following theorem, we obtain sufficient condition for the sum of
n-binormal operators be n-binormal( refer [7]).
Theorem 1.7 Let S and T be commuting [nBN ] operators such that
n −1 n
  n−k k
∗
 S T . Then (S + T ) is n-binormal operator.
commutes
with
(S + T )
∑
k =1  k 
Proof. Consider (S + T )∗ (S + T )n (S + T )n (S + T )∗
n

 n  n 
n
∗
∗
=  (S + T ) ∑   S n−k T k   ∑   S n−k T k (S + T ) 
k =0  k 

  k =0  k 

n −1 n
n −1 n


 
 
∗
∗
∗
∗
∗
∗
=  (S + T ) S n + (S + T ) ∑   S n − k T k + (S + T ) T n   S n (S + T ) + ∑   S n − k T k (S + T ) + T n (S + T ) 
k =1  k 
k =1  k 



n −1 n
n −1 n
 ∗



 
 
∗
∗
=  S + T ∗ S n + (S + T ) ∑  S n−k T k + S ∗ + T ∗ T n  S n S ∗ + T ∗ + ∑  S n−k T k (S + T ) + T n S ∗ + T ∗ 
k =1  k 
k =1  k 



n
−
1
n
−
1



n
 n
∗
∗
=  S ∗ S n + T ∗ S n + (S + T ) ∑  S n−k T k + S ∗T n + T ∗T n   S n S ∗ + S nT ∗ + ∑   S n−k T k (S + T ) + T n S ∗ + T nT ∗ 
k =1  k 
k =1  k 



(
)
(
)
(
)
(
)
Since S and T are commuting [nBN] operators such that (S + T )∗ commutes
n −1 n
 
with ∑  S n−k T k .
k =1  k 
n −1 n
n −1 n



 
 
∗
=  S n S ∗ + S n T ∗ + ∑   S n − k T k (S + T )∗ +T n S ∗ + T n T ∗  S ∗ S n + T ∗ S n + (S + T ) ∑   S n − k T k + S ∗T n + T ∗T n 
k =1  k 
k =1  k 



n −1 n

 
∗
=  S n S ∗ + T ∗ + ∑   S n− k T k (S + T ) + T n S ∗ + T ∗
k
k =1  

(
)
(
)  (S

∗
)
+ T ∗ S n + (S + T )
∗
n −1
n
k =1
 
∑  k S
n −k
n −1 n
n −1 n





 
 
=   S n + ∑  S n−k T k + T n  S ∗ + T ∗   S ∗ + T ∗  S n + ∑   S n−k T k + T n  



k =1  k 
k =1  k 





n
n
 n 
 n 
∗
∗
= ∑   S n − k T k (S + T ) (S + T ) ∑   S n − k T k
k =0  k 
k =0  k 
(
) (
n
∗
∗
n
= (S + T ) (S + T ) (S + T ) (S + T ) . Hence
)
(S +T ) is n-binormal

T k + S ∗ + T ∗ T n 

(
)
On n-binormal Operators
5
Theorem 1.8 Let T ∈ L (H ) with the Cartesian decomposition T = A + iB . Then
T is binormal if and only if (i) AB 3 + B 3 A = A3 B + BA3 and
(ii) A 2 BA + ABA 2 = B 2 AB + BAB 2 .
Proof. Since T is binormal then T ∗TTT ∗ = T T ∗T ∗T .
T ∗TTT ∗ = ( A − iB )( A + iB )( A + iB )( A − iB )
(
)(
= A 2 + iAB − iBA + B 2 A 2 − iAB + iBA + B 2
)
= A 4 − iA3 B + iA 2 BA + A 2 B 2 + iABA 2 + ABAB − ABBA + iAB 3
− iBA 3 − BAAB + BABA − iBAB 2 + B 2 A 2 − iB 2 AB + iB 3 A + B 4
TT ∗T ∗T = ( A + iB )( A − iB )( A − iB )( A + iB )
(
)(
= A 2 − iAB + iBA + B 2 A 2 + iAB − iBA + B 2
)
= A + iA B − iA BA + A B − iABA + ABAB − ABBA − iAB 3
4
3
2
2
2
2
+ iBA3 − BAAB + BABA + iBAB 2 + B 2 A 2 + iB 2 AB − iB 3 A + B 4
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.
1 1 
 on R 2 , which is 3-binormal but not
Example 1.9 Consider the operator T = 
1 0 
3-isometry.
 1 1
 on R 2 , which is 3-isometry but
Example 1.10 Consider the operator T = 
0
1


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 (µ )→ L (µ ) is defined by Cφ f = f φ for
p
p
every f ∈ Lp (µ ) . In case Cφ is continuous, we call it a composition operator
dµφ −1
induced by φ . Cφ is bounded operator if and only if
= f 0 . This is the
dµ
Radon- Nikodym derivative of the measure µφ −1 w.r.to the measure µ and it is
6
S. Panayappan et al.
essentially bounded. For more details about composition operators refer [1]
and[6].
A kernel
K ∈ Lp (µ × µ ) always induces a bounded integral operator
TK : Lp (µ ) → Lp (µ )
defined by (TK f )(x ) = ∫ K (x, y ) f ( y )dµ ( y ) .
Given a kernel K and a non-singular measurable function φ : X → X , the
composite integral operator TKφ induced by (K ,φ ) is a bounded linear operator
TK : Lp (µ ) → Lp (µ ) defined by
(T
Kφ
)
f ( x ) = ∫ K ( x, y ) f (φ ( y ))dµ ( y )
= ∫ K φ ( x, y ) f ( y )dµ ( y )
We note that
(T f )(x )= ∫ K
(x, y ) f (φ ( y ))dµ ( y )
= ∫ K φn (x, y ) f ( y )dµ ( y )
n
Kφ
n
where
K φn ( x, y ) = ∫ ∫ ∫ .....∫ K φ ( x, z1 )K φ (z1, z 2 ).......K φ ( z n −2 , z n −1 )K φ (z n −1, y )dz1dz 2 .....dz n −1 .
Theorem 2.1 Let K φ ∈ L2 (µ × µ ) . Then TKφ is n-binormal if and only if
∫∫∫ K φ (x, y )Kφ ( y, z )K φ (z, t )Kφ (t, p )dµ ( y )dµ (z )dµ (t )
∗
n
∗
n
= ∫∫∫ K φn ( x, y )K φ∗ ( y, z )K φ∗ ( z , t )K φn (t , p )dµ ( y )dµ ( z )dµ (t ) .
Proof. Suppose the condition is true . For f , g ∈ L2 (µ ) , we have
(
)
TK∗φ TKnφ TKnφ TK∗φ f , g = ∫ TK∗φ TKnφ TKnφ TK∗φ f ( x )g ( x )dµ ( x )
[
(
= ∫∫ K ( x, y ) (∫ K
]
)
= ∫∫ K φ∗ ( x, y ) TKnφ TKnφ TK∗φ f ( y )dµ ( y ) g (x )dµ ( x )
∗
φ
n
φ
( y, z )(TKn TK∗ f )(z )dµ (z )) dµ ( y )g (x )dµ (x )
= ∫∫∫ K φ∗ ( x, y )K φn ( y, z )
φ
(∫ K
n
φ
φ
(z, t )TK∗φ f (t )dµ (t )) dµ (z )dµ ( y )g (x )dµ (x )
(
)
= ∫ ∫ ∫ ∫ K φ∗ ( x, y )K φn ( y, z )K φn ( z , t ) ∫ K φ∗ (t , p ) f ( p )dµ ( p ) dµ (t )dµ ( z )dµ ( y )g ( x )dµ ( x )
= ∫ ∫ ∫ ∫ ∫ K φ∗ ( x, y )K φn ( y, z )K φn ( z , t )K φ∗ (t , p ) f ( p )dµ ( p )dµ (t )dµ ( z )dµ ( y )g (x )dµ ( x )
On n-binormal Operators
7
= ∫ ∫ ∫ ∫ ∫ K φ∗ ( x, y )K φn ( y, z )K φn ( z , t )K φ∗ (t , p )dµ ( y )dµ ( z )dµ (t ) f ( p )dµ ( p )g ( x )dµ ( x )...(1)
(
)
TKnφ TK∗φ TK∗φ TKnφ f , g = ∫ TKnφ TK∗φ TK∗φ TKnφ f ( x )g ( x )dµ ( x )
and
= ∫ ∫ ∫ ∫ ∫ K φn ( x, y )K φ∗ ( y, z )K φ∗ ( z , t )K φn (t , p ) f ( p )dµ ( p )dµ (t )dµ ( z )dµ ( y )g (x )dµ ( x )
= ∫ ∫ ∫ ∫ ∫ K φn ( x, y )K φ∗ ( y, z )K φ∗ ( z , t )K φn (t , p )dµ ( y )dµ ( z )dµ (t ) f ( p )dµ ( p )g ( x )dµ ( x )...(2 )
It follows from (1) and (2) that TKφ is n-binormal .
Conversely suppose TKφ is n-binormal .Take f = χ E and g = χ F we see that from
(1) and (2)
∫∫∫ Kφ (x, y )Kφ ( y, z )Kφ (z, t )Kφ (t , p )dµ ( y )dµ (z )dµ (t )
n
n
n
∗
EF
= ∫ ∫ ∫ K φn ( x, y )K φ∗ ( y , z )K φ∗ ( z , t )K φn (t , p )dµ ( y )dµ ( z )dµ (t )
EF
for all E , F ∈ S × S . Hence the required condition holds.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
A. Gupta and B.S. Komal, Binormal and idempotent integral operators,
Int. J. Contemp. Math. Sciences, 6(34) (2011), 1681-1689.
J. Agler and M. Stankus, m-isometries transformations of Hilbert space,
Integral Equations and Operator Theory, 21(1995), 383- 427.
A.A.S. Jibril, On n-power normal operators, The Arabian Journal for
Science and Engineering, 33(2A),(2008), 247-25.
A.A.S. Jibril, On 2-normal operators, Dirasat, 23(2) (1996), 190-194.
A.A.S. Jibril, On operators for which T ∗ T 2 = (T ∗T ) , International
2
2
Mathematical Forum, 5(46) (2010), 2255-2262.
R.K. Singh and J.S. Manhas, Composition Operators on Function Spaces,
North Holland Mathematics Studies 179, Elsevier Sciences Publishers
Amsterdam, New York, (1993).
S.A. Alzuraiqi and A.B. Patel, On n-normal operators, Gen. Math. Notes,
1(2) (2010), 61-73.
8
[8]
S. Panayappan et al.
S.M. Patel, 2-isometric operators, Glasnik Matematicki, 37(57) (2002),
143-147.