TT kn 0 )1 - Tierärztliche Praxis

QUASI - n - BINORMAL OPERATORS

Dr.R.Jeetendra1 D.Kavitha*

1Assistant Professor, Department of Mathematics, Kongu Engineering college, Perundurai, Erode-

638060, Tamil Nadu , India

*Associate Professor, Department of Mathematics , JCT College of Engineering and Technology,

Pichanur, Coimbatore -641105,Tamilnadu,India

* denotes Corresponding Author

Abstract : In this paper, we investigate some basic properties of quasi – n – binormal operators acting

on a Hilbert space H. Further we study quasi – n – binormal Composite integral operators.

Subject Classification: 47B15, 47B20, 47B25

Keywords: Binormal operator, n – binormal , quasi n – binormal, n - Normal

1 Introduction

Let H be a Hilbertspace L(H) be the algebra of all bounded linear operators acting on H. An operator

)(HLT is called normal if ** TTTT n – normal if ** TTTT nn binormal if TT * commute with

*TT . Isometry if ITT * .

2 – isometry if 02 *22* ITTTT

3 isometry if 033 *22*33* ITTTTTT

n – isometry if if 0)1( *

0

knknn

k

k TTk

n

(or)

0)1(1

)1(.............21

*122*11**

ITT

n

nTT

nTT

nTT nnnnnnnn

n – binormal if if nnnn TTTTTTTT **** .Quasi n – binormal if

TTTTTTTTTT nnnn )()( ****

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Theorem 1.1 If QnBNT then so are

1) KT far any real number k

2) Any )(HLs is unitarily equivalent to T.

3) The restriction T/M of T to any closed subspace M of H that reduces T.

Proof: 1. The proof is straight forward.

2. Let )(HLs be unitarily equivalent to T, then there is a unitary operator )(HLu

Such that ***** , UUTSUUTSUTUS nn

If T is quasi – n – binormal then TTTTTTTTTT nnnn )()( ****

Consider *********** UTTTUTUUTUUTUUTUUTSSSS nnnnnn

)()( ********* UUTUUTUUTUUTUTUSSSSS nnnn = *** UTTTUTT nn ……………….(1)

)2.....(..........*************** TUTTTUTTUTTTUTUTUUUTUUTUUTUUTSSSSS nnnnnnnn

Hence S is unitarily equivalent to T.

4) If T is quasi n – binormal the (T/M) (Quasi nBN)

Theorem 1.2 If )(HLT is n – Binormal then T (Quasi nBN)

Proof: If T is n – binormal, then nnnn TTTTTTTT ****

Post multiply by T

TTTTTTTTTT nnnn )()( ****

Hence T is quasi n – binormal.

Theorem 1.3 Let T )(QnBN and S )(QnBN . If T and S are doubly commuting then TS is quasi n

Binormal.

Proof:

****** )()()()())(( TSTSTSTSTSTSTSTSTSTs nnnnnn

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= ST **** STSTSTST nnnn

= nnnn SSTTTTSSTS ****

= nnnn SSTTTTTSSS ****

= nnnn SSTTTSTTSS ****

= **** SSTTTTSTSS nnnn

= STTSTSTSTS nnnn ****

= ))()()()(( **** TSTSTSTSTS nnnn

= )()()()()( ** TSTSTSTSTS nn

Hence TS is quasi - n – binormal.

Theorem 1.4 : Let S andT be commuting (QnBN) operators such that nTS )( commute with

kknn

k

TSk

n

1

1

. Then (S+T) is n – binormal operator.

Proof

n

k

kknN

K

kknnn TSTSk

nTS

k

nTSTSTSTSTSTSTS

0

*

0

*** )()()()()()())((

=

1

1

***1

1

***)()()()()()(

n

k

nkknnn

k

nkknn TSTTSTSk

nTSSTTSTS

k

nTSSTSTS

1

1

*****1

1

***** )()()()()(n

k

nkknnn

k

nkknn TSTTSTSk

nTSSTTSTS

k

nTSSTSTS

1

1

*****1

1

***** )()()(n

k

nnkknnnn

k

nnkknnn TTSTTSTSk

nTSSSTTTSTS

k

nTSSTSSTS

Since S and T are commuting Quasi n Binormal operators such that *)( TS commute with

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kknn

k

TSk

n

1

1

.

=

`

)())()((1

1

********1

1

**

n

K

nnKKnnnnnkknn

k

nn TTTSTSK

nTSSTSSTTSTTSTS

k

nTSSSTS

1

1

********

1

1

****1

1

**

)()(

*)(

n

k

nnkknnnnnkkn

n

k

nnnnKKnn

K

nn

TTTSTSk

nTSSTSSTTTSTTTSTS

k

nTTTSSTSTSTSSTTSTS

K

nSTSSSSS

1

1

*******1

1

*

**1

1

*****

)())(

()(

n

k

nnkknnnnnkknn

k

nnn

k

nnkknnn

TTTSTSk

nTSSTSSTTTSTS

k

nTS

STSSTTTSTTSTSk

nTSSSS

nn

k

nkknnn

n

k

nnkknnn

TTTSTSTSSTSSTS

XTTSTTSTSk

nTSSS

1

1

*****

1

1

*****

)(

)(

= ))()()()((1

1

****1

1

nkknn

k

nnkknn

k

n TTSk

nSTSTSTSTTS

k

nS

= )()()(0

** TSTSk

nTSTSTS

k

nkkn

n

k

kknn

ok

= TSTSTSTSTS nn ()()()() **

Hence (S+T) is quasi n – binormal operator.

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Example:1.1 Let

10

11T be an operator on 2R which is 3QBN but either 3 binormal nor

normal. Let

11

01T be 3 Quasi binormal operator.Then TTTTTTTTTT )( 3**33**3

10

01

10

11

10

112T

10

11

10

11

10

01.23 TTT

11

12

11

01

10

11*3TT

21

11

10

11

11

013*TT

10

11

21

11

11

12

10

113**3 TTTTT

10

11

10

11

21

11

11

12)( 3**3 TTTTT

Example 1.2 Let

01

11

10

01TS be not commuting (2QBN) operators. Then ST is 2 quasi

Binormal operators.

Solution: Let

10

01

10

01TS be two operators.

Then

10

01

10

01

10

01ST

10

01

10

01

10

012ST

10

01)( *ST

10

01)()()()( 2**2

STSTSTSTST

10

01)()()()()( 2**2 STSTSTSTST .Hence ST is quasi n binormal operators.

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Example: 1.3 If T is quasi Binormal operator on 2R , then *T is also quasi Binormal operator on 2R .

Solution Consider the operator T =

11

21 Then it can be easily verified that

TTTTTTTTTT )(99

189)( ****

and

199

189)()( ****** TTTTTTTTTT

Hence *T is also quasi binormal operator on 2R .

Example: 1.4 )(HBT be an invertible operator on the real Hilbert Space H. If

11

02T .

Then both T and *T are quasi normal op erators. Also T and *T are quasi binormal operators.

Solution :

11

02T and

21

01*T implies that TTTTTT )(

22

04)( **

. Hence

T is quasi normal operator. Further **** )(

42

02)( TTTTTT

. Hence *T is also quasi normal

operator.

Implies that T is quasi binormal operator.

****** )(84

04)( TTTTTTTTTT

.Hence *T is also quasi binormal operator.

Theorem: 1.5 Let )(HLT with the Cartesian decomposition T = A+iB .Then T is binormal if and only if

3333 BABAABAB and 2222 BABABBABABAA

Proof: Since T is quasi binormal .Then TTTTTTTTTT )()( ****

))()()()(()( ** iBAiBAiBAiBAiBATTTTT

= ))()()(( 22 BAiBAiBAiBA

TTTTTTTTTT )(40

08)( ****

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= 53242323234325 iBBiAABBABAAiBBiAABA

))()()()(()( ** iBAiBAiBAiBAiBATTTTT

= ))()()(( 22 BAiBAiBAiBA

= ))()(( 3223 iBBiAABAiBAiBA

= ))(( 322322 iBBiAABABA

= 532432324235 iBBiAABABBiABiABAA

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

Theorem : 1.6 If TC is quasi n binormal operator on a Hilbert space and sC is unitarily equivalent to

TC , then SC is quasi n binormal operator.

Proof:

SC is unitarily equivalent to TC then an unitary operator U such that *UUCC TS

Consider sS

n

S

n

SSS

n

S

n

SSS CCCCCCCCCC )()(****

))()()())((()( *********UUCUUCUUCUUCUUCCCCCC T

n

T

n

TTTS

n

S

n

SSS

= ******* UUCUUCUUCUUCUUC T

n

T

n

TTT

= ***UCCCCUC T

n

T

n

TTT

= *22* )( UCCCUn

TTT ………..(1)

)()()()()(()( *********UUCUUCUUCUUCUUCCCCCC TT

n

T

n

TTSS

n

S

n

SS

= *******UUCUUCUUCUUCUUC TT

n

T

n

TT

= ***UCCCCUC TT

n

T

n

TT

= *22*)( UCCCU

n

TTT ……………(2)

From( 1) and( 2) SC quasi - n – binormal composition operator.

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2. Quasi –n - Binormal composite integral operators

),,( SXLet be a finite measure space and Let XX : be a non singular measurable

transformation. 0)(0)( 1 EE Then a Composition transformation for p1

)()(: pp LLC is defined by fofC for every )(pLf

In case C is continuous .We call it a composition operator induced by . C is bounded operatorif and

only if 0

1

fd

d

.This Random Nikodym derivative of the measure 1 w.r.t measure and it is

essentially bounded.

A kernel )( xLK p always induces a bounded integral operator )()(: pp

k LLT

defined by )()(),())(( ydyfyxKxfTk

Given a kernel k and a nonsingular measurable function XX : the composite integral operator

kT induced by ),( k is a bounded linear operator )()(: pp

k LLT defined by

)())((),())(( ydyfyxKxfTk

= )()(),( ydyfyxk

)())((),())(( ydyfyxKxfT nn

k

= )()(),( ydyfyxk n

121,11,2211 ...)()()....,(),(....),( nnnn

n dzdzdzyzKzzkzzkzxkyxk

Theorem 2.1 Let ).(2 xLk Then kT is quasi n binormal if and only if

)(()(()()()(),(),(),(),()),(**

sdpdtdzdydspkptktzkzyKyxknn =

)(()()()(()((),(),(),(),(),(**

sdpdtdzdydspkptktzkzykyxknn

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Proof: Suppose the condition is true. For )(, 2 Lgf .We have

)()())((,**** xdxgxfTTTTTgfTTTTT k

n

k

n

kkkk

n

k

n

kkk

= )()()())()(((**

xdxgydyfTTTTxyk k

n

k

n

kk

= )()()()())(,((),(**

xdxgydzdzfTTTzykyxk k

n

k

n

k

= )()()()()())(),(),()( **xdxgydzdtdtfTTtzkzykxyk k

n

k

n

= )()()()()()()(),()(,(),(),(*

xdxgydzdtdpdpfptktzkzykyxknn

))()()()()()()()()(),(),(),(),(),(**

xdxgydzdtdpdpfsdsfspkptktzkzykyxknn

)()()()(

)()()()()()()()(),(),(),(),(),(**

xdxfydyf

zdzftdtfpdpfsdsfspkptktzkzykyxknn

...(1)

And )()())((),(

****xdxgxfTTTTTgfTTTTT k

n

kkk

n

kk

n

kkk

n

k

=

)()(

)()()()()()()()()()(),(),(),(),(),(**

xdxf

ydyfzdzftdtfpdpfsdsfspkptktzkzykyxknn

…….(2)

It follows from (1) &(2) that kT is quasi n binormal operator.

Conversely suppose that kT is quasi n – binormal take Ef and

Fg Gh JI we

See that )()()()(),(),(),(),(),(**

tdzdydxdspkptktzkzykyxknn

I E F G

= ),(),(),(),(),(**

spkptktzkzykyxknn

I E F G

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Hence the required condition holds.

Quasi – n – binormal operators in Fockspace

Let F be a Fock Space and C is quasi – n – binormal iff CCCCCCCCCCnnnn

).()(****

Theorem: 3 Let C be a Composition operator on F. Then C is quasi - n –binormal if and only if

if and only if CCCCCCCCCCnnnn

).()(****

Proof:

Consider ).()( )()()(

**nn

Bn

B ooKoK

nnCMCMCCCCCC

= ookooK n

onBon

BoCMCM )(

)(.. ………………………(1)

CCMCCCMCCCCCBB K

nn

K

nn )()(**

= CCMCM nn

Bn

B ooKoK ).( )()()(

= oooOKooK nn

Bn

BoCMCM )()()( ... ………………………………(2)

Hence the result.

Conclusion

In this paper some basic properties of quasi – n – binormal operators acting on a Hilbert space H are

characterized.

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.

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ISSN: 0303-6286

[5] A.A.S. Jibril, On operators for which 2*22* )(, TTTT 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.

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