Volume 8, Number 2, 2022, Pages 337-342 https://doi.org/10.34198/ejms.8222.337342
Received: December 24, 2021; Accepted: February 12, 2022 2010 Mathematics Subject Classification: 47B20.
Keywords and phrases: normal operator, quasi normal operator, binormal operator, Hilbert space.
-Power Quasi Binormal Operator
Alaa Hussein Mohammed
Department of Mathematics, College of Education, University of Al-Qadisiyah, Diwaniya, Iraq e-mail: [email protected]
Abstract
In this paper we introduce a new class of operators on Hilbert space called -power quasi binormal operator. We study this operator and give some properties of it.
Introduction
Consider ( ) be the algebra of all bounded linear operators on Hilbert space . An operator is called normal if ∗ = ∗. Quasi normal operator was introduced by Brown in 1953 [1]. In [3] Campbell introduced the class binormal of operator which is defined as ∗ ∗= ∗ ∗ .
In [5] Sid Ahmed generalize quasi normal operator to -power quasi normal operator. In this paper we define a new class of operators on Hilbert space as ( ∗ ∗) = ( ∗ ∗ ) called q-power quasi binormal operator and study some properties of it.
1. Main Results
Definition 1.1. Let be bounded operator. Then is called q-power quasi binormal operator if and only if ( ∗ ∗) = ( ∗ ∗ ) , where is a nonnegative integer.
Proposition 1.2. If S is a self adjoint and q-power quasi binormal operator, then ∗ is a q-power quasi binormal operator.
Proof. Since is -power quasi binormal operator, ( ∗ ∗) = ( ∗ ∗ ) .
Let
( ∗) (( ∗)∗ ∗ ∗( ∗)∗) = ( ∗) ( ∗ ∗ ), since is a self adjoint
= ( ∗ ∗), since is -power quasi binormal
= ( ∗ ∗ ) , since is a self adjoint
= ( ∗( ∗)∗( ∗)∗ ∗)( ∗) . Hence, ∗ is -power quasi binormal operator.
Proposition 1.3. If S is a q-power quasi binormal operator, and if exist, then is a q-power quasi binormal operator.
Proof. Since is -power quasi binormal operator, ( ∗ ∗) = ( ∗ ∗ ) . Let
( ) ( )∗ ( )∗) = ( ) ( ∗) ( ∗) )
= ( ) ( ∗) ( ∗ ) )
= ( ) ( ∗ )( ∗)
= ( ∗ ∗) , since is binormal
= ( ∗ ∗ ) ,
since is a -power quasi binormal,
= ( ∗ ∗) , since is binormal
= ( ∗ ∗ )
= ( ∗ ∗ ) ( )
= ( ∗ ) ( ∗) ( )
= ( ∗) ( ∗) ( )
= ( )∗( )∗ ( ) . Hence, is -power quasi binormal operator.
Definition 1.4 [4]. If , are bounded operator on Hilbert space . Then , are unitary equivalent if there is an isomorphism : → such that = ∗.
Proposition 1.5. If S is q-power quasi binormal operator and if ∈ ( ) is unitary equivalent to S, then R is q-power quasi binormal operator.
Proof. Since is unitary equivalent to , = ∗, ( ∗) = ∗ and since is -power quasi binormal operator, ( ∗ ∗) = ( ∗ ∗ ) .
Let
( ∗ ∗) = ( ∗) ( ∗)∗( ∗)( ∗)( ∗)∗
= ( ∗) ( ∗ ∗) ( ∗)( ∗)( ∗ ∗)
= ( ∗ ∗) ∗,since is -power quasi binormal operator
= ( ∗ ∗ ) ∗
= ( ∗)( ∗ ∗)( ∗ ∗)( ∗) ( ∗) = ( ∗ ∗ ) . Hence is -power quasi binormal operator.
Theorem 1.6. The set of all q-power quasi binormal operators on H is a closed subset of ( ) under scalar multiplication.
Proof. Let
( ) = ∈ ( ): is -power quasi binormal operator on for some nonnegative integer
Let ∈ ( ), then we have is -power quasi binormal operator and thus ( ∗ ∗) = ( ∗ ∗ ) .
Let ! be a scalar, hence
"! # (! )∗(! )(! )(! )∗ = ! !̅ ∗(! )(! ))!̅ ∗
= ! !̅!!!% ( ∗ ∗)
= ! !̅!!!% ( ∗ ∗ )
= (! )(! )∗(! )∗(! ) "! # . Thus ! ∈ ( ).
Let & be a sequence in ( ) and converge to , then we can get that
‖ ( ∗ ∗) − ( ∗ ∗ ) ‖
= ) ( ∗ ∗) − &( &∗ & & &∗) + ( & &∗
&∗
&) & − ( ∗ ∗ ) )
≤ ) ( ∗ ∗) − &( &∗ & & &∗)) + )( & &∗
&∗
&) & − ( ∗ ∗ ) ) → 0 as / →∞.
Hence, ( ∗ ∗) = ( ∗ ∗ ) .
Therefore ∈ ( ). Then, ( ) is closed subset.
Theorem 1.7. If 0 and are normal, -power quasi binormal operators on , and let commute with 0, then ( 0) is -power quasi binormal operator on .
Proof. Since T and S are -power quasi binormal operators ( ∗ ∗) = ( ∗ ∗ ) and 0 (0∗000∗) = (00∗0∗0)0 ,
( 0) (( 0)∗( 0)( 0)( 0)∗) = (0 ) (0∗ ∗)( 0)( 0)(0∗ ∗)
= (0 ) 0∗ ∗ 0 0 0∗ ∗
= (0 ) 0∗ ∗0 0∗0 ∗
= (0 ) 0∗ ∗0 0∗ ∗0
= (0 ) 0∗0 ∗ 0∗ ∗0
= (0 ) 00∗ ∗ 0∗ ∗ 0
= 0 00∗ ∗ 0∗ ∗ 0
= 0 00∗ ∗ 0∗ ∗ 0
= 0 0 0∗ ∗ 0∗ ∗ 0
= 00 0∗ ∗ 0∗ ∗ 0
⋮
= ( 0)(0∗ ∗)(0∗ ∗)( 0) (0 )
= (( 0)( 0)∗( 0)∗( 0))( 0) . Then ( 0) is -power quasi binormal operator.
Theorem 1.8. Let 0 , 02, … , 0& are q-power quasi binormal operators on H. Then the direct sum ( 0 ⨁02⨁ … ⨁0&) is q-power quasi binormal operator on .
Proof. Since every operator of 0 , 02, … , 0& is -power quasi binormal,
05 (05∗0505 ∗)
= (0505∗05∗05)05 for all 6 = 1,2, … , /
= (0 ⨁02⨁ … ⨁0&) (0 ⨁02⨁ … ⨁0&)∗(0 ⨁02⨁ … ⨁0&)
× (0 ⨁02⨁ … ⨁0&)(0 ⨁02⨁ … ⨁0&)∗
= ( 0 ⨁02 ⨁ … ⨁0& ) ( 0 ∗⨁02∗⨁ … ⨁0&∗)(0 ⨁02⨁ … ⨁0&)
× (0 ⨁02⨁ … ⨁0&)( 0 ∗⨁02∗⨁ … ⨁0&∗)
= 0 (0∗0 0 0∗)⨁02 (02∗020202∗)⨁ … ⨁0& (0&∗0&0&0&∗)
= (0 0∗0∗0 )0 ⨁(0202∗02∗02)02 ⨁ … ⨁(0&0&∗0&∗0&)0&
= (0 ⨁02⨁ … ⨁0&)(0 ⨁02⨁ … ⨁0&)∗(0 ⨁02⨁ … ⨁0&)∗(0 ⨁02⨁ … ⨁0&)
× (0 ⨁02⨁ … ⨁0&)
Thus, (0 ⨁02⨁ … ⨁0&) is -power quasi binormal operator on .
Theorem 1.9. Let 0 , 02, … , 0& are -power quasi binormal operators on . Then the tenser product ( 0 ⨂02⨂ … ⨂0&) is -power quasi binormal operator on .
Proof. Since every operator of , 2, … , & is a -power quasi binormal, 05 (05∗0505 ∗)
= (0505∗05∗05)05 for all 6 = 1,2, … , /
= (0 ⨂02⨂ … ⨂0&) (0 ⨂02⨂ … ⨂0&)∗(0 ⨂02⨂ … ⨂0&)
× (0 ⨂02⨂ … ⨂0&)(0 ⨂02⨂ … ⨂0&)∗ (;<⨂;=⨂ … ⨂;>)
= ( 0 ⨂02 ⨂ … ⨂0& ) ( 0 ∗⨂02∗⨂ … ⨂0&∗)(0 ⨂02⨂ … ⨂0&)
× (0 ⨂02⨂ … ⨂0&)( 0 ∗⨂02∗⨂ … ⨂0&∗) (;<⨂;=⨂ … ⨂;>)
= 0 (0∗0 0 0∗);<⨂02 (02∗020202∗);=⨂ … ⨂ 0& (0&∗0&0&0&∗);>
= 0 (0 ∗0 0 0 ∗) ;<⨂ (0202∗02∗02)02 ;=⨂ … ⨂ (0&0&∗0&∗0&)0& ;>
= (0 ⨂02⨂ … ⨂0&)(0 ⨂02⨂ … ⨂0&)∗(0 ⨂02⨂ … ⨂0&)∗(0 ⨂02⨂ … ⨂0&)
× (0 ⨂02⨂ … ⨂0&) (;<⨂;=⨂ … ⨂;>).
Thus (0 ⨂02⨂ … ⨂0&) is -power quasi binormal operator.
References
[1] A. Brown, On a class of operators, Proc. Amer. Math. Soc. 4 (1953), 723-728.
https://doi.org/10.1090/S0002-9939-1953-0059483-2
[2] S. K. Berberian, Introduction to Hilbert Space, 2nd ed., Chelsea Publishing Co., New York, 1976.
[3] Stephen L. Campbell, Linear operators for which 0∗0 and 00∗commute, Proc. Amer.
Math. Soc. 34 (1972), 177-180. https://doi.org/10.2307/2037922
[4] J. B. Conway, A course in functional analysis, Graduate Texts in Mathematics, 96, Springer-Verlag, New York, 1985. https://doi.org/10.1007/978-1-4757-3828-5
[5] Ould Ahmed Mahmoud Sid Ahmed, On the class of n-power quasi-normal operators on Hilbert space, Bull. Math. Anal. Appl. 3(2) (2011), 213-228.
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