The class ?(??) −operators

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International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056

Volume: 11 Issue: 02 | Feb 2024 www.irjet.net p-ISSN:2395-0072

The class ��(����) operators

Department of Mathematics, College of Basic Education, University of Kufa, Najaf, Iraq

Abstract: This article presents a new category of operators called ��(����) operators, which operate on a complex Hilbert space H. An operator �� ∈��(��) is considered a ��(����) operators if the equation (��∗ �� )���� =��(��∗��)�� holds, where �� is a positive integer greater than 1, and ��∗ is the adjoint of the operator ��, We will explore the fundamental characteristics of these operators and provide examples for better understanding

Keywords: Hilbert space, normal operators, ��(��) operators, adjoint operators, bounded linear operators.

1-Introduction

In this article, ��(��) refers to the algebra consisting of all bounded linear operators on a complex Hilbert space ��. A Hilbertspaceisamathematicalspacethathasaninnerproductandisalsocharacterizedbyitscompletenesswithrespect tothe norm induced by this inner product An operator ��∗ is definedas the adjoint of ��ifand onlyif the inner product (����,��)is equivalent to (��,��∗��) for all ��and �� belonging to the set �� A normal operator , which maps from a Hilbert space �� to itself, is defined by the equation ��∗�� =����∗ The theory of operators in Hilbert space has been thoroughly analysedbyseveralauthors,asseenbythereferences[1,2,3].In2021,Elaf.S.A. conductedanexaminationoftheclassof operators ��(��) as described in [4], and presented the fundamental characteristics of this class in a Hilbert space. An operator�� ∈��(��)isreferredtoasa��(��) operatorsifthereexists�� ∈��(��)suchthat�� , and��∗���� =����∗��

This article presents the class ��(����) operators as an extension of the class ��(��) operators and examines its essential characteristics.WeshallestablishtheconditionsunderwhichanoperatorTisconsideredtobe ��(����).Tobeclassifiedin thiscategory,theadditionandmultiplicationoftwooperators��(����)mustmeetcertainconditions,whichwewillexamine

The representation of any operator �� in cartesian form is �� =��+����, where �� and �� represent the real and imaginary components of ��, respectively. The real component of ��, represented as ������, is calculated as the average of ��and its complex conjugate, which is ��+��∗ 2 .On the other hand, the imaginary component of ��, written as ����, is determined by takingthedifferencesbetween��anditscomplexconjugate,dividedby2��,resultingin �� ��∗ 2�� .

2-Main Results

Theobjectiveoftheworkistointroduceanewcategoryofoperatorsknownas��(����) operatorsandanalyze fundamentalpropertiesofthiscategory.

2-1 Definition:

Let��beaboundedoperatorfromacomplexHilbertspace��toitself,the��issaidtobeaclass��(����) operators ifthere existsisanoperator�� ∈��(��)suchthat ��(��∗������)=(��

��

)��,where��isapositiveintegergreaterthan1.

2-2 Example:

Theoperators �� and��aretwooperatorsinthetwo-dimensionalHilbertspace₵2 , where �� =*2�� 22��+and �� =* �� �� 1+

So �� ∈��(��2) operators

International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056

Volume: 11 Issue: 02 | Feb 2024 www.irjet.net p-ISSN:2395-0072

If ��isequalto3,then ��(��∗3 ��3)=*

Therefore,��isnota��(��3)operator.

Iftheoperator�� ∈ ��(����) operators,itisnotnecessarilyanoperatorin��(����+1)for�� >1

Thefollowingstatementsmaybededucedfromthedefinitionsofthenormaloperatorandthe��(����) operators.

2-3 Remarks:

1. Whenthevalueof��=1,itfollowsthat��isa��(��) operator.

2. ��������������������������������������,��ℎ����������ℎ����������

Thesubsequenttheoremshowsalinkbetweenthe��(��) ��������(����) operatorsofthetwoclasses.

2-3 Theorem:

If�� ∈��(��) operators,then�� ∈ ��(����) operatorsforeach�� >1.

Proof:

since�� ∈��(��)

Therefore,weconcludethat��isanoperatorbelongingtothe��(����)class.

The example below demonstrates that theconverseoftheaboveclaimisnottrue.

2-4 Example:

Considertheoperators ��

inthetwo-dimensionalHilbertspace₵2 . Theoperator��belongstotheclass��(����)where��(��

but��(��

��)��. Thus,��isnotamemberofclass��(��)

Theoperator�� canbeclassifiedasamemberofclassD(����)bysatisfyingparticular conditions,asprovenbythe followingtheorem.

2-5 Theorem:

Anoperator ��belongstotheclass��(����)ifandonlyif��∗��commuteswith������and ����

Proof:

Suppose ��isanelementof ��(����),i.e.,��(��∗��)��=(��∗��)����,then��

Ontheotherhand,theequation��

and

��holdtrue. Hence,��

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Volume: 11 Issue: 02 | Feb 2024 www.irjet.net

2-6 Propositions:

Let��:�� →��bea��(����) operator,then

1. ���� alsobelongstotheclassof��(����) operators,where�� >1

2. ����belongstotheclass��(����) operatorswith��∈��.

3. If�� 1 exists,then�� 1 belongstotheclass��(����) operators.

Proof:

1-Todemonstratethat���� isanoperatorinsidetheclassof��(����) operators.Wewillnowproceedwiththeinduction onthevariable��.If�� =1,thentheresultistrue,somoreproofisunnecessary. Letusassumethattheoutcome‘’true’’isvalidfor�� =��

Followingthat,wewillprovethatitholdstruewhen�� =��+1

Theinductionproofisnowcompleted. Therefore,���� ∈��(����) operators.

Thesubsequentassertionsinthisstatementmaybedemonstratedstraightforwardlybasedonthedefinitionofthe ��(����) operatorsasfollows:

2-7 Theorem:

If��and��areinvertibleoperatorsand

operators,

alsobelongstotheclass

) operators. Proof:

International Research Journal of Engineering and

Volume:

Therefore, �� 1∗ belongstotheclassof��(����) operators

3-Properties of ��(����) operators.

Thissectionfocusesonanalyzingalgebraiccharacteristicsthatareassociatedwithourclassof��(����) operators.

3-1 Remark:

If��1,��2 and��areoperatorsbelongingtotheclassof��(����) operators,itisnotcertainthat ��1 +��2 willalsobelongto be��(����) Thefollowingexamplescanhelptoillustratethis.

3-2 Example:

Let��1 =* 1 1 1 1+,��2 =*1 2 2 +aretwooperatorsofclass��(��2)and�� =*1 1 1+∈��(��).

3-3 Example:

Theoperators��1 =[1 �� 1 �� 1]and��2 =[1

aretwoHilbertspace₵ 3 whichareintheclass ��(����) operatorswithaboundedlinearoperator��

Now,thefollowingtheoremholdstrueiftheconditionsnecessaryforremark(3-1)aresatisfied.

3-6 Theorem:

Let��1 and��2 betwo��(����) operatorsonaHilbertspace��suchthat��1��

Since

operators.

Moreover,itisimportanttomentionthattheproductof

Therefore,itcanbeproventhatbyexaminingtheexampleof(3

International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056

Volume: 11 Issue: 02 | Feb 2024 www.irjet.net p-ISSN:2395-0072

Thesubsequenttheoremestablishesthat,undercertainconditions,themultiplicationoftwooperatorsinsidethe specifiedclassislikewiseamemberofthisclass.

3-7 Theorem:

Consider��1 and��2 astwo��(����) operatorsonaHilbertspace��,where

and

2 Inthiscase,it canbeconcludedthat��1��2 isalsoa��(����)operator Proof

Thus,��1��2 isa��(����) operator.

4- Conclusion:

This paper introduces a new class of operators known as ��(����) operators and examines its features through illustrative examples The paper presents the primary findings, which are:

1. If�� ∈��(��) operators,then�� ∈ ��(����) operatorsforevery�� >1

2. Consider��1 and��2 betwo��(����) operatorsonaHilbertspace��.

alsoa��(����) operator.

3. Consider��1 and��2 betwo��(����) operatorsonaHilbertspace��.If ��

alsoa��(����) operator.

:References-4

1. A. Brown, on a Class of Operator, Proce. Amer. Math.Soc,4(1953), 723-728.

2. Berberian, S.K., Introduction to Hilbert space. Chelsea Publishing Company, New Yourk, (1976)

3. Conway,J.,ACourseinFunctionalAnalysis,SpringerVerlag.NewYork,(1985)

4 Elaf,S.A.,TheClassof��(��) operatorsonHilbertSpaces,Int.J.NonlinearAnal.Appl,12(2021),1293-1298.

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