# α,β-Normal Operators in Several Variables

Authors: Sid Ahmed Ould Ahmed Mahmoud; Ahmed Himadan Ahmed; Ahmad Sarosh
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

We consider an extension of the concept ofα,β-normal operators in single variable operator to tuples of operators, similar to those extensions of the concepts of normality to joint normality, hyponormality to joint hyponormality, and quasi-hyponormality to joint quasi-hyponormality.


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## Body

## 1. Introduction

Through this paper, we denote Hilbert space over the field of complex numbersℂ by ℋ,.|.. We write ℬℋ for the set of all bounded linear operators on ℋ. For an operator T∈ℬℋ, the adjoint, the kernel, and the range of T are denoted by T∗, NT, and ℛT, respectively. For T,S∈ℬℋ,T is said to be positive if Tx|x≥0 for all x∈ℋ and T≥S if and only if T−S≥0. We let T,S:=TS−ST.The class of normal operators on Hilbert spaces plays a critical role in operator theory. This class of operators has been generalized by many authors to non-normal operators.Letα,β∈ℝ2 such that 0≤α≤1≤β. An operator T∈ℬℋ is called(1)
Normal ifT∗,T=0⇔Tx=T∗x,∀x∈ℋ [1].(2)
Hyponormal ifT∗,T≥0⇔Tx≥T∗x,∀x∈ℋ [2, 3].(3)
α,β-normal 0≤α≤1≤β [4] if(1)α2T∗T≤TT∗≤β2T∗T,or equivalently(2)αTx≤T∗x≤βTx,for all x∈ℋ. It should be noted that for α=1=β, T is a normal operator. For α=1, we observe that T∗ is hyponormal and for β=1, we obtain that T is hyponormal. Interested readers can find more details on α,β-normal operators in [4–9]. The concept of p-α,β-normal operators was introduced by Senthilkumar and Shanthi [10]. An operator T∈ℬℋ is said to be p-α,β-normal operator for 0<p≤1 if(3)α2T∗Tp≤TT∗p≤β2T∗Tp,0≤α≤1≤β.Whenp=1, this coincides with α,β-normal operators.The study of tuples of commuting operators has received great attention in recent years by many authors, and the interested reader can refer to [11–20] for complete details.LetT=T1,…,Tm∈ℬℋm be a m-tuple of operators. T is said to be jointly normal if TiTj=TjTi and every Ti is a normal operator for all i,j=1,…,m (see [21]).Given anm-tuple T:=T1,…,Tm∈ℬℋm, let T∗,T∈ℬℋ⊕…⊕ℋ denote the self-commutator of T, defined by T∗,Ti,j:=Tj∗,Ti for all i,j∈1,…,m. In [22], the author has introduced the concept of joint hyponormality of operators. An m-tuple T=T1,…,Tm of operators is called joint hyponormal, if the operator matrix(4)T∗,T=Tj∗,Tii,j=1m=T1∗,T1T2∗,T1⋯Tm∗,T1,T1∗,T2T2∗,T2⋯Tm∗,T2⋮⋮⋮⋮T1∗,TnT2∗,Tn⋯Tm∗,Tm,is positive on the direct sum of m copies of ℋ (i.e., ⊕mℋ:=ℋ⊕…⊕ℋ ) or equivalently if ∑1≤i,j≤mTi∗,Tjui|uj≥0, for each finite collection u1,…,um of ℋ (see [22]).Recently, the first named author in [23] has introduced the concept of joint m-quasi-hyponormal tuple of operators as follows. An m-tuple of operators T=T1,⋯,Tm∈ℬℋm is said to be joint m-quasi-hyponormal if and only if T satisfies(5)∑1≤i,j≤mTj∗Tj∗,TiTiuj|ui≥0,for u1⋯,um∈ℋ. Or equivalently, is joint m-quasi-hyponormal tuple if the operator matrix(6)T∗T∗,TT=Tj∗Tj∗,TiTii,j=1m=T1∗T1∗,T1T1T2∗T2∗,T1T1⋯Tm∗Tm∗,T1T1T1∗T1∗,T2T2T2∗T2∗,T2T2⋯Tm∗Tm∗,T2T2⋮⋮⋮⋮T1∗T1∗,TmTmT2∗TT2∗,TmTm⋯Tm∗Tm∗,TmTm,is positive on the space ⊕k=1mℋ.ForT∈ℬℋ, the numerical radius ωT of an operator T∈ℬℋ is defined by(7)ωT=supTx|x:x=1.These concepts were later generalized by Dekker [24] to joint numerical radius of T=T1,…,Tm∈ℬℋm as follows.The joint numerical radius ofT∈ℬℋm is defined [25] by(8)ωT=sup∑1≤k≤mTkx|x21/2,x∈ℋ:x=1.Now, for a givenm-tuple of operators T=T1,…,Tm, we consider the joint operator norm of T defined by(9)T=sup∑1≤k≤mTkx1/2,x∈ℋ:x=1.Note that‖T‖ was introduced by Cho and Takaguchi in [26].It was proved in [6] that if T∈ℬℋ is an α,β-normal operator and λ∈ℂ, then(10)αT2≤ωT2+2βT−λT∗21+λα2,and in [7],(11)1+α2T2≤12T−T∗2+ωT2.The study of classes of operators and related topics is one of the hottest areas in operator theory in Hilbert spaces. In this work, we are going to consider an extension of the concept ofα,β-normal operators in single variable to tuples of operators, similar to those extensions of the concepts of normality to joint normality, hyponormality to joint hyponormality, and quasi-hyponormality to joint quasi-hyponormality (for more details, see [21–23]).The outline of the paper is as follows. Section2 is devoted to the study of our new class of multioperators. We give several remarks and examples, which try to clarify the context of the concept of joint α,β-normal tuples. We show that the product of a joint α,β-normal tuple by an m-tuple of operators satisfying suitable conditions is joint α,β-normal tuple. In Section 3, some inequalities involving joint operator norms and joint numerical radius for joint α,β-normal tuples are proved under suitable conditions.


## 2. Jointα,β-Normal Operators

This section is devoted to the study of the class of jointlyα,β-normal operators acting on complex Hilbert spaces. We show some basic results related to this class of operators.LetT=T1,…,Tm∈ℬℋm and set(12)T=T1∗T1T2∗T1⋯Tm∗T1T1∗T2T2∗T2⋯Tm∗T2⋮⋮⋮⋮T1∗TmT2∗Tm⋯Tm∗TmandT∗=T1T1∗T1T2∗⋯TmT1∗T1T2∗T2T2∗⋯T1T∗⋮⋮⋮⋮TmT1∗TmT2∗⋯TmTm∗.Definition 1.
LetT=T1,…,Tm∈ℬℋm be an m-tuple of operators and let 0≤α≤1≤β. We say that T is a joint α,β-normal tuple of operators if the operator matrix T:=Ti∗Tj1≤i,j≤m satisfies(13)α2T≤T∗≤β2Torβ2T≥T∗≥α2T,on the direct sum of m-copies of ℋ or equivalently(14)β2∑1≤i,j≤mTjxi|Tixj−∑1≤i,j≤mTi∗xi|Tj∗xj≥0,∑1≤i,j≤mTi∗xi|Tj∗xj−α2∑1≤i,j≤mTjxi|Tixj≥0,forx1,⋯,xm∈ℋ.Remark 1.
Ifm=1, (14) coincides with (1).Remark 2.
LetT=T1,…,Tm∈ℬℋm be a joint normal tuple of operators; then, T is a joint 1,1-normal tuple.Proposition 1.
LetT=T1,…,Tm∈ℬℋm such that TiTj∗=0 for i≠j,1≤i,j≤m. Then, T is joint α,β-normal if and only if(15)α2∑1≤k≤mTkxk2≤∑1≤k≤mTk∗xk2≤β2∑1≤k≤mTkxk2,for x1,…,xm in ℋ.Proof.
Under the assumptions thatTiTj∗=0 for i≠j and 1≤i,j≤m, it follows that for each finite collections x1,…,xm∈ℋ,(16)β2∑1≤i,j≤mTjxi|Tixj−∑1≤i,j≤;mTi∗xi|Tj∗xj≥0,β2∑1≤k≤mTkxk2−∑1≤k≤mTk∗xk2≥0,and(17)∑1≤i,j≤mTi∗xi|Tj∗xj−α2∑1≤i,j≤mTjxi|Tixj≥0,∑1≤k≤mTk∗xk2−α2∑1≤k≤mTkxk2≥0.
Forz∈ℂ, let Re(z) denote the real part of z.Theorem 1.
LetT=T1,…,Tm∈ℬℋm be an m-tuple of operators. If T is a joint α,β-normal operator, then(18)α2Re∑1≤i,j≤mTjxi|Tixj≤Re∑1≤i,j≤mTi∗xi|Tj∗xj≤β2Re∑1≤i,j≤mTjxi|Tixj,for x1,…,xm∈ℋ.Proof.
SinceT is a joint α,β-normal operator, it follows from (14)that(19)β2Re∑1≤i,j≤mTjxi|Tixj−Re∑1≤i,j≤;mTi∗xi|Tj∗xj≥0,Re∑1≤i,j≤mTi∗xi|Tj∗xj−α2Re∑1≤i,j≤mTjxi|Tixj≥0,forx1,…,xm∈ℋ.
We deduce from the above inequalities that(20)β2Re∑1≤i,j≤mTjxi|Tixj−Re∑1≤i,j≤;mTi∗xi|Tj∗xj≥0,Re∑1≤i,j≤mTi∗xi|Tj∗xj−α2Re∑1≤i,j≤mTjxi|Tixj≥0,forx1,…,xm∈ℋ,from which (18) follows.
Question. Assume that T satisfies (18); is it then true that T is a joint α,β-normal operator?Remark 3.
IfT=T1,…,Tm∈ℬℋm is joint 0,1-normal, then T is joint hyponormal. In fact, taking α=0 and β=1 in (14), we get(21)∑1≤i,j≤mTi∗Tjxi|xj−∑1≤i,j≤mTjTi∗xi|xj≥0,for x1,…,xm∈ℋ, or equivalently(22)∑1≤i,j≤mTi∗Tj−TjTi∗xi|xj=∑1≤i,j≤mTi∗,Tjxi|xj≥0,which yields T∗,T≥0 on ⊕mℋ. Therefore, T is joint hyponormal.Remark 4.
LetT=T1,…,Tm∈ℬℋm be a commuting tuple of operators. If T is joint 1,1-normal, then T is a joint normal operator.
In fact, sinceT is joint 1,1-normal, it follows from (14) that(23)∑1≤i,j≤mTjxi|Tixj−∑1≤i,j≤mTi∗xi|Tj∗xj≥0,and(24)∑1≤i,j≤mTi∗xi|Tj∗xj−∑1≤i,j≤mTjxi|Tixj≥0.
We deduce that(25)∑1≤i,j≤mTi∗xi|Tj∗xj=∑1≤i,j≤mTjxi|Tixj.
This implies that(26)∑1≤i,j≤mTjTi∗x|x=∑1≤i,j≤mTi∗Tjx;|x,for each x∈ℋ. In particular,(27)TjTi∗−Ti∗Tjx|x=0,∀x∈ℋand∀i,j=1,…,m.
Consequently,TiTi∗=Ti∗Ti for i=1,…,m. So, T is joint normal.Example 1.
ConsiderT=T1,T2 where T1=001010100 and T2=−1000−1000−1. A simple calculation shows that T is joint α,β-normal for all α,β:0≤α≤1≤β.
The following example shows that there exists am-tuple of operators T=T1,…,Tm∈ℬℋm such that each Tj is an α,β-normal operator for all j=1,…,m, but T=T1,…,Tm is not joint α,β-normal.Example 2.
Letℋ=ℂ2 and eii=12 be the standard orthonormal basis of ℂ2. Consider T=T1,T2∈ℬℋ2 where T1=1011 and T2=10−11. It was observed that T1 and T2 are α,β-normal with α=3−5/2 and β=3+5/2 (see [4, 8]). However, a short calculation shows that(28)∑1≤i,j≤2Ti∗ei|Tj∗ej−3+52∑1≤i,j≤2Tiej|Tjei≥0.
Consequently,T is not joint 3−5/2,3−5/2-normal.Example 3.
LetT∈ℬℋ be α,β-normal. Then, T=T,…,T∈ℬℋm is joint α,β-normal. Indeed, Since T is α,β-normal, it follows that(29)α2Tx|Tx≤T∗x|T∗x≤β2Tx|Tx,∀x∈ℋ.
Letx1,x2,…,xm∈ℋ. Using these inequalities for x=∑1≤i≤mxi, we can write(30)β2T∑1≤i≤mxi|T∑1≤i≤mxi−T∗∑1≤i≤mmxi|T∗∑1≤i≤mxi≥0,and(31)T∗∑1≤i≤mxi|T∗∑1≤i≤mxi−α2T∑1≤i≤mxi|T∑1≤i≤mxi≥0.
Moreover, one has(32)β2∑1≤i,j≤mTxi|Txj−∑1≤i,j≤mT∗xi|T∗xj≥0,and(33)∑1≤i,j≤mT∗xi|T∗xj−α2∑1≤i,j≤mTxi|Txj≥0,for x1,…,xm∈ℋ. This means that T is joint α,β-normal.Proposition 2.
LetT∈ℬℋ and consider T=T,…,T∈ℬℋm. Then, T is joint α,β-normal if and only if T is α,β-normal.Proof.
Assume thatT is α,β-normal. In view of Example 3, we know that T is joint α,β-normal. Conversely, assume that T is joint α,β-normal. From Definition 1, it follows that(34)β2∑1≤i,j≤mTxi|Txj−∑1≤i,j≤mT∗xi|T∗xj≥0,and(35)∑1≤i,j≤mT∗xi|T∗xj−α2∑1≤i,j≤mTxi|Txj≥0,for each collection forxi1≤i≤m∈ℋ. In particular, for x=x1=⋯=xm, we get(36)β2Tx|Tx−T∗x|T∗x≥0,and(37)T∗x|T∗x−α2Tx|Tx≥0,or equivalently,(38)α2Tx|Tx≤T∗x|T∗x≤β2Tx|Tx,for all x∈ℋ. So, T is α,β-normal.
In the general case, we have the following theorem.Theorem 2.
LetT=T1,…,Tm be m-tuple of operators on ℋ and let 0≤α≤1≤β. Assume that(39)β2Ti∗Tj−TjTi∗=0fori≠j,TjTi∗−α2Ti∗Tj=0fori≠j.Then,T=T1,…,Tm is joint α,β normal tuple if and only if each Ti is α,β normal for i=1,…,m.Proof.
We have thatT=T1,…,Tm is joint α,β normal tuple if and only if(40)β2T−T∗≥0,T∗−α2T≥0.
However, by using (37), we get(41)β2T−T∗≥0,T∗−α2T≥0.⇔β2∑1≤k≤mTkx2−∑1≤k≤mTk∗x2≥0,∀x∈ℋ∑1≤k≤mTk∗x2−α2∑1≤k≤m‖Tkx‖2≥0,∀x∈ℋ⇔∑1≤k≤mβ2Tkx2−Tk∗x2≥0,∀x∈ℋ.∑1≤k≤mTk∗x2−α2Tkx2≥0∀x∈ℋ.⇔β2Tkx2−Tk∗x2≥0,∀x∈ℋTk∗x2−α2Tkx2≥0,∀x∈ℋ⇔Tkisα,β−normalfork=1,…,m.Proposition 3.
LetT=T1,…,Tm∈ℬℋm and let α,β∈ℝ2 such that 0<α≤1≤β. If TiTj∗=0 for all i≠j, then T is joint α,β-normal if and only if T∗=T1∗,…,Tm∗ is joint 1/β,1/α-normal.Proof.
Assume thatT is joint α,β-normal and prove that T∗ is joint 1/β,1/α-normal. To do so, we have the following in view of (15):(42)α2∑1≤k≤mTkx2≤∑1≤k≤mTk∗x2≤β2∑1≤k≤mTkx2,∀x∈ℋ.from which it follows that(43)1β2∑1≤k≤mTk∗x2≤∑1≤k≤mTkx2≤1α2∑1≤k≤mTk∗x2,∀x∈ℋ.
Therefore,T∗ is joint 1/β,1/α-normal tuple. Conversely, if T∗ is joint 1/β,1/α-normal, we get(44)1β2∑1≤k≤mTk∗x2≤∑1≤k≤mTkx2≤1α2∑1≤k≤mTk∗x2,∀x∈ℋ.
This means that(45)α2∑1≤k≤mTkx2≤∑1≤k≤mTk∗x2≤β2∑1≤k≤mTkx2,∀x∈ℋ.
Hence,T is a joint α,β-normal tuple by (13).
The following corollary is an immediate consequence of Proposition3.Corollary 1.
LetT=T1,…,Tm∈ℬℋm. Let α,β∈ℝ2 such that 0<α≤1≤β and αβ=1. If TiTj∗=0 for i≠j, then T is joint α,β-normal if and only if T∗ is joint α,β-normal.Proposition 4.
LetT=T1,…,Tm∈ℬℋm be joint α,β-normal. The following statements hold.(1)
μT:=μ1T1,…,μmTm is joint α,β-normal for all μ:=μ1,…,μm∈ℂm.(2)
IfV∈ℬℋ is an isometry, then VTV∗=VT1V∗,…,VTmV∗ is joint α,β-normal.Proof.
(1) SinceT is joint α,β-normal, it follows for each collection x1,…,xm∈ℋ that(46)=β2∑1≤i,j≤mμjTjxi|μiTixj−∑1≤i,j≤mμiTi∗xi|μjTj∗xj=β2∑1≤i,j≤mTjμi¯xi|Tiμj¯xj−∑1≤i,j≤mTi∗μi¯xi|Tj∗μj¯xj=β2∑1≤i,j≤mTjui|Tiuj−∑1≤i,j≤mTi∗ui;|Tj∗ujui=μi¯xi≥0.
On the other hand, for same reason, we have(47)=β2∑1≤i,j≤mμjTjxi|μiTixj−∑1≤i,j≤mμiTi∗xi|μjTj∗xj=β2∑1≤i,j≤mTjμi¯xi|Tiμj¯xj−∑1≤i,j≤mTi∗μi¯xi|Tj∗μj¯xj=β2∑1≤i,j≤mTjui|Tiuj−∑1≤i,j≤mTi∗ui;|Tj∗ujui=μi¯xi≥0.
Consequently,(48)α2μT≤μT∗≤β2μT.
Thus,μT is joint α,β-normal as required. (2) Under the assumption that V is an isometry, we have for each collection x1,…,xm∈ℋ,(49)∑1≤i,j≤mVTjV∗∗xi|VTiV∗∗xj−α2∑1≤i,j≤mVTiV∗xi|VTjV∗xj=∑1≤i,j≤mTi∗V∗xi|Tj∗V∗xj−α2∑1≤i,j≤mTjV∗xi|TiV∗xj=∑1≤i,j≤mTi∗yi|Tj∗yj−α2∑1≤i,j≤mTjyi|Tiyjyi=V∗xi≥0.
On the other hand,(50)β2∑1≤i,j≤mVTjV∗xi|VTiV∗xj−∑1≤i,j≤mVTiV∗∗xi|VTjV∗∗xj=β2∑1≤i,j≤mTjV∗xi|TiV∗xj−∑1≤i,j≤mTi∗V∗xi|Tj∗V∗xj=β2∑1≤i,j≤mTjyi|Tiyj−∑1≤i,j≤mTi∗yi|Tj∗yj≥0.
More precisely, we have shown thatVTV∗ satisfies(51)α2VTV∗≤VTV∗∗≤β2VTV∗.
It is obvious that ifT∈ℬℋ is α,β-normal, then NT=NT∗. The following lemma extended this result to the multivariable case.Lemma 1.
LetT=T1,…,Tm∈ℬℋm be joint α,β-normal. Then,(52)∩1≤k≤mNTk=∩1≤k≤mNTk∗.Proof.
SinceT is joint α,β-normal, it follows that(53)β2∑1≤i,j≤mTjxi|Tixj−∑1≤i,j≤mTi∗xi|Tj∗xj≥0,and(54)∑1≤i,j≤mTi∗xi|Tj∗xj−α2∑1≤i,j≤mTjxi|Tixj≥0,for x1⋯,xm∈ℋ. In particular, for x∈ℋ, we get(55)β2∑1≤i,j≤mTjx|Tix−∑1≤i,j≤mTi∗x|Tj∗x≥0,and(56)∑1≤i,j≤mTi∗x|Tj∗x−α2∑1≤i,j≤mTjx|Tix≥0.
If we takex∈∩1≤k≤mNTk, one can see that(57)∑1≤i,j≤mTj∗x|Ti∗x=0.
This implies thatTiTj∗x=0;∀i,j=1,…,m, and we deduce that Tj∗x=0 for all j=1,…,m. So, x∈∩1≤k≤mNTk. Conversely, let x∈∩1≤k≤mNTk. From the above inequalities, we infer immediately that(58)∑1≤i,j≤mTjx|Tix=0.
Hence, we always haveTi∗Tjx=0;∀i,j=1,…,m, and we deduce that Tjx=0 for all j=1,…,m. This yields x∈∩1≤k≤mNTk.Proposition 5.
LetT=T1,…,Tm∈ℬℋm and let S=S1,…,Sm∈ℬℋm such that Sj∗Ti=0 for i,j=1,…,m. If T and S are joint α,β-normal, then so is T+S=T1+S1,…,Tm+Sm.Proof.
Letx1,…,xm∈ℋ. Under the assertions that Sj∗Ti=0 for i,j=1,…,m, we get(59)Tj+Sjxi|Ti+Sixj=Tjxi|Tixj+Sjxi|Sixjfori,j=1,…,m,and(60)Ti+Si∗xi|Tj+Sj∗xj=Ti∗xi|Tj∗xj+Si∗xi|Sj∗xjfori,j=1,…,m.
SinceT and S are α,β-normal, it follows that(61)=∑1≤i,j≤mTi+Si∗xi|Tj+Sj∗xj−α2∑1≤i,j≤mTj+Sjxi|Ti+Sixj∑1≤i,j≤mTi∗xi|Tj∗xj−α2∑1≤i,j≤mTjxi|Tixj︸≥0+∑1≤i,j≤mSi∗xi|Sj∗xj−α2∑1≤i,j≤mSjxi|Sixj︸≥0≥0,and(62)β2∑1≤i,j≤mTj+Sjxi|Ti+Sixj−∑1≤i,j≤mTi+Si∗xi|Tj+Sj∗xj=β2∑1≤i,j≤mTixj|Tjxi−∑1≤i,j≤mTi∗xi|Tj∗xj︸≥0+β2∑1≤i,j≤mSjxi|Sixj−∑1≤i,j≤mSi∗xi|Sj∗xj︸≥0≥0.
From the previous calculations, we see thatT+S is a joint α,β-normal operator.
In [26, Theorem 4], the authors have proved that if T∈ℬℋ is an α,β-normal operator and S∈ℬℋ is an α,β-normal operator such that T∗S=ST∗, then TS and ST are αα′,ββ′-normal operators. The following example proves that even if T=T1,…,Tm and S=S1,…,Sm are joint α,β-normal operators, their product TS=T1S1,…,TmSm is not in general a joint α,β-normal operator.Example 4.
(1)
ConsiderT=001010100 and S=−1000−1000−1 which are α,β-normal and their product TS=00−10−10−100 is α,β-normal. In view of Proposition 2, it follows that T=T,…,T, S=S,…,S, and TS=TS,…,TS are joint α,β-normal operators.(2)
ConsiderT1=1011 and S1=−100−1 which are α,β-normal, whereas their product T1S1=−10−1−1 is not α,β-normal. Hence, by applying Proposition 2, we observe that T1=T1,…,T1 and S1=S1,…,S1 are joint α,β-normal operators. However, T1S1=T1S1,…,T1S1 is not a joint α,β-normal operator.
In the following theorems, we show that the product of a jointα,β-normal tuple by an m-tuple of operators satisfying suitable conditions is joint α,β-normal tuple.Theorem 3.
LetT=T1,…,Tm∈ℬℋm be joint α,β-normal and let A=A1,…,Am∈ℬℋm be joint normal operators such that(63)TjAi=AiTjfor allj,i=1,…,m.Then,TA=T1A1,…,TmAm is a joint α,β-normal operator.Proof.
SinceA is joint normal and AiTj=TjAi for all i,j=1,…,m, it follows from Fuglede theorem [1] that(64)Ai∗Aj=AjAi∗andAi∗Tj=TjAi∗∀i,j=1,…,m.
By elementary calculations based on the above arguments, we get for alli,j=1,…,m and for x1,…,xm∈ℋ,(65)TjAjxi|TiAixj=TjAjxi|TiAixjAjTjxi|AiTixjAi∗AjTjxi|TixjTjAi∗xi|TiAj∗xj.
We have(66)∑1≤i,j≤mTiAi∗xi|TjAj∗xj−α2∑1≤i,j≤mTjAjxi|TiAixj=∑1≤i,j≤mTi∗Ai∗xi|Tj∗Aj∗xj−α2∑1≤i,j≤mTjAi∗xi|TiAj∗xj=∑1≤i,j≤mTj∗zi|Ti∗zj−α2∑1≤i,j≤mTjzi|Tizjzi=Ai∗xi≥0,and(67)β2∑1≤i,j≤mTjAjxi|TiAixj−∑1≤i,j≤mTiAi∗xi|TjAj∗xj=β2∑1≤i,j≤mTjAi∗xi|TiAj∗xj−∑1≤i,j≤mTi∗Ai∗xi|Tj∗Aj∗xj=β2∑1≤i,j≤mTjzi|Tizj−∑1≤i,j≤mTj∗zi|Ti∗zjzi=Ai∗xi≥0.
So,TA satisfies (14). Thus, the proof is complete.Corollary 2.
LetT=T1,…,Tm∈ℬℋm be joint α,β-normal and let A=A1,…,Am∈ℬℋm be a commuting tuple of operators such that each Ai is self-adjoint for i=1,…,m. If TjAi=AiTj for all i,j=1,…,m, then TA=T1A1,…,TmAm is joint α,β-normal.Proof.
This is an immediate consequence of Theorem3.Theorem 4.
LetT=T1,…,Tm∈ℬℋm be a joint α,β-normal tuple and S=S1,…,Sm∈ℬℋm such that Sl∗Sk=SkSl∗ and Sl∗Tk=TkSl∗ for k,l∈1,…,m2; then, ST:=S1T1,…,SmTm is a joint α,β-normal tuple.Proof.
Letx1,…,xm∈ℋ; under the assumptions(68)Sl∗Sk=SkSl∗andTlSk∗=Sk∗Tl,∀k,l∈1,…,m2and the fact that T is jointly α,β-normal, it follows that(69)β2∑1≤i,j≤mSjTjxi|SiTixj−∑1≤i,j≤mSiTi∗xi|SjTj∗xj=β2∑1≤i,j≤mTjSi∗xi|TiSj∗xj−∑1≤i,j≤mTi∗Si∗xi|Tj∗Sj∗xj≥β2∑1≤i,j≤mTjyi|Tiyj−∑1≤i,j≤mTi∗yi|Tj∗yjyi=Si∗xi≥0.
Similarly,(70)∑1≤i,j≤mSiTi∗xi|SjTj∗xj−α2∑1≤i,j≤mSjTjxi|SiTixj=∑1≤i,j≤mTi∗Si∗xi|Tj∗Sj∗xj−α2∑1≤i,j≤mTjSi∗xi|TiSj∗xj≥β2∑1≤i,j≤mTi∗yi|Tj∗yj−α2∑1≤i,j≤mTjyi|Tiyjyi=Si∗xi≥0.


## 3. Some Inequalities for Jointα,β-Normal Operators

Finding some inequalities involving norms and numerical radius of Hilbert space operators is a topic which has been considered by many mathematicians in recent years. Recently, a generalization of the numerical radius, known as joint numerical radius, has been introduced and studied and several inequalities containing numerical radius are extended.Proposition 6.
LetT=T1,…,Tm∈ℬℋm such that TiTj=TiTj∗=0 for i≠j,1≤i,j≤m. If T is joint α,β-normal tuple, then(71)ωT2≤12βT+ωT2,where T2:=T12,…,Tm2.Proof.
By taking into account the well-known inequality (see [6]),(72)u|w0w0|v≤12uv+u|v,for any u,v,w0∈ℋ with w0=1. Let x∈ℋ with x=1 and set u=∑1≤j≤mTjx and v=∑1≤j≤mTjx. We get(73)∑1≤j≤mTjx|xx|∑1≤j≤mTj∗x≤12∑1≤j≤mTjx∑1≤j≤mTj∗x+∑1≤j≤mTjx|∑1≤j≤mTj∗x=12∑1≤j≤mTjx21/2∑1≤j≤mTj∗x21/2+∑1≤j≤mTj2x|x≤12∑1≤j≤mTjx21/2β2∑1≤j≤mTjx21/2+∑1≤j≤mTj2x|x≤12β∑1≤j≤mTjx2+∑1≤j≤mTj2x|x,from which it follows that(74)∑1≤j≤mTjx|x2≤12β∑1≤j≤mTjx2+∑1≤j≤mTj2x|x,so that(75)sup‖x‖=1∑1≤j≤mTjx|x2≤12βsup‖x‖=1‖∑1≤j≤mTjx‖2+sup‖x‖=1∑1≤j≤mTj2x|x,ωT2≤12βT+ωT2.Theorem 5.
LetT=T1,…,Tm∈ℬℋm be a joint α,β-normal tuple. If TiTj=TiTj∗=0 for i≠j,1≤i,j≤m, then(76)αT2≤ωT2+2β1+αλ2T−λT∗2,λ∈ℂ.Proof.
Taking into account the following inequality ([6]):(77)uv≤|u|v|+uvu−v2u+v2,u,v∈ℋu,v≠0,for x1,…,xm∈ℋ, set u=∑1≤k≤mTkxk and v=λ∑1≤k≤mTk∗xk.
SinceTiTj=0 for i≠j, we get(78)∑1≤k≤mTkxkλ∑1≤k≤mTk∗xk=∑1≤k≤mTkxk21/2λ∑1≤k≤mTk∗xk21/2=∑1≤k≤mTkxk21/2λ∑1≤k≤mTk∗xk21/2≥αλ∑1≤k≤mTkxk2by2.2.
Similarly, we get from (13),(79)∑1≤k≤mTkxkλ∑1≤k≤mTk∗xk≤λβ∑1≤k≤mTkxk2.
Moreover,(80)∑1≤k≤mTkxkλ∑1≤k≤mTk∗xk=λ∑1≤k≤mTk2x|x≤λ∑1≤k≤mTk2x|x,and(81)∑1≤k≤mTkxk+λ∑1≤k≤mTk∗xk2=∑1≤k≤mTkxk21/2+λ∑1≤k≤mTk∗xk21/22=∑1≤k≤mTkxk21/2+λ∑1≤k≤mTk∗xk21/22≥1+λα2∑1≤k≤mTkxk2by2.2.
From the above calculations, we get(82)αλ∑1≤k≤mTkxk2≤λ∑1≤k≤mTk2xk|xk+2λβ1+λα2∑1≤k≤mTk−λTk∗xk.
Forx=x1=⋯=xm, we get(83)αλ∑1≤k≤mTkx2≤λ∑1≤k≤mTk2x|x+2λβ1+λα2∑1≤k≤mTk−λTk∗x.
Taking the supremum in (83) over x∈ℋ, ‖x‖=1, we get the desired inequality (76).



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