# Inequalities InvolvingA-Numerical Radius and Operator A-Norm for a Class of Operators Related to α,β−A-Normal Operators

Authors: Sid Ahmed Ould Ahmed Mahmoud; Sidi Hamidou Jah
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, we introduce and study a new class of operators, larger thanα,β−A-normal operators and different than α,β−A-normal operators, named m-quasi-α,β−A-normal operators. Considering the semi-inner product induced by a positive operator A, the m-quasi-α,β−A-normal operators turn into a generalization (for this new structure) of classical m-quasi-α,β-normal operators. Several results concerning properties of this kind of operators are presented in the paper. Several inequalities for the A-numerical radius and A-operator norm for members of this class are established.


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## Body

## 1. Introduction and Preliminaries

Letℬbℋ denote the C∗-algebra of all bounded linear operators on an infinite dimension complex Hilbert space ℋ. ℬbℋ+ denotes the cone of positive operators of ℋ. It is well known that A∈ℬbℋ+⇔Aw|w≥0∀w∈ℋ.ForR∈ℬbℋ, we denote by R∗ the adjoint operator of R, range R by ranR, and its null space by kerR. If R∈ℬbℋ, then ranR¯ stands for its closure in the norm topology of ℋ. We denote by PranA¯ the orthogonal projection onto the closed linear subspace ranA¯.IfA is a positive operator on ℋ, the bilinear functional ⋅|⋅:ℋ×ℋ⟶ℂ defines a positive semidefinite sesquilinear form given by w,v↦w|vA=Aw|v.Notice that(1)w|wA=0⇔Aw|w=0⇔wA2=0⇔w∈kerA.An operatorR∈ℬbℋ is said to be A-bounded if there exists C>0 such that(2)RwA≤CwA,∀w∈ℋ.The set of allA-bounded operators on ℋ will be denoted by ℬbKA.For anyR∈ℬbℋA, we take from [1],(3)RA≔supRwAwA,w∈ranA¯=supRwA,u∈ℋ:wA=1.In [1], the authors have introduced the concept of A-self-adjoint for an operator R∈ℬbℋ. An S∈ℬbℋ is called an A-adjoint operator of R if R and S satisfy(4)Rw|vA=w|SvA,∀w,v∈ℋ.We observe that(5)Rw|vA=w|SvA,∀w,v∈ℋ⇔AR=S∗A.R is an A-self-adjoint operator if R is an A-adjoint of itself, that is, AR=R∗A.It was observed by Douglas Theorem [2] that an operator R∈ℬbℋ admits an A-adjoint if and only if ranR∗A⊆ranA. In the following, the set of all operators R∈ℬbℋ which admit an A-adjoint is denoted by ℬbℋA. Throughout this paper, we consider that A∈ℬbℋ+ with ranA=ranA¯. In [1], it was observed that if R∈ℬbℋA, the solution of the equation AX=R∗A is a distinguished A-adjoint operator of R, which is denoted by R♯. However, R♯ is given by R♯=A†R∗A where A† is the Moore–Penrose inverse of A. The A-adjoint operator R♯ satisfies the following axioms:(6)AR♯=R∗A,ranR♯⊆ranAandkerR♯=kerR∗A.The authors in [1, 3, 4] have studied the following useful properties of R♯ for R∈ℬℋA.ForR∈ℬbℋA, then the following axioms hold:(1)
R♯∈ℬbℋA.(2)
R♯♯=PranARPranA.(3)
IfS∈ℬbℋA, then RS♯=S♯R♯.In recent years, the theory of normal operators has known many extensions due to the work carried out by several authors ([5–8]). The study of operators in semi-Hilbertian spaces is of significant interest and is currently being done by a number of mathematicians around the world. Some developments toward this subject have been done in [1, 3, 4, 9–18].Any operatorR∈ℬbℋ is A-positive symbolicallyR≥A0 if AR∈ℬℋ+, that is,(7)ARw,w=Rw,wA≥0,∀w∈ℋ.ForR,S∈ℬbℋ:R≥AS⇔AR−S≥0⇔Rw,wA≥Sw,wA∀w∈ℋ.The classes isometry, unitary, normal, andn,m-normal operators in Hilbert space have been generalized to the concepts of so called A-isometry, A-unary, and A-normal and n,m−A-normal operators by many authors. An operator R∈ℬℋA is called(1)
A-isometry if R∗AR=A (RuA=uA,∀u∈ℋ [19]),(2)
A-unitary if R♯R=R♯♯R♯=PranA ([1]),(3)
A-normal if R♯R=RR♯ ([12, 17]),(4)
α,β−A-normal, if β2R♯R≥ARR♯≥Aα2R♯R, for 0≤α≤1≤β ([11]),(5)
n,m−A-normal if RnR♯m=R♯mRn, for some positive integers n and m [9].Of course, these extensions are not trivial since many difficulties arise.Recently, the authors in [20] have introduced the concept of m-quasi totally-α,β-normal operators as a generalization of totally-α,β-normal operators in Hilbert space. An element R∈ℬℋ is called m-quasi totally-α,β-normal if R satisfies(8)αR−λRmw≤R−λ∗Rm≤βR−λRmw,∀w∈ℋ,∀λ∈ℂ.In this paper, our goal is to introduce a new class of operators, larger thanα,β−A-normal operators, named m-quasi-α,β−A-normal operators in semi-Hilbertian spaces. First of all, we introduce notations and consider a few preliminary results which are useful to prove the main result of the paper. We give sufficient conditions on two m-quasi α,β−A-normal operators defined on a semi-Hilbert space, which make their product and their tensor product m-quasi α,β−A-normal. The inspiration for our investigation comes from [11, 20]. Moreover, we established various inequalities between the A-numerical radius ωA⋅ and A-operator norm ⋅A of m-quasi-α,β−A-normal operators.


## 2.m-Quasi-α,β−A-Normal Operators

In this section, we define a class of operators in semi-Hilbertian spaces, i.e., spaces generated by positive semidefinite sesquilinear forms. This kind of spaces appears in many problems concerning linear and bounded operators on Hilbert spaces and is intensively studied in the present. The concept of operator theory is semi-Hilbertian spaces have attracted attention. Recently, many extensions of some concepts of Hilbert space operators to semi-Hilbertian space operators have attracted much attention of various authors in several papers [1, 3, 4, 9–18]. In this framework, we are interested to introducing a new concept of quasinormality in semi-Hilbertian spaces known as m-quasi-α,β−A-normal operators. We investigate various structural properties of this class of operators and study some relations about it.Definition 1.
An operatorR∈ℬℋA is called m-quasi-α,β−A-normal for 0≤α≤1≤β if(9)β2R♯m+1Rm+1≥AR♯mRR♯Rm≥Aα2R♯m+1Rm+1,for some positive integer m, or equivalently, if(10)βRm+1uA≥R♯RmuA≥αRm+1uA,for allu∈ℋ.Remark 1.
The reader will easily see from Definition1 that(i)
Anα,β−A-normal is an m-quasi-α,β−A-normal operator.(ii)
Everym-quasi-α,β−A-normal operator is m+1-quasi-α,β−A-normal operator.Remark 2.
Notice that the converse of the statement (i) in Remark1 is not true in general, as shown in the following example:Example 1.
Let us considerℋ=ℂ2 and A=1003 and R=1301. Direct calculations show that ranR∗A⊆ranA.(11)3R2wA≥R♯RwA≥R2wA,∀w∈ℂ2,R♯w0A≥3Rw0A,for somew0∈ℂ2.
This shows thatR is a quasi-1,3−A-normal, but it is not a 1,3−A-normal.Remark 3.
From Example1, it is obvious that the class of m-quasi-α,β−A-normal operators contained the class of α,β−A-normal operators as a proper subset.Theorem 1.
LetR∈ℬbℋA be an m-quasi-α,β−A-normal operator. If Rm has dense range, then R is α,β−A-normal.Proof.
SinceRm has a dense range, it follows that ranRm¯=ℋ. Let u∈ℋ, then there exists a sequence un in ℋ such that Rmun⟶u as n⟶∞.
From the fact thatR is m-quasi-α,β−A-normal operator, we have(12)αRm+1wA≤R♯RmwA≤βRm+1wA,for all w∈ℋ.
In particular, we have(13)αRm+1unA≤R♯RmunA≤βRm+1unA.
Consequently,(14)αRu≤R♯u≤βRuA,∀u∈ℋ.
Therefore,R is α,β−A-normal operator.Corollary 1.
LetR∈ℬbℋA be an m-quasi-α,β−A-normal operator. If Rm≠0 and if R has no nontrivial Rm-invariant closed subspace, then R is α,β−A-normal.Proof.
From the fact thatRm has no nontrivial invariant closed subspace, it follows that Rm has no nontrivial hyperinvariant subspace. But, kerRm and ranRm¯ are hyperinvariant subspaces, and Rm≠0, hence kerRm=0 and ranRm¯=ℋ. Therefore, R is α,β−A-normal operator by [11], Definition 6.The following theorem gives a characterization ofm-quasi-α,β−A-normal operators.Theorem 2.
LetR∈ℬbℋA, then R is m-quasi-α,β−A-normal operator if and only if(15)λ2R♯mRR♯Rm+2λα2R♯m+1Rm+1+R♯mRR♯Rm≥A0,λ2R♯m+1Rm+1+2λR♯mRR♯Rm+β4R♯m+1Rm+1≥A0,for all λ∈ℝ.Proof.
By using elementary properties of real quadratic forms, we infer(16)λ2R♯mRR♯Rm+2λα2R♯m+1Rm+1+R♯mRR♯Rm≥A0⇔λ2R♯RmuA2+2λα2Rm+1uA2+R♯RmuA2≥0,∀u∈ℋand∀λ∈ℝ⇔αRm+1uA≤R♯RmuA,∀u∈ℋ.
Similarly,(17)λ2R♯m+1Rm+1+2λR♯mRR♯Rm+β4R♯m+1Rm≥A0⇔λ2Rm+1uA2+2λR♯RmuA2+β4Rm+1uA2≥0,∀u∈ℋand∀λ∈ℝ⇔βRmuA≥R♯RmuA,∀u∈ℋ.
Therefore,R is the m-quasi-α,β−A-normal operator.In the following theorem, we show the relationship betweenm-quasi-α,β−A-normal operator and it is A-adjoint.Theorem 3.
Letα,β∈ℝ2 such that 0<α≤1≤β and let R∈ℬbℋA. The following statements hold.(1)
IfR is an m-quasi-α,β−A-normal such that ranRm=ranR♯m, then R♯ is m-quasi-1/β,1/α−A-normal.(2)
IfR♯ is m-quasi-1/β,1/α−A-normal such that ranRm=ranR♯m and RkerA⊥⊆kerA⊥, then R is an m-quasi-α,β−A-normal.Proof.
(1) IfR is m-quasi-α,β−A-normal, it follows that(18)αRRmxA≤R♯RmxA≤βRRmxA,∀x∈ℋ.
Taking into account the assumptionranRm=ranR♯m, we infer(19)αRR♯muA≤R♯R♯muA≤βRR♯muA,∀u∈ℋ.
Combining these inequalities,(20)1αR♯m+1uA≥RR♯muA≥1βR♯m+1uA.
So,R♯ is m-quasi-1/β,1/α−A-normal as required.
(2) SinceR♯ is m-quasi-α,β−A-normal operator, we have for all w∈ℋ,(21)1βR♯m+1wA≤R♯♯R♯mwA≤1αR♯m+1wA.
After rearranging it further, we get(22)1βR♯R♯mwA≤PranARPranAR♯mwA≤1αR♯R♯mwA.
From the assumption thatRkerA⊥⊆kerA⊥, we observe that kerA is a reducing subspace of R and it follows that RPranA=PranAR and APranA=PranAA=A. Thus, we have(23)1βR♯R♯mwA≤RR♯mwA≤1αR♯R♯mwA.
From the assumption thatranRm=ranR♯m, we infer(24)1βR♯RmuA≤Rm+1uA≤1αR♯RmwA,∀u∈ℋ.
This gives(25)αRm+1uA≤R♯RmuA≤βRm+1uA,then T is always m-quasi-α,β−A-normal.Corollary 2.
Under the same conditions of Theorem3, if αβ=1, then R is m-quasi-α,β−A-normal if and only if R♯ is m-quasi-α,β−A-normal.Theorem 4.
LetR∈ℬbℋA be m-quasi-α,β−A-normal such that αβ=1, then(26)β‖R♯Rmu‖A≥‖Rm+1u‖A≥α‖R♯Rmu‖A,∀u∈ℋ.Proof.
By using them-quasi-α,β−A-normality of R and given condition, we have the following:
R is m-quasi-α,β−A-normal if and only if(27)β2R♯m+1Rm+1≥AR♯mRR♯Rm≥Aα2R♯m+1Rm+1.
Therefore,(28)α2β2R♯m+1Rm+1≥Aα2R♯mRR♯Rm≥Aα4R♯m+1Rm+1,β4R♯m+1Rm+1≥Aβ2R♯mRR♯Rm≥Aβ2α2R♯m+1Rm+1.
Combining these inequalities, we get(29)β2R♯mRR♯Rm≥AR♯m+1Rm+1≥Aα2R♯mRR♯Rm,and so,(30)β2R♯mRR♯Rmu,uA≥R♯m+1Rm+1u,uA≥α2R♯mRR♯Rmu,uA.
This means that(31)βR♯RmuA≥Rm+1uA≥αR♯RmuA,∀u∈ℋ.
Hence, the result holds.Theorem 5.
LetR,S∈ℬbℋA be two m-quasi-α,β−A-normal, then T⊕S is also m-quasi-α,β−A⊕A-normal, but tensor product R⊗S is m-quasi-α2,β2−A⊗A- normal.Proof.
SinceR and S are m-quasi-α,β−A-normal, we have(32)β2R⊕S♯m+1R⊕Sm+1=β2R♯m+1Rm+1⊕β2S♯m+1Sm+1≥A⊕AR♯Rm+1⊕S♯Sm+1=R⊕S♯R⊕Sm+1.
Moreover,(33)R⊕S♯R⊕Sm+1=R♯Rm+1⊕S♯Sm+1≥A⊕Aα2R♯m+1Rm+1⊕α2S♯m+1Sm+1=α2R⊕S♯m+1R⊕Sm+1.
Therefore,(34)β2R⊕S♯m+1R⊕Sm+1≥A⊕AR⊕S♯R⊕Sm+1≥A⊕Aα2R⊕S♯m+1R+Sm+1.
Thus,R⊕S is m-quasi-α,β−A⊕A-normal operator.
To prove thatR⊗S is m-quasi-α2,β2−A⊗A-normal operator, we observe that(35)β4A⊗AR⊗S♯m+1R⊗Sm+1=AR♯m+1Rm+1⊗β2AS♯m+1Sm+1≥AR♯Rm+1⊗AS♯Sm+1=A⊗AR⊗S♯R⊗Sm+1.
We get(36)β4R⊗S♯m+1R⊗Sm+1≥A⊗AR⊗S♯R⊗Sm+1.
Similarly,(37)A⊗AR⊗S♯R⊗Sm+1=AR♯Rm+1⊗AS♯Sm+1≥α2AR♯m+1Rm+1⊗α2AS♯m+1Sm+1=α4A⊗AR⊗S♯m+1R⊗Sm+1.
Therefore,(38)R⊗S♯R⊗Sm+1≥A⊗Aα4A⊗AR⊗S♯m+1R⊗Sm+1.
Combining (36) and (38), we get(39)β4R⊗S♯m+1R⊗Sm+1≥A⊗AR⊗S♯R⊗Sm+1≥A⊗Aα4R⊗S♯m+1R⊗Sm+1.Theorem 6.
The set(40)Γα,β=R∈ℬbℋA:β2R♯m+1Rm+1≥AR♯mRR♯Rm≥Aα2R♯m+1Rm+1,and 0≤α≤1≤β is Arcwise connected for m∈ℕ.Proof.
LetR∈Γα,β and λ∈ℂ,λ≠0. We need to prove that λR is m-quasi-α,β−A-normal. Now, for u∈ℋ,(41)β2λR♯m+1λRm+1u,uA=λ2m+1β2R♯m+1Rm+1u,uA≥λ2m+1R♯mRR♯Rmu,uA=λm+1λ¯m+1R♯mRR♯Rmu,uA=λR♯mλRλR♯λRmu,uA.
Otherwise,(42)λR♯mλRλR♯λRmu,uA=λm+1λ¯m+1R♯mRR♯Rmu,uA≥λm+1λ¯m+1α2R♯m+1Rm+1u,uA|=α2λR♯m+1λRm+1u,uA.
Hence,(43)βλRm+1A≥λR♯λRmA≥αλRm+1A.
This means that the class ofm-quasi-α,β−A-normal operator is Arcwise connected.Proposition 1.
LetR,V∈ℬbℋA such that RkerA⊥⊆kerA⊥ and VkerA⊥⊆kerA⊥. If R is m-quasi- α,β−A-normal for 0≤α≤1≤β and V is an A-isometry, then VRV♯ is m-quasi-α,β−A-normal operator.Proof.
In view of Definition1 and the fact that V is an A-isometry ([1], Proposition 3.6), we have(44)β2R♯m+1Rm+1≥AR♯mRR♯Rm≥Aα2R♯m+1Rm+1andV♯V=PranA.
Moreover, from the fact thatkerA is a reducing subspace for both R and V, it follows that(45)PranAR=RPranB,R♯PranA=PranAR♯,VPranA=PranAV,V♯PranA=PranAV♯.
Direct calculations show that(46)VRV♯k=VRV♯VRV♯⋅⋯VRV♯︸k−times=VRPranBRV♯⋯VRV♯=PranAVR2V♯⋯VRV♯=⋮=PranAVRkV♯.
This implies(47)β2VRV♯♯m+1VRV♯m+1=β2V♯♯R♯V♯m+1VRV♯m+1=PranAVPranAR♯V♯m+1VRV♯m+1=β2PranAVR♯m+1V♯PranAVRm+1V♯=β2VPranAR♯m+1Rm+1VPranA♯≥AVPranAR♯mRR♯RmVPℛA¯♯by11,Lemma1=VRV♯♯mVRV♯VRV♯♯VRV♯m.
Similarly, we have(48)VRV♯♯mVRV♯VRV♯♯VRV♯m=VPranAR♯mRR♯RmVPℛA¯♯≥α2VPranAR♯m+1Rm+1VPℛA¯♯≥α2VRV♯♯m+1VRV♯m+1.
The conclusion holds.Proposition 2.
LetR,S∈ℬbℋA be commuting and such that R is m-quasi-α,β−A-normal operator. The following statements hold.(1)
IfS is A-self-adjoint and R♯S=SR♯, then RS is m-quasi-α,β−A-normal.(2)
IfS is A-unitary, R♯S=SR♯ and kerA is invariant subspace for R, then RS is m-quasi-α,β−A-normal.Proof.
(1) SinceR is m-quasi-α,β−A-normal, S is A-normal and R♯S=SR♯, it follows that for u∈ℋ,(49)αRSm+1uA=αRm+1Sm+1uA≤R♯RmSm+1uA=SR♯RmSmuA=S♯R♯RmSmuA=RS♯RSmuA.
On the other hand,(50)RS♯RSmuA=S♯R♯RmSmuA=SR♯RmSmuA=R♯RmSm+1uA≤βRm+1Sm+1uA=βRSm+1uA.
Consequently,RS is the m-quasi-α,β−A-normal operator.
(2) SincekerA is invariant subspace for R, we observe that RPranA=PranAR and R♯PranA=PranAR♯. In view of the fact that S is A-unitary, we have S♯S=SS♯=PranA.
Now, direct calculations give(51)β2RS♯m+1RSm+1=β2R♯m+1S♯m+1Sm+1Rm+1=β2R♯m+1PranARm+1=β2PranAR♯m+1Rm+1PranA.
SinceR is m-quasi-α,β−A-normal, it follows immediately from [11], Lemma 1 that(52)β2RS♯m+1RSm+1≥APranAR♯mRR♯RmPranA︸1≥Aα2PranAR♯m+1Rm+1PranA︸2.
From the quantity in (1), we get(53)PranAR♯mRR♯Rm+1PranA=RmPranARR♯RmPranA=RS♯mRSRS♯RSm.
Similarly by (2), we have(54)PranAR♯m+1Rm+1PranA=R♯m+1S♯m+1Sm+1Rm+1=RS♯m+1RSm+1.
This means that(55)β2RS♯m+1RSm+1≥ARS♯mRSRS♯RSmα2RS♯m+1RSm+1.
Therefore,RS is m-quasi-α,β−A-normal operator.Theorem 7.
LetR,S∈ℬbℋA be commuting and R♯S=SR♯. If R is an m-quasi-α,β−A-normal and S is an m-quasi-α′,β′−A-normal for 0≤α,α′≤1≤β,β′, then RS is an m-quasi-αα′,ββ′−A-normal.Proof.
By using them-quasi-α,β−A-normality of R and m-quasi-α′,β′−A-normality of S and given condition, we have(56)αα′RSm+1uA=αα′Rm+1Sm+1uA≤α′R♯RmSm+1uA≤S♯R♯RmSmuA,∀u∈ℋ.
Otherwise,(57)S♯SmR♯RmuA≤β′SSmR♯RmuA=β′R♯RmSSmuA≤ββ′Rm+1Sm+1uA,αα′RSm+1uA≤SR♯TmSmuA≤ββ′Tm+1ASm+1uA,∀u∈ℋ.Hence, the desired result is obtained.


## 3. Inequalities InvolvingA-Numerical Radius and Operator A-Norm

In this section, we are interested to study some inequalities concerning theA-numerical radius and A-norm of m-quasi-α,β−A-normal operators. It is followed by several inequalities, refer to [21–29]. The authors in [20] have given various inequalities between the operator norm and the numerical radius of m-quasi-α,β-normal operators in Hilbert spaces. Motivated by this work, we will extend some of these inequalities to A-operator norm and A-numerical radius for m-quasi-α,β−A-normal operators in semi-Hilbertian spaces by using some known results for vectors in inner product spaces.Definition 2 (see [17]).
TheA-numerical radius of an operator R∈ℬbℋ is defined by(58)ωAR=supRu,uA,u∈ℋ:uA=1.Theorem 8.
IfR∈ℬbℋA is m-quasi-α,β−A normal operator, then the following inequality holds:(59)1+αβ2sRm+1A2≤2βωAR♯mRm+2+s2β2βRm+1−R♯RmA2,ifs≥1,2βωAR♯mRm+2+1β2βRm+1−R♯RmA2,ifs<1.Proof.
From [28], we have inspired the following inequalities:(60)w2s+v2s−2ws−1vs−1Rew,v≤s2w2s−2w−v2,ifs≥1,v2s−2w−v2,ifs<1,for s∈ℝ and w,v∈ℋ with w≥v.
By replacingw by A1/2w and v by A1/2v for w,v∉kerA1/2, we get(61)A1/2w2s+A1/2v2s−2A1/2ws−1A1/2vs−1ReA1/2w,A1/2v≤s2A1/2w2s−2A1/2w−A1/2v2,ifs≥1,A1/2v2s−2A1/2w−A1/2v2,ifs<1,for s∈ℝ and w,v∈ℋ with w,v∉kerA and A1/2w≥A1/2v.
This means(62)wA2s+vA2s−2wAs−1vAs−1Rew,vA≤s2wA2s−2w−vA2,ifs≥1,vA2s−2w−vA2,ifs<1,for s∈ℝ and w,v∈ℋ with w,v∉kerA and wA≥vA.
If we takew=βRm+1u,v=R♯Rmu, we obtain the following inequality:(63)βRm+1wA2s+R♯RmwA2s≤s2βRm+1wA2s−2βRm+1w−R♯RmwA2+2βRm+1wAs−1R♯RmwAs−1ReβRm+1w,R♯RmwA,for any w∈ℋ with wA=1 and s≥1.
Hence,(64)α2s+β2sRm+1wA2s=α2sRm+1wA2s+β2sRm+1wA2s≤R♯RmwA2s+β2sRm+1wA2s≤s2β2s−2Rm+1wA2s−2βRm+1w−R♯RmwA2+2β2s−1Rm+1wA2s−2R♯mRm+2w,wA.
Applyingsup over w∈ℋ, wA=1 of both sides of the above inequality, we infer(65)α2s+β2sRm+1A2s≤s2β2s−2Rm+1A2s−2βRm+1−R♯RmA2+2β2s−1Rm+1A2s−2ωAR♯mRm+2.
This proves the first inequality.
Analogously, by considering similar techniques, we get the second inequality fors<1.Theorem 9.
IfR∈ℬbℋA is m-quasi-α,β−A- normal operator and if λ∈ℂ, then(66)αRm+1A2≤ωAR♯mRm+2+2β1+λα2Rm+1−λR♯RmA2.Proof.
We give our proof depending on the following inequality inspired from [27],(67)wv≤w,v+2wvw−v2w+v2,for w,v∈ℋ\0.
By choosingw=A1/2Rm+1u and v=λA1/2R♯Rmu, we infer(68)Rm+1uAλR♯RmuA≤Rm+1u,λR♯RmuA+2Rm+1uAλR♯RmuARm+1u−λR♯RmuA2Rm+1uA+λR♯RmuA2.
This gives(69)αRm+1uA2≤Rm♯R2Rmu,uA+2β1+λα2Rm+1u−λR♯RmuA2.
Taking thesup of both sides of the last inequality for uA=1, we infer the desired result.Theorem 10.
IfR∈ℬbℋA is m-quasi-α,β−A-normal operator and if λ∈ℂ\0, then(70)α2−1λ+β2Rm+1A4≤ωARm♯Rm+22.Proof.
We give our proof depending on the following inequality inspired from [24]:(71)w2v2≤w,v2+1λ2w2w−λv2,provided w,v∈ℋ and λ∈ℂ\0. By choosing w=A1/2Rm+1u,v=A1/2R♯Rmu, we infer(72)Rm+1uA2R♯RmuA2≤Rm+1u,R♯RmuA2+1λ2Rm+1uA2Rm+1u−λR♯RmuA2.
This gives(73)α2Rm+1uA4−1λ2Rm+1uA41+λβ2≤R♯mR2Rmu,uA2.
Taking thesup of both sides of the last inequality for uA=1, the desired result will be obtained.Theorem 11.
IfR∈ℬbℋA is m-quasi-α,β−A-normal operator, then(74)2ωARm+1ωAR♯Rm≤βRm+1A2+ωAR♯mRm+2.Proof.
In the following inequality [21],(75)2w,uu,v≤wv+w,v,where w,v,u∈ℋ and u=1.
Choosingu=A1/2x, w=A1/2Rm+1x, and v=A1/2R♯Rmx with xA=1, it follows(76)2A1/2Rm+1x,xx,A1/2R♯Rmx≤Rm+1xAR♯RmxA+Rm+1x,R♯RmxA.
Hence,(77)2Rm+1x,xAR♯mRx,xA≤βRm+1A2+R♯mRm+2x,xA.
Taking thesup of both sides of the last inequality for xA=1, the required inequality will be obtained.Theorem 12.
IfR∈ℬbℋA is m-quasi-α,β−A-normal operator and if p≥2, then(78)1+αpRm+1Ap≤12Rm+1+R♯RmAp+Rm+1−R♯RmAp.Proof.
We use the following inequality [26]:(79)wp+vp≤12w+vp+w−vp,for any w,v∈ℋ and p≥2.
Now, if we choosew=A1/2Rm+1u,v=A1/2R♯Rmu, we get(80)A1/2Rm+1up+A1/2R♯Rmup≤12A1/2Rm+1u+A1/2R♯Rmup+A1/2Rm+1u−A1/2R♯Rmup.
That means(81)Rm+1uAp+R♯RmuAp≤12Rm+1u+R♯RmuAp+Rm+1u−R♯RmuAp.
Hence, sinceR is m-quasi-α,β−A normal operator, we have(82)1+αpRm+1uAp≤Rm+1uAp+R♯RmuAp≤12Rm+1u+R♯RmuAp+Rm+1u−R♯RmuAp.
Taking thesup over all u∈ℋ with uA=1 in the above inequality, we get(83)1+αpRm+1Ap≤12Rm+1+R♯RmAp+Rm+1−R♯RmAp.Theorem 13.
LetR∈ℬbℋA be an m-quasi-α,β−A-normal operator and p≥2. Then,(84)ωAR♯m+1Rm+1+R♯mRR♯Rm2p/2≤1+βp41+αpRm+1+R♯RmAp+Rm+1−R♯RmAp.Proof.
Using the following elementary inequality,(85)21−qa+bq≤aq+bq,for a,b≥0 and q≥1.
For anyw∈ℋ with wA=1, take a=A1/2Rm+1w2,b=A1/2R♯Rmw2, and q=p/2 in the above inequality to get(86)21−p/2A1/2Rm+1w2+A1/2R♯Rmw2p/2≤A1/2Rm+1w2p/2+A1/2R♯Rmw2p/2.
Then,(87)21−p/2Rm+1wA2+R♯RmwA2p/2≤Rm+1wA2p/2+R♯RmwA2p/2,whence(88)Rm+1w2+R♯RmwA22p/2≤12Rm+1wAp+R♯RmwAp.
SinceR is m-quasi-α,β−A-normal, we have(89)Rm+1wAp+R♯RmwAp≤1+βpRm+1wAp.
Hence, by Theorem12, we obtain(90)Rm+1w2+R♯RmwA22p/2≤1+βp41+αpRm+1+R♯RmAp+Rm+1−R♯RmApwA.
But,(91)R♯m+1Rm+1+R♯mRR♯Rm2w,wAp/2=Rm+1w2+R♯RmwA22p/2.
Therefore,(92)R♯m+1Rm+1+R♯mRR♯Rm2w,wAp/2≤1+βpwA41+αpRm+1+R♯RmAp+Rm+1−R♯RmAp.
Taking thesup of both sides of the last inequality for wA=1, we infer the desired result.Theorem 14.
LetR∈ℬbℋA be an m-quasi-α,β−A-normal operator. If p∈ℝ with 1<p<2 and maxλ−μβ,αμ−λ≥0, then(93)λ+αμpRm+1Ap+maxλ−μβ,αμ−λpRm+1Ap≤λRm+1+μR♯RmAp+λRm+1−μR♯RmAp.Proof.
We use the following inequality inspired from [26],(94)w+vp+w−vp≤w+vp+w−vp,for any w,v∈ℋ and p∈ℝ:1<p<2.
Putw=λA1/2Rm+1u and v=μA1/2R♯Rmu for u∈ℋ to get(95)λRm+1uA+μR♯RmuAp+λRm+1uA−μRm+1uAp≤λRm+1u+μR♯RmuAp+λRm+1u−μR♯RmuAp.
SinceR is m-quasi-α,β−A-normal operator, it follows that(96)λ+αμpRm+1uAp≤λRm+1uA+μR♯RmuAp,λ−μβRm+1uA≤λRm+1uA−μR♯RmuA≤λ−αμRm+1uA.
By observing that ifmaxλ−μβ,αμ−λ=λ−μβ≥0, then(97)λ−μβRm+1uA≤λRm+1uA−μR♯RmuA=λRm+1uA−μR♯RmuA,and therefore(98)λ−μβpRm+1uAp≤λRm+1uA−μR♯RmuAp.
However, ifmaxλ−μβ,αμ−λ=μα−λ≥0, then(99)μR♯RmuA−λRm+1uA≥αμ−λRm+1uA≥0,and therefore(100)λRm+1uA−μR♯RmuAp=μR♯RmuA−λRm+1uAp≥μα−λpRm+1uAp.
From either case, we get(101)λRm+1uA−μR♯RmuAp≥maxλ−μβ,αμ−λpRm+1Ap.
Hence,(102)λ+αμpRm+1uAp+maxλ−μβ,αμ−λRm+1uAp≤λRm+1uA+μR♯RmuAp+λRm+1uA−μRm+1uAp≤μRm+1u+μR♯RmuAp+λRm+1u−μR♯RmuAp.
Taking thesup of both sides over u∈ℋ with uA=1, we will deduce the desired result.Theorem 15.
LetR∈ℬbℋA be an m-quasi-α,β−A-normal operator and s≥0. If(103)λR♯Rm−Rm+1A≤s≤infλR♯RmuA,u∈ℋ,uA=1,then the following inequality holds:(104)α2Rm+1A4≤ωARm♯Rm+22+s2λ2Rm+1A2.Proof.
We use the following inequality [22]:(105)w2v2≤Rew,v2+s2v2,provided w−v≤s≤v.
Settingv=λA1/2R♯Rmu and w=A1/2Rm+1u for u∈ℋ, we get(106)Rm+1uA2λR♯RmuA2≤ReRm+1u,λR♯RmuA2+s2Rm+1uA2.
Hence,(107)λ2α2Rm+1uA4≤λ2ReRm♯Rm+2u,uA2+s2Rm+1A2.
Taking thesup on both sides of the last inequality on u∈ℋ, uA=1, we get the desired result.Theorem 16.
LetR∈ℬbℋA be an m-quasi-α,β−A-normal operator, s≥0.
IfλR♯Rm−Rm+1A≤s for λ∈ℂ,λ≠0, then the following inequality holds:(108)αRm+1A2≤ωAR♯mRm+2+s22λ.Proof.
Using the following reverse of the Schwarz inequality inspired from [23],(109)wv≤Rew,v+s22,provided w−v≤s.Setting v=λA1/2R♯Rmu and w=A1/2Rm+1u for u∈ℋ, we get(110)Rm+1uAλR♯RmuA≤ReRm+1u,λR♯RmuA+s22.
From this, we obtain(111)λαRm+1uA2≤λReRm♯Rm+2u,uA+s22.
Taking thesup of both sides of the last inequality for uA=1, we infer the required result.


## 4. Conclusion

The study of classes of operators and related topics is one of the hottest areas in operator theory in Hilbert and semi-Hilbert spaces. In the work, we introduce a new class of operators known asm-quasi-α,β−A-normal operators, which is a generalization of α,β−A-normal and m-quasi-α,β-normal operators. Several properties are proved by exploiting the special kind of structure associated with such operators. In the course of our investigation, we find some properties of α,β-normal and m-quasi-α,β-normal operators which are retained by m-quasi-α,β−A-normal operators. Some inequalities between the A-operator norm and A-numerical radius of m-quasi-α,β−A-normal operators are proved.



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*Source: 1506330-2022-05-20.xml*