arXiv:1610.03462v1 [math.FA] 11 Oct 2016 eety hs lse foeaoshv engnrlzdb ma by generalized [2 (see been considered have is product operators semi-inner of additional classes of these sets larger Recently, the to isometry, sense, partial some inve isometry, in was hyponormal, generalized, operators been has these class of This theory The studies. intensive some of nHletsae h ls fnra prtr ( operators normal of bounded all class of the algebra space, the of Hilbert subclasses on important most the of One ebul,ClfAgra ..11HyEslm he 02000 chlef Essalem, Hay 151 B.P. Algeria. Chlef Benbouali, [1] [2] NRDCINADPRELIMINARIES AND INTRODUCTION 1 bekdrBenali Abdelkader Keywords. eiinrproduct semi-inner oiieoeaos ( operators, positive on ahmtc ujc Classification: Subject rv oeporeiso ( of proprieties some prove tutrs utemr esaevrosieulte bet inequalities various state we Furthermore structures. and aut fsine ahmtc eatetUiest of Department,University Mathematics science, of Faculty ( ahmtc eatet olg fSine lofUniver Aljouf Science. of College Department, Mathematics H Let ,β α, A . nmrclrdu f( of radius -numerical hsmakes This H EIHLETA SPACES SEMI-HILBERTIAN eaHletsaeadlet and space Hilbert a be ) eiHleta space, Semi-Hilbertian - A NRA PRTR IN OPERATORS -NORMAL H ,β α, h u noasm-ibrinsae nti ae eitoueand introduce we paper this In space. semi-Hilbertian a into | -omloperators. )-normal lof21.SuiArabia Saudi 2014. Aljouf [1] v ,β α, i [email protected] A ,β α, n udAmdMhodSdAhmed Sid Mahmoud Ahmed Ould and [email protected] -omloeaosacrigt eiHleta space semi-Hilbertian to according operators )-normal := coe 2 2016 12, October -omloeaosi eiHleta spaces. Hilbertian semi in operators )-normal h Au RESULTS A Abstract m A slajitoperators, -selfadjoint | ioere prtr nHletspaces. Hilbert on operators -isometries eapstv one prtron operator bounded positive a be v i rmr 60,Scnay47A05. Secondary 46C05, Primary ,v u, , T T , 3 ∗ H ∈ , 4 = , 17 T nue semi- a induces ∗ , enthe ween T 18 .hyhv enteobject the been have ).They A , 1 n te papers. other and ) 21] nra operators, -normal tgtdi 5 n [20]. and [5] in stigated ieroeaosacting operators linear ocle quasinormal, so-called A yatoswe an when authors ny oeao norm -operator Algeria. , H Hassiba The . k . sity k A [2] A - Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 2

In this framework, we show that many results from [7, 8, 12] remain true if we consider an additional semi-inner product defined by a positive semi-definite operator A. We are interested to introducing a new concept of normality in semi-Hilbertian spaces. The contents of the paper are the following. In Section 1, we give notation and results about the concept of A-adjoint operators that will be useful in the sequel. In Section 2 we introduce the new concept of normality of operators in semi-Hilbertian space ( , . . A), called (α, β)-A-normality and we investigate various structural properties of thHish class| i of operators. In Section 3,we state various inequalities between the A-operator norm and A-numerical radius of (α, β)-A-normal operators. We start by introducing some notations. The symbol stands for a complex with inner product . . and norm . . We denoteH by ( ) the of all bounded linear operatorsh | i on , I = kI k being the identityB H operator. ( )+ is the H H B H cone of positive (semi-definite) operators, i.e., ( )+ = A ( ): Au, u 0, u . For every T ( ) its range is B H { ∈ B H h | i ≥ ∀ ∈H} ∈ L H denoted by (T ), its null space by (T ) and its adjoint by T ∗. If is a closed R N M ⊂ H subspace, PM is the orthogonal projection onto . The subspace is invariant for T if T . We shall denote the set of all complexM numbers and theM complex conjugate M⊂M of a complex number λ by C and λ, respectively. The closure of (T ) will be denoted by (T ), and we shall henceforth shorten T λI by T λ. In addition,R if T,S ( ) then RT S means that T S 0. . − − ∈ B H ≥ − ≥ Any A ( )+ defines a positive semi-definite , denoted by ∈ B H . . : ( ) ( ) C, u v = Au v . h | iA B H × B H → h | iA h | i 1 1 We remark that u v A = A 2 u A 2 v . The semi-norm induced by . . A, which is h | i h | i 1 1 h | i denoted by . , is given by u = u u 2 = A 2 u . This makes into a semi- k kA k kA h | iA k k H Hilbertian space. Observe that u A = 0 if and only if u (A). Then . A is a norm if and only if A is an injective operator,k k and the semi-normed∈N space ( ( ), k. k ) is complete B H k kA if and only if (A) is closed. Moreover A induced a on a certain subspace of ( ), namely,R on the subset of all Th|i ( ) for witch there exists a constant c > 0 suchB thatH Tu c u for every u ∈( BT His called A-bounded). For this operators it k kA ≤ k kA ∈ H holds Tu T = sup k kA < . k kA u ∞ u∈R(A),u6=0 k kA It is straightforward that

T = sup Tu v : u, v and u 1, v 1 . k kA {|h | iA| ∈ H k kA ≤ k kA ≤ } Definition 1.1. ([2]) For T ( ), an operator S ( ) is called an A-adjoint of T if for every u, v ∈ B H ∈ B H ∈ H Tu v = u Sv , h | iA h | iA i.e., AS = T ∗A. If T is an A-adjoint of itself, then T is called an A-selfadjoint operator AT = T ∗A .  Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 3

It is possible that an operator T does not have an A-adjoint, and if S is an A-adjoint of T we may find many A-adjoints; In fact, in AR = 0 for some R ( ),then S + R is an ∈ B H A-adjoint of T . The set of all A-bounded operators which admit an A-adjoint is denoted by ( ). By Douglas Theorem (see [6, 10] ) we have that BA H ( )= T ( ) / (T ∗A) (A) . BA H ∈ B H R ⊂ R  If T A( ), then there exists a distinguished A-adjoint operator of T , namely,the reduced∈ B solutionH of equation AX = T ∗A, i.e., A†T ∗A. This operator is denoted by T ♯. Therefore, T ♯ = A†T ∗A and AT ♯ = T ∗A, (T ♯) (A) and (T ♯)= (T ∗A). R ⊂ R N N Note that in which A† is the Moore-Penrose inverse of A. For more details see [2, 3, 4]. In the next proposition we collect some properties of T ♯ and its relationship with the seminorm . . For the proof see [2, 3, 4] . k kA Proposition 1.1. Let T ( ). Then the following statements hold. ∈ BA H (1) T ♯ ( ), (T ♯)♯ = P TP and (T ♯)♯)♯ = T ♯. ∈ BA H R(A) R(A) (2) If S ( ) then TS ( ) and (TS)♯ = S♯T ♯. ∈ BA H ∈ BA H (3) T ♯T and T T ♯ are A-selfadjoint. 1 1 (4) T = T ♯ = T ♯T 2 = T T ♯ 2 . k kA k kA k kA k kA (5) S = T ♯ for every S ( ) which is an A-adjoint of T. k kA k kA ∈ B H (6) If S ( ) then TS = ST . ∈ BA H k kA k kA We recapitulate very briefly the following definitions. For more details, the interested reader is referred to [2, 4, 21] and the references therein. Definition 1.2. Any operator T ( ) is called ∈ BA H (1) A-normal if T T ♯ = T ♯T. ♯ (2) A-isometry if T T = PR(A). ♯ ♯ (3) A-unitary if T T = T T = PR(A). In [21], the A- of an operator T ( ), denoted r (T ) is defined as ∈ B H A 1 n n rA(T ) = lim sup T A n−→∞ k k and the A-numerical radius of an operator T ( ), denoted by ω (T ) is defined as ∈ B H A w (T ) = sup Tu u , u , u =1 . A |h | iA| ∈ H k kA  It is a generalization of the concept of numerical radius of an operator. Clearly, ωA defines a seminorm on ( ). Furthermore, for every u , B H ∈ H Tu u ω (T ) u 2 . |h | iA| ≤ A k kA Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 4

Remark 1.1. If T ( ) is A-selfadjoint,then T = w (T ) (see [21]). ∈ BA H k kA A Theorem 1.1. ([21], Theorem 3.1 ) A necessary and sufficient condition for an operator T ( ) to be A-normal is that ∈ BA H (1) (T T ♯) (A) and R ⊂ R (2) T ♯Tu = T T ♯u for all u . k kA k kA ∈ H

2 PROPERTIES OF (α, β)-A-NORMAL OPERA- TORS

In this section we define the class of (α, β)-A-normal operators according to semi-Hilbertian space structures and we give some their proprieties. Let (α, β) R2 such that 0 α 1 β, an operator T ( ) is said to be ∈ ≤ ≤ ≤ ∈ B H (α, β)-normal [7, 19] if α2T ∗T T T ∗ β2T ∗T, ≤ ≤ which is equivalent to the condition

α Tu T ∗u β Tu k k≤k k ≤ k k for all u . For α =1= β is a normal operator. For α = 1, we observe from the left inequality∈ that H T ∗ is hyponormal and for β =1, from the right inequality we obtain that T is hyponormal. In recent work, Senthilkumar [22] introduced p-(α, β)-normal operators as a generalization of (α, β)- normal operators . An operator T ( ) is said to be p-(α, β)- normal operators for 0

As a generalization of A-normal and A-hyponormal operators, we introduce (α, β)-A- normal operators. Definition 2.3. An operator T ( ) is said to be (α, β)-A-normal for 0 α 1 β, ∈ BA H ≤ ≤ ≤ if β2T ♯T T T ♯ α2T ♯T, ≥A ≥A which is equivalent to the condition β Tu T ♯u α Tu , for all u . k kA ≥k kA ≥ k kA ∈ H When A = I (the identity operator), this coincide with (α, β)-normal operator. For α =1= β is a A normal operator. • For β =1, we observe from the right inequality that T is A-hyponormal. • For α =1, and (A) is invariant subspace for T from the right inequality we obtain • that T ♯ is A-hyponormal.N Remark 2.1. (1) Every A-normal operator is (α, β)-A-normal operator. (2) If A is injective,then (1, 1)-A-normal is A-normal operator. (3) If (T T ♯) (A),then (1, 1)-A-normal is A-normal operator. R ⊂ R We give an example of (α, β)-A-normal operator which is neither A-normal nor A- hyponormal. 1 0 1 2 Example 2.1. Let A = and T = (C2). It easy to check that  0 2   0 1  ∈ B 1 0 A 0, (T ∗A) (A)and T ♯ = , T ♯T = T T ♯and Tu T ♯u . ≥ R ⊂ R  1 1  6 k kA 6≥ k kA T is neither A-normal nor A-hyponormal. Moreover 1 10T ♯T T T ♯ T ♯T. ≥A ≥A 6 1 So T is ( , √10)-A-normal operator. √6 The following theorem gives a necessary and sufficient conditions that an operator to be (α, β)-A-normal. It is similar to [14, Theorem 2.3].

2 Theorem 2.1. Let T A( ) and (α, β) R such that 0 α 1 β. Then T is (α, β)-A-normal if and∈ only B ifH the following conditions∈ are satisfied≤ ≤ ≤ λ2T T ♯ +2α2λT ♯T + T T ♯ 0, for all λ R (1) ≥A ∈    and  2 ♯ ♯ 4 ♯ R  λ T T +2λT T + β T T A 0, for all λ (2).  ≥ ∈  Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 6

Proof. Assume that the conditions (1) and (2) are satisfied and prove that T is (α, β)-A- normal. In fact we have by using elementary properties of real quadratic forms

2 ♯ 2 ♯ ♯ λ T T +2α λT T + T T A 0 2 ♯ 2 ♯ ≥♯ (λ T T +2α λT T + T T )u u A 0, u , λ R ⇔ 2 | 2≥ ∀ ∈ H ∀ ∈ λ 2 T ♯u +2α2λ Tu 2 + T ♯u 0, u , λ R ⇔ A k kA A ≥A ∀ ∈ H ∀ ∈ α Tu T ♯u , u . ⇔ k kA ≤ A ∀ ∈ H

Similarly

2 ♯ ♯ 4 ♯ λ T T +2λT T + β T T A 0 2 ♯ ♯ 4 ♯ ≥ (λ T T +2λT T + β T T )u u A 0, u , λ R ⇔ 2 | ≥ ∀ ∈ H ∀ ∈ λ2 Tu 2 +2λ T ♯u + β4 Tu 2 0, u , λ R ⇔ k kA A k kA ≥A ∀ ∈ H ∀ ∈ T ♯u β Tu , u . ⇔ A ≤ k kA ∀ ∈ H

Consequently α Tu T ♯u β Tu , u . k kA ≤ A ≤ k kA ∀ ∈ H

So T is (α, β)-A-normal as desired. The proof of the converse seems obvious.

Proposition 2.2. Let T A( ) such that (A) is invariant subspace for T and let (α, β) R2 such that 0 <α∈ B 1 H β . Then T Nis an (α, β)-A-normal if and only if T ♯ is 1 1 ∈ ≤ ≤ ( , )-A-normal operator. β α Proof. First assume that T is (α, β)-A-normal operator. We have for all u ∈ H α Tu T ♯u β Tu . k kA ≤ A ≤ k kA

It follows that 1 1 T ♯u Tu and Tu T ♯u . β A ≤k kA k kA ≤ α A

On the other hand,since ( A) is invariant subspace for T we observe that TP = N R(A) PR(A)T and APR(A) = PR(A)A = A and it follows that

♯ ♯ T u = PR(A)TPR(A)u = Tu A . A A k k  Consequently 1 ♯ ♯ ♯ 1 ♯ T u A T u T u A β ≤ A ≤ α  1 1 for all u . Therefore T ♯ is ( , )-A-normal operator. ∈ H β α Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 7

1 1 Conversely assume that T ♯ is ( , )-A-normal operator. We have β α

1 ♯ ♯ ♯ 1 ♯ T u A T u T u A β ≤ A ≤ α 

for all u , and from which it follows that ∈ H 1 1 T ♯u Tu T ♯u β A ≤k kA ≤ α A

for all u . Hence ∈ H α Tu T ♯u β Tu , u . k kA ≤ A ≤ k kA ∀ ∈ H

This completes the proof. The following corollary is a immediate consequence of Proposition 2.2.

Corollary 2.1. Let T ( ) such that (A) is invariant subspace for T and let ∈ BA H N (α, β) R2 such that 0 < α 1 β and αβ = 1. Then T is an (α, β)-A-normal if and only if∈T ♯ is (α, β)-A-normal≤ operator.≤

Remark 2.2. (α, β)-A-normality is not translation invariant, more precisely, there exists an operator T A( ) that T is (α, β)-A-normal,but T + λ is not (α, β)-A-normal for some λ C. The∈ B followingH example shows that such operators exist: ∈ 1 0 1 2 Example 2.2. Consider the operators A = ,T = and S = T + I =  0 2   0 1  2 2 1 (R2). It is easily to check that T is ( , √10)-A-normal, but S is not  0 2  ∈ B √6 1 ( , √10)-A-normal. So (α, β)-A-normality is not translation-invariant. √6 Similarly to [12], we define the following quantities

Re Tu u 1 µ1 (T ) = inf h | iA , u =1, Tu/ (A 2 ) A  Tu k kA ∈N  k kA and Re Tu u 1 µ2 (T ) = sup h | iA , u =1, Tu/ (A 2 ) . A  Tu k kA ∈N  k kA A.Saddi [21, Corollary 3.2 ] have shown that if T is A-normal operator such that (A) is invariant subspace for T , then T λ is A-normal. In [18, Theorem 2.7 ] the authorsN proved this property for A-hyponormal− operators. In the following theorem we extend these results to (α, β)-A-normal operators. This is a generalization of [12, Theorem 2.1]. Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 8

Theorem 2.2. Let T A( ) such that (A) is invariant subspace for T and 0 α 1 β. The following statements∈ B H hold. N ≤ ≤ ≤ (1) If T is (α, β)-A-normal, then λT is (α, β)-A-normal for λ C . ∈ (2) If T is (α, β)-A-normal, then T + λ for λ C is (α, β)-A-normal, if one of the following conditions holds: ∈ (i) µ1 (λT ) 0 A ≥ (ii) µ1 (λT ) < 0, λ 2 +2 λ T µ1 (λT ) > 0. A | | | |k kA A Proof. (1) Since (A) is invariant subspace for T we observe that TP = P T and N R(A) R(A) APR(A) = PR(A)A = A. Let T be (α, β)-A-normal then β2T ♯T T T ♯ α2T ♯T β2 λ 2T ♯T λ 2T T ♯ λ 2α2T ♯T ≥A ≥A ⇔ | | ≥A | | ≥A | | AλT ♯λT AλT λT ♯ α2AλT ♯λT ⇔ ≥ ≥ AP λT ♯λT AP λT λT ♯ α2AP λT ♯λT ⇔ R(A) ≥ R(A) ≥ R(A) β2A(λT )♯(λT ) A(λT )(λT )♯ α2A(λT )♯(λT ) ⇔ ≥ ≥ β2(λT )♯(λT ) (λT )(λT )♯ α2(λT )♯(λT ). ⇔ ≥A ≥A Therefore λT is (α, β)-A-normal operator. (2) Assume that T is (α, β)-A-normal and the condition (i) holds. We need to prove that ♯ ♯ α2 T + λ T + λ u u T + λ T + λ u u | A ≤ | A  D   E D   E  (2.1)  ♯ ♯ T + λ T + λ u u β2 T + λ T + λ u u . | A ≤ | A  D   E D   E In order To verify (2.1) we have

2 ♯ 2 ♯ ♯ α T + λ T + λ u u = α T Tu u + λT u u + λPR(A)Tu u | A  | A | A | A D   E D E 2 + λ PR(A)u u | | D | EA  = α2 T ♯Tu u +2Re λTu u + λ 2 u 2  | A | A | | k kA 

α2 T ♯Tu u + α2 2Re λTu u + λ 2 u 2 . ≤ | A  | A | | k kA 

The condition (i) implies that 2Re λT u u 0 and it follows that | A ≥

2 ♯ ♯ 2 2 α T + λ T + λ u u T T u u +2Re λTu u + λ u A | A ≤  | A | A | | k k  D   E ♯ = T + λ T + λ u u | A D   E = T T ♯u u +2Re λTu u + λ 2 u 2  | A | A | | k kA 

♯ β2 T + λ T + λ u u ≤ | A D   E Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 9

and hence T + λ is (α, β)-A-normal. On the other hand if the condition (ii) is satisfied then we have for λ =0 6 λ 2 +2 λ T µ1 (λT ) | | | |k kA A Re λTu u 1 = λ 2 +2 λ sup Tu inf | A , u =1, Tu/ (A 2 ) | | | | k kA  λ Tu k kA ∈N  kukA=1 | |k kA 2 λ + 2 inf Re λTu u A ≤ | | kukA=1 |

λ 2 +2Re λTu u . ≤ | | | A

A similar argument used as above shows that T + λ is (α, β)-A-normal.

Corollary 2.2. Let T A( ) be an (α, β)-A-normal operator. The following statement hold ∈ B H (1) If µ1 (T ) 0 then T + λ is (α, β)-A-normal for every λ> 0. A ≥ (1) If µ2 (T ) 0 then T + λ is (α, β)-A-normal for every λ< 0. A ≤ 1 1 1 Proof. (1) For every λ > 0 we have µA(λT ) = µA(λT ) = µA(T ) 0.By using Theorem 2.2 (i) we have that T + λ is an (α, β)-A-normal. ≥ 1 2 (2) For every λ < 0 we have µA(λT ) = µA(T ) 0.By using Theorem 2.2 (ii) we have that T + λ is an (α, β)-A-normal. − ≥ Lemma 2.1. ([18], Lemma 2.1) Let T,S ( ) such that T S and let R ( ). ∈ B H ≥A ∈ BA H Then the following properties hold (1) R♯T R R♯SR. ≥A (2) RT R♯ RSR♯. ≥A (3) If R is A-selfadjoint then RT R RSR. ≥A Proposition 2.3. Let T,V ( ) such that (A) is invariant subspace for both T ∈ BA H N and V .If T is an (α, β)-A-normal (0 α 1 β) and V is an A-isometry , then VTV ♯ is an (α, β)-A-normal operator. ≤ ≤ ≤ Proof. Assume that β2T ♯T T T ♯ α2T ♯V and V ♯V = P . This implies ≥A ≥A R(A)

♯ β2 VTV ♯ VTV ♯ = β2 V ♯)♯T ♯V ♯VTV ♯     = β2 P VP T ♯P TV ♯  R(A) R(A) R(A) 

♯ = β2 VP T ♯T VP  R(A) R(A)   ♯ VP T T ♯ VP (by Lemma 2.1) ≥A R(A) R(A) ♯  VTV ♯ VTV ♯ . ≥A   Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 10

Similarly, we have

♯ ♯ ♯ ♯ ♯ VTV VTV = VPR(A)T T VPR(A)    ♯ α2VP T ♯T VP (by Lemma 2.1) ≥A R(A) R(A) ♯  α2 VTV ♯ VTV ♯ . ≥A   The conclusion holds.

Proposition 2.4. Let T,S A( ) such that T is (α, β)-A-normal and S is A-selfadjoint. If T ♯S = ST ♯ then TS is (α,∈ β B)-AH-normal. Proof. Since T is (α, β)-A-normal we have for u ∈ H α TSu T ♯Su β TSu . k kA ≤ A ≤ k kA

On the other hand 2 2 T ♯Su = T ♯Su T ♯Su = AST ♯u ST ♯u = (TS)♯u (TS)♯u = (TS)♯u . A | A | | A A

This implies α TSu (TS)♯u β TSu . k kA ≤ A ≤ k kA

Proposition 2.5. Let T,S A( ) such that T is (α, β)-A-normal and S is A-unitary. If TS = ST and (A) is invariant∈ B H subspace for T then TS is (α, β)-A-normal. N

Proof. Since (A) is invariant subspace for T we observe that TPR(A) = PR(A)T and ♯ N ♯ ♯ ♯ T PR(A) = PR(A)T . Let S be A-unitary then S S = SS = PR(A). Now it is easy to see that

β2 (TS)♯(TS) = β2 T ♯S♯ST = β2 T ♯P T = β2 P T ♯TP .      R(A)   R(A) R(A) By using the fact that T is (α, β)-A-normal,it follows immediately from Lemma 2.1 that

β2 (TS)♯(TS) P T T ♯P α2 (P T ♯TP .   ≥A  R(A) R(A) ≥A R(A) R(A)

(1) (2) | {z } | {z } Notice that (1) gives

♯ ♯ ♯ ♯ ♯ (PR(A)T T PR(A) = TPR(A)T = TSS T = TS(TS) and similarly (2) gives

♯ ♯ ♯ ♯ ♯ (PR(A)T TPR(A) = T PR(A)T = T S ST =(TS) (TS). So β2(TS)♯(TS) TS(TS)♯ α2(TS)♯(TS). ≥A ≥A Hence TS is (α, β)-A-normal operator. Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 11

The following example proves that even if T and S are (α, β)-A–normal operators, their product TS is not in general (α, β)-A-normal operator. 0 0 1 1 0 0 − Example 2.3. (1) Consider T =  0 1 0  and S =  0 1 0  which are 1 0 0 0− 0 1    −  (α, β)-I3-normal and their product is (α, β)-I3-normal. 1 0 1 0 (2) Consider T = and S = − which are (α, β)-I2-normal whereas  1 1   0 1  − 1 0 their product TS = − is not (α, β)-I2-normal.  1 1  − − Theorem 2.3. Let T,S ( ) such that T is (α, β)-A-normal (0 α 1 β) and ∈ BA H ≤ ≤ ≤ S is (α′, β′) -A-normal (0 α′ 1 β′). Then the following statements hold: ≤ ≤ ≤ (1) If T ♯S = ST ♯,then TS is (αα′, ββ′)-A-normal operator. (2) If S♯T = TS♯,then ST is (αα′, ββ′)-A-normal operator. Proof. (1) Since T is (α, β)-A-normal and S is (α′, β′)- A-normal, it follows that for all u ∈ H αα′ TSu α′ T ♯Su = α′ ST ♯u k kA ≤ A A S ♯T ♯u ≤ A ♯ = (TS) u A β ′ ST ♯u ≤ A ′ ♯ = β T Su A ββ ′ TSu . ≤ k kA It follows that αα′ STu (TS)♯u ββ′ TSu . k kA ≤ A ≤ k kA

The proof of the second assertion is completed in much the same way as the first assertion.

The following example shows that the power of (α, β)-A-normal operator not neces- sarily an (α, β)-A-normal. 1 0 1 2 Example 2.4. Let A = and T = (C2). By Example 2.1,T  0 2   0 1  ∈ B 1 is ( , √10)-A-normal.However by direct computation one can show that T 2 is is nei- √6 1 1 1 ther ( , √10)-A-normal nor ( , 10)-A-normal. But it is ( , 100)-A-normal i.e., T 2 is √6 6 36 1 22 2 , (√10)2 -A-normal. √6   n n2 n2 Question. If T A( ) which is (α, β)-A-normal operator, is that true T is (α , β )- A-normal operator?∈ B H Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 12

Remark 2.3. Let T ( ) ,then ∈ BA H (1) If T is A-normal, then r (T )= T (see,[21, Corollary 3.2]). A k kA (2) If T is A-hyponormal, then r (T )= T (see,[18, Theorem 2.6]). A k kA The following theorem presents a generalization of these results to (α, β)-A-normal. Our inspiration cames from [12, Theorem 2.5]. 2n Theorem 2.4. Let T A( ) be an (α, β)-A-normal such that T is (α, β)-A-normal for every n N,too. Then,∈ B weH have ∈ 1 T r (T ) T . β k kA ≤ A ≤k kA Proof. It is we know that if T ( ) then ∈ BA H T ♯T = T T ♯ = T 2 A A k kA and if T is A-selfadjoint then T 2 = T 2 . A k kA

From the definition of (α, β)-A-normal operator and Lemma 2.1.1 we deduce that β2 T ♯ 2T 2 T ♯T 2 α2 T ♯ 2T 2 ≥A ≥A    and so 1 sup T ♯ 2T 2u u sup T ♯T 2u u . | A ≥ β2 | A kukA=1 D  E kukA=1 D  E Hence 2 2 ♯ 2 2 1 ♯ 2 1 4 T T T T = T A . A ≥ β2 A β2 k k   Now using a mathematical induction, we observe that for every positive integer number n, n n ♯ 2 2n 1 2 +1 T T n+1 T A . A ≥ β2 −2 k k  We have 1 n 2n n 1 2 ♯ ♯ 2 2 2n rA(T ) = rA(T )rA(T ) = lim sup T lim sup T A n−→∞ A n−→∞  1 n n n 2 ♯ 2 2 lim T T A ≥ n−→∞  A   1 n n n 2 lim T ♯ 2 T 2 ≥ n−→∞  A   1 T 2 . ≥ β2 k kA Therefore, we get 1 T r (T ) T . β k kA ≤ A ≤k kA This completes the proof. Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 13

Let denote the completion, endowed with a reasonable uniform crose-norm, of the algebraicH⊗H tensor product of with . Given non-zero T,S ( ), let H⊗H H H ∈ B H T S ( ) denote the tensor product on the Hilbert space , when T S is defined⊗ ∈ as B followsH⊗H H⊗H ⊗

T S(ξ η ) (ξ η ) = Tξ ξ Sη η . h ⊗ 1 ⊗ 1 | 2 ⊗ 2 i h 1| 2ih 1| 2i The operation of taking tensor products T S preserves many properties of T,S ( ), but by no means all of them. Thus, whereas⊗ T S is normal if and only if T and∈ BS Hare ⊗ normal [15], there exist paranormal operators T and S such that T S is not paranormal [1]. In [9], Duggal showed that if for non-zero T,S ( ), T ⊗S is p-hyponormal if and only if T and S are p-hyponormal. Thus result was∈ B extendedH ⊗ to p-quasi-hyponormal operators in [16] .

Recall that for T A( ) and S B( ), T S is (α, β)-(A B)-normal operator with 0 α 1 β∈, B if H ∈ B H ⊗ ⊗ ≤ ≤ ≤ ♯ ♯ ♯ β2 T S T S T S T S α2 T S T S ⊗ ⊗ ≥A⊗B ⊗ ⊗ ≥A⊗B ⊗ ⊗       or equivalently

♯ α T S u v T S u v β T S u v , ⊗ ⊗ A⊗B ≤ ⊗ ⊗ A⊗B ≤ ⊗ ⊗ A⊗B      

for all u, v . ∈ H Lemma 2.2. ( [18], Lemma 3.1) + Let Tk,Sk ( ), k = 1, 2 and Let A, B ( ) , such that T1 A T2 A 0 and S S ∈0 B,H then ∈ B H ≥ ≥ 1 ≥B 2 ≥B T S T S 0. 1 ⊗ 1 ≥A⊗B 2 ⊗ 2 ≥A⊗B   Proposition 2.6. ( [18], Proposition 3.2 ) Let T1, T2,S1,S2 ( ) and let A, B + ∈ B H ∈ ( ) such that Tk is A- positive and Sk is B-positive for k = 1, 2. If T1 = 0 and SB H=0,then the following conditions are equivalents 6 1 6 (1) T S T S 2 ⊗ 2 ≥A⊗B 1 ⊗ 1 (2) there exists d> 0 such that dT T and d−1S S . 2 ≥A 1 2 ≥B 1 The following theorem gives a necessary and sufficient condition for T S to be (α, β)- -A B-normal operator when T and S are both nonzero operators. ⊗ ⊗ Proposition 2.7. Let T A( ) and let S B( ) with T = 0 and S = 0. Let (α, β) R2 and (α′, β′) R∈2 BsuchH that 0 α, α∈′ B 1 Hand 1 β, 61 β′. The6 following ∈ ∈ ≤ ≤ ≤ ≤ properties hold: (1) If T is an (α, β)-A normal and S is an (α′, β′)-B-normal ,then T S is a (αα′, ββ′)- A B-normal operator. ⊗ ⊗ (2) If T S is an (α, β)-A B-normal, then there exist two constants d> 0 and d0 > 0 ⊗ ⊗ 1 such that T is d−1α, √dβ -A-normal and S is √d , -B-normal operator. 0 0 √ q  d Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 14

Proof. Assume that T is an (α, β)-A normal and S is an (α′, β′)-B-normal. By assumptions we have β2T ♯T T T ♯ α2T ♯T ≥A ≥A and β′2S♯S SS♯ α′2S♯S. ≥B ≥B It follows from the inequalities above and Lemma 2.2 that

β2β′2T ♯T S♯S T T ♯ SS♯ α2α′2T ♯T S♯S ⊗ ≥A⊗B ⊗ ≥A⊗B ⊗ and so

♯ ♯ ♯ (ββ′)2 T S T S T S T S (αα′)2 T S T S . ⊗ ⊗ ≥A⊗B ⊗ ⊗ ≥A⊗B ⊗ ⊗       Hence,T S is a (αα′, ββ′)-A B-normal operator. ⊗ ⊗ Conversely assume that T S is a (α, β)-A B-normal operator. ⊗ ⊗ We have β2T ♯T S♯S T T ♯ SS♯ α2T ♯T S♯S. ⊗ ≥A⊗B ⊗ ≥A⊗B ⊗ So β2T ♯T S♯S T T ♯ SS♯ (2.2) ⊗ ≥A⊗B ⊗ and T T ♯ SS♯ α2T ♯T S♯S (2.3) ⊗ ≥A⊗B ⊗ We deduce from inequality (2.2) and Proposition 2.6 that there exists a constant d > 0 such that dβ2T ♯T T T ♯ ≥A    and  −1 ♯ ♯  d S S B SS  ≥  dβ2 sup T ♯Tu u sup T T ♯u u | A ≥ | A kukA=1 kukA=1 and so dβ2 T ♯T T T ♯ . A ≥ A 2 −1 Thus, dβ 1. Similarly, we obtain d 1. ≥ ≥ On the other had by inequality (2.3) and we can find a constant d0 > 0 satisfies

d T T ♯ α2T ♯T 0 ≥A    and  −1 ♯ ♯  d0 SS B S S  ≥ It easily to see that  −1 d0 α 1 and d0 1. q ≤ ≤ Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 15

Consequently we have

√dβ 2T ♯T T T ♯ d−1α 2T ♯T ≥A ≥A 0  q  and 1 2 2 S♯S SS♯ d S♯S. √ ≥B ≥B 0 d p  This proof is completes. ′ ′ Theorem 2.5. Let T,S A( ) such that T is (α, β)-A-normal and S is (α , β )-A - normal operators with 0 ∈α B H1 β and 0 α′ 1 β′. The following statements hold: ≤ ≤ ≤ ≤ ≤ ≤ (1) If T ♯S = ST ♯, then TS T is (α2α′, β2β′)-(A A)-normal operator and TS S is (αα′2, ββ′2)-A A-normal⊗ operator ⊗ ⊗ ⊗ (2) If S♯T = TS♯,then ST T is (α′α2, β′β2)-(A A)-normal operator and ST S is (α′2α, β′2β)-A A-normal⊗ operator. ⊗ ⊗ ⊗ Proof. The proof is an immediate consequence of Theorem 2.3 and Proposition 2.7.

3 INEQUALITIES INVOLVING A-OPERATOR NORMS AND A-NUMERICAL RADIUS OF (α, β)-A-NORMAL OPERATORS Drogomir and Moslehian [7] have given various inequalities between the operator norm and the numerical radius of (α, β)-normal operators in Hilbert spaces. Motivated by this work, we will extended some of these inequalities to A-operator norm and A-numerical radius ωA of (α, β)-A-normal in semi-Hilbertian spaces by employing some known results for vectors in inner product spaces. We start with the following lemma reproduced from [13]. R Lemma 3.1. Let r and u, v such that u A v A and u,v / (A) ,then the following inequalities∈ hold ∈ H k k ≥k k ∈N r2 u 2r−2 u v 2 if r 1 k kA k − kA ≥ Re u v  u 2r + v 2r 2 u r v r h | iA  and (3.1) k kA k kA − k kA k kA u v ≤  k kA k kA  v 2r−2 u v 2 if r< 1.  A A  k k k − k Theorem 3.1. T ( ) be an (α, β)-A-normal operator. Then ∈ BA H 2βrω (T 2)+ β2r−2 βT T ♯ 2 if r 1 A − A ≥  2r 2r 2  α + β T A  and (3.2)   k k ≤  r 2 ♯ 2  2β ωA(T )+ βT T if r< 1.  − A   Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 16

Proof. Firstly , assume that r 1 and let u with u A = 1. Since T is (α, β)-A- normal ≥ ∈ H k k 2 α2 Tu 2 T ♯u β2 Tu 2 k kA ≤ A ≤ k kA

we have 2r α2r + β2r Tu 2r β2r Tu 2r + T ♯u .   k kA ≤ k kA A

♯ Applying Lemma 3.1 with the choices u0 = β Tu and v0 = T u we get

2r r−1 2 βTu 2r+ T ♯u 2 βTu r−1 T ♯u Re βTu T ♯u r2 βTu 2r−2 βTu T ♯u . k kA A − k kA A | A ≤ k kA − A (3.3)

From which , it follows that

r−1 α2r + β2r Tu 2r 2 βTu r−1 T ♯u βT 2u u   k kA ≤ k kA A | A

2 +r2 βTu 2 r−2 βTu T ♯u . (3.4) k kA − A

Taking the supremum in (3.4) over u , u = 1 and using the fact that ∈ H k kA sup T 2u u = ω (T 2) | iA A kukA=1

we get

2 α2r + β2r T 2r 2βr T 2r−2 ω (T 2)+ r2β2r−2 T 2r−2 βT T ♯ .   k kA ≤ k kA A k kA − A

So 2 α2r + β2r T 2 2β2rω (T 2)+ r2β2r−2 βT T ♯   k kA ≤ A − A

which is the first inequality in (3.2). By employing a similar argument to that used in the first inequality in (3.1) , gives the second inequality of (3.2).

Theorem 3.2. Let T ( ) be an A(α.β)-normal operator. Then ∈ BA H 1 ω (T )2 β T 2 + ω (T 2) . (3.5) A ≤ 2 k kA A 

Proof. Since for all u, v and e ∈ H u v u e e v u e e v u v |h | iA −h | iA h | iA|≥|h | iA h | iA|−|h | iA| we have by applying the inequalities reproduced from [11]

u v u v u e e v + u e e v u v k kA k kA ≥ |h | iA −h | iA h | iA| |h | iA h | iA|≥|h | iA| Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 17

that 1 u e e v u v + u v (3.6) |h | iA||h | iA| ≤ 2 k kA k kA |h | iA|  for all u,v,e with e =1. ∈ H k kA Let x with x = 1 and choosing in (3.6) u = T x , v = T ♯x and e = x we get ∈ H k kA 1 T x x x T ♯x T x T ♯x + T x T ♯x . (3.7) |h | iA| | A ≤ 2 k kA A | A 

Since T is (α, β)-A-normal, it follows that

1 T x x 2 β T x 2 + T 2x x . (3.8) |h | iA| ≤ 2 k kA | A 

Tanking the supremum over x x =1, we get the desired inequality in (3.5). ∈Hk kA Theorem 3.3. Let T ( ) be an (α, β)-A-normal operator and λ C. Then ∈ BA H ∈ 2 2β T λT ♯ α T 2 ω (T 2)+ − A . (3.9) A A 2 k k ≤ 1+ λ α | |  Proof. For λ = 0, the inequality (3.9) is obvious. Assume that λ = 0.From the following 6 inequality [13]

1 u v u + v u v , ; u, v 0 2 k k k k  u − v ≤k − k ∈H−{ }

k k k k which is well known in the literature as the Dunkl-Williams inequality, it follows that

1 u v u + v u v for all u, v /u,v / (A). 2 k kA k kA u − v ≤k − kA ∈ H ∈N  A A A k k k k A simple computation shows that

u v 2 Re u v 4 u v 2 =2 2 h | iA k − kA u − v − u v ≤ 2 A A A A A u A + v A k k k k k k k k k k k k  which shows that

2 u A v A u v A 2 u v A k k k k − |h | i | k − k 2 , for all u, v /u,v / (A) u A v A ≤ u + v ∈ H ∈N k k k k k kA k kA  and so

2 u A v A 2 u A v A u v A + k k k k 2 u v A . k k k k ≤ |h | i | u + v k − k k kA k kA  Abdelkader Benali and Ould Ahmed Mahmoud Sid Ahmed 18

1 ♯ 2 Let x with x A = 1 and consider u = T x and v = λT x with x / (A T ) = 1 ∈♯ H k k ∈ N (A 2 T ) we obtain N ♯ 2 T x λT x 2 T x λT ♯x T x λT ♯x + k kA A T x λT ♯x . A A A 2 A k k ≤ | T x + λT ♯x − k kA k kA  Since T being (α, β)-A-normal operator, we get

2 2 2 2β T x A ♯ 2 α T x A T x x + k k T x λT x . k k ≤ | A T x + α λ T x 2 − A k kA | |k kA  Tanking the supremum over x ; x =1, we get the desired inequality in (3.9). ∈ H k kA

Theorem 3.4. Let T ( ) be an (α, β)-A-normal operator and λ C. Then ∈ BA H ∈ 1 α2 + β 2 T 4 ω (T 2). (3.10)  − λ  k kA ≤ A | |  Proof. We apply the following inequality inspired from [8] 1 0 u 2 v 2 u v u 2 u v 2 (3.11) ≤k kA k kA − |h | iA| ≤ λ 2 k kA k − kA | | for all u, v and λ C ,λ =0. ∈ H ∈ 6 Let x and set u = T x and v = T ♯x in (3.11) we get ∈ H 1 α2 T x 4 T 2x x 2 + T x 2 1+ λ β 2 T x 2 . (3.12) k kA ≤ | A λ 2 k kA | | k kA  | | Tanking the supremum over x x =1, we get the desired inequality in (3.10). ∈Hk kA References

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