SEMI-HILBERTIAN SPACES

Transcription

1 arxiv: v1 [mathf] 11 Oct 2016 α, β)--norml OPERTORS IN SEMI-HILBERTIN SPCES bdelkader Benali [1] and Ould hmed Mahmoud Sid hmed [2] [1] Faculty of science, Mathematics Department,University of Hassiba Benbouali, Chlef lgeria BP 151 Hay Essalem, chlef 02000, lgeria [2] Mathematics Department, College of Science ljouf University ljouf 2014 Saudi rabia October 12, 2016 bstract Let H be a Hilbert space and let be a positive bounded operator on H The semi-inner product u v := u v, u,v H induces a semi-norm on H This makes H into a semi-hilbertian space In this paper we introduce and prove some proprieties ofα, β)-normal operators according to semi-hilbertian space structures Furthermore we state various inequalities between the -operator norm and -numerical radius of α, β)-normal operators in semi Hilbertian spaces Keywords Semi-Hilbertian space, -selfadjoint operators,-normal operators, - positive operators, α, β)-normal operators Mathematics Subject Classification: Primary 46C05, Secondary INTRODUCTION ND PRELIMINRIES RESULTS One of the most important subclasses of the algebra of all bounded linear operators acting on Hilbert space, the class of normal operators TT = T T)They have been the object of some intensive studies The theory of these operators was investigated in [5] and [20] This class has been generalized, in some sense, to the larger sets of so-called quasinormal, hyponormal, isometry, partial isometry, m-isometries operators on Hilbert spaces Recently, these classes of operators have been generalized by many authors when an additional semi-inner product is considered see [2,3,4,17,18,21] ) and other papers

2 bdelkader Benali and Ould hmed Mahmoud Sid hmed 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 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 -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 H, ), called α, β)--normality and we investigate various structural properties of this class of operators In Section 3,we state various inequalities between the -operator norm and -numerical radius of α, β)--normal operators We start by introducing some notations The symbol H stands for a complex Hilbert space with inner product and norm We denote by BH) the Banach algebra of all bounded linear operators on H, I = I H being the identity operator BH) + is the cone of positive semi-definite) operators, ie, BH) + = { BH) : u, u 0, u H } For every T LH) its range is denoted by RT), its null space by NT) and its adjoint by T If M H is a closed subspace, P M is the orthogonal projection onto M The subspace M is invariant for T if TM M We shall denote the set of all complex numbers and the complex conjugate of a complex number λ by C and λ, respectively The closure of RT) will be denoted by RT), and we shall henceforth shorten T λi by T λ In addition, if T,S BH) then T S means that T S 0 ny BH) + defines a positive semi-definite sesquilinear form, denoted by : BH) BH) C, u v = u v We remark that u v = 1 2u 1 2v The semi-norm induced by, which is denoted by, is given by u = u u 1 2 = 2u 1 This makes H into a semi- Hilbertian space Observe that u = 0 if and only if u N) Then is a norm if and only if is an injective operator, and the semi-normed space BH), ) is complete if and only if R) is closed Moreover induced a seminorm on a certain subspace of BH), namely, on the subset of all T BH) for witch there exists a constant c > 0 such that Tu c u for every u H T is called -bounded) For this operators it holds Tu T = sup < u R),u 0 u It is straightforward that T = sup{ Tu v : u,v H and u 1, v 1 } Definition 11 [2]) For T BH), an operator S BH) is called an -adjoint of T if for every u,v H Tu v = u Sv, ie, S = T If T is an -adjoint of itself, then T is called an -selfadjoint operator T = T )

3 bdelkader Benali and Ould hmed Mahmoud Sid hmed 3 It is possible that an operator T does not have an -adjoint, and if S is an -adjoint of T we may find many -adjoints; In fact, in R = 0 for some R BH),then S +R is an -adjoint of T The set of all -bounded operators which admit an -adjoint is denoted by B H) By Douglas Theorem see [6, 10] ) we have that B H) = { T BH) / RT ) R) } If T B H), then there exists a distinguished -adjoint operator of T, namely,the reduced solution of equation X = T, ie, T This operator is denoted by T Therefore, T = T and T = T, RT ) R) and NT ) = NT ) Note that in which is the Moore-Penrose inverse of 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] Proposition 11 Let T B H) Then the following statements hold 1) T B H),T ) = P R) TP R) and T ) ) = T 2) If S B H) then TS B H) and TS) = S T 3) T T and TT are -selfadjoint 4) T = T = T T 1 2 = TT 1 2 5) S = T for every S BH) which is an -adjoint of T 6) If S B H) then TS = ST We recapitulate very briefly the following definitions For more details, the interested reader is referred to [2,4,21] and the references therein Definition 12 ny operator T B H) is called 1) -normal if TT = T T 2) -isometry if T T = P R) 3) -unitary if T T = TT = P R) In [21], the -spectral radius of an operator T BH), denoted r T) is defined as r T) = limsup T n 1 n n and the -numerical radius of an operator T BH), denoted by ω T) is defined as w T) = sup { Tu u, u H, u = 1 } It is a generalization of the concept of numerical radius of an operator Clearly, ω defines a seminorm on BH) Furthermore, for every u H, Tu u ω T) u 2

4 bdelkader Benali and Ould hmed Mahmoud Sid hmed 4 Remark 11 If T B H) is -selfadjoint,then T = w T) see [21]) Theorem 11 [21], Theorem 31 ) necessary and sufficient condition for an operator T B H) to be -normal is that 1) RTT ) R) and 2) T Tu = TT u for all u H 2 PROPERTIES OF α, β)--norml OPER- TORS In this section we define the class ofα, β)--normal operators according to semi-hilbertian space structures and we give some their proprieties Let α,β) R 2 such that 0 α 1 β, an operator T BH) is said to be α,β)-normal [7,19] if α 2 T T TT β 2 T T, which is equivalent to the condition α Tu T u β Tu for all u H For α = 1 = β is a normal operator For α = 1, we observe from the left inequality that 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 n operator T BH) is said to be p-α,β)- normal operators for 0 < p 1 if α 2 T T) p TT ) p β 2 T T) p,0 α 1 β When p = 1, this coincide with α, β)-normal operators Now we are going to consider an extension of the notion of α,β) -normal operators, similar to those extensions of the notion of normality to -normality and hyponormality to -hyponormality see [18, 21]) Definition 21 [18]) Let BH) + and T BH) We say that T is an -positive if T BH) + which is equivalent to the condition We note T 0 Tu u 0 u H Definition 22 [18]) n operator T B H) is said to be -hyponormal if T T TT is -positive ie, T T TT 0 Proposition 21 [18]) Let T B H) Then T is -hyponormal if and only if Tu T u for all u H

5 bdelkader Benali and Ould hmed Mahmoud Sid hmed 5 s a generalization of -normal and -hyponormal operators, we introduce α, β)-normal operators Definition 23 n operator T B H) is said to be α,β)--normal for 0 α 1 β, if β 2 T T TT α 2 T T, which is equivalent to the condition β Tu T u α Tu, for all u H When = I the identity operator), this coincide with α, β)-normal operator For α = 1 = β is a normal operator For β = 1, we observe from the right inequality that T is -hyponormal For α = 1, and N) is invariant subspace for T from the right inequality we obtain that T is -hyponormal Remark 21 1) Every -normal operator is α, β)--normal operator 2) If is injective,then 1, 1)--normal is -normal operator 3) If RTT ) R),then 1,1)--normal is -normal operator We give an example of α, β)--normal operator which is neither -normal nor - hyponormal ) ) Example 21 Let = and T = BC ) It easy to check that ) 1 0 0,RT ) R)and T =,T T TT and Tu 1 1 T u T is neither -normal nor -hyponormal Moreover So T is 1 6, 10)--normal operator 10T T TT 1 6 T T The following theorem gives a necessary and sufficient conditions that an operator to be α, β)--normal It is similar to [14, Theorem 23] Theorem 21 Let T B H) and α,β) R 2 such that 0 α 1 β Then T is α, β)--normal if and only if the following conditions are satisfied λ 2 TT +2α 2 λt T +TT 0, for all λ R 1) and λ 2 T T +2λTT +β 4 T T 0, for all λ R 2)

6 bdelkader Benali and Ould hmed Mahmoud Sid hmed 6 Proof ssume that the conditions 1) and 2) are satisfied and prove that T is α,β)-normal In fact we have by using elementary properties of real quadratic forms λ 2 TT +2α 2 λt T +TT 0 λ 2 TT +2α 2 λt T +TT )u u 0, u H, λ R λ 2 T u 2 +2α2 λ Tu 2 + T u 2 0, u H, λ R α Tu T u, u H Similarly λ 2 T T +2λTT +β 4 T T 0 λ 2 T T +2λTT +β 4 T T)u u 0, u H, λ R λ 2 Tu 2 +2λ T u 2 +β4 Tu 2 0, u H, λ R T u β Tu, u H Consequently α Tu T u β Tu, u H So T is α,β)--normal as desired The proof of the converse seems obvious Proposition 22 Let T B H) such that N) is invariant subspace for T and let α,β) R 2 such that 0 < α 1 β Then T is an α,β)--normal if and only if T is 1 β, 1 )--normal operator α Proof First assume that T is α,β)--normal operator We have for all u H It follows that α Tu T u β Tu 1 T u β Tu and Tu 1 T u α On the other hand,since N) is invariant subspace for T we observe that TP R) = P R) T and P R) = P R) = and it follows that T ) u = P R) TP R) u = Tu Consequently 1 T u β T ) u 1 T u α for all u H Therefore T is 1 β, 1 )--normal operator α

7 bdelkader Benali and Ould hmed Mahmoud Sid hmed 7 Conversely assume that T is 1 β, 1 )--normal operator We have α 1 T u β T ) u 1 T u α for all u H, and from which it follows that for all u H Hence This completes the proof 1 T u β Tu 1 T u α α Tu T u β Tu, u H The following corollary is a immediate consequence of Proposition 22 Corollary 21 Let T B H) such that N) is invariant subspace for T and let α,β) R 2 such that 0 < α 1 β and αβ = 1 Then T is an α,β)--normal if and only if T is α,β)--normal operator Remark 22 α, β)--normality is not translation invariant, more precisely, there exists an operator T B H) that T is α,β)--normal,but T +λ is not α,β)--normal for some λ C The following example shows that such operators exist: ) ) Example 22 Consider the operators =,T = and S = T +I = ) 2 2 BR ) It is easily to check that T is 1, 10)--normal, but S is not 6 1 6, 10)--normal So α,β)--normality is not translation-invariant and Similarly to [12], we define the following quantities µ 1 T) = inf{ Re Tu u Tu, u = 1, Tu / N 1 2 ) { } Re Tu µ 2 T) = sup u, u = 1, Tu / N 1 2 ) Tu Saddi [21, Corollary 32 ] have shown that if T is -normal operator such that N) is invariant subspace for T, then T λ is -normal In [18, Theorem 27 ] the authors proved this property for -hyponormal operators In the following theorem we extend these results to α, β)--normal operators This is a generalization of [12, Theorem 21] }

8 bdelkader Benali and Ould hmed Mahmoud Sid hmed 8 Theorem 22 Let T B H) such that N) is invariant subspace for T and 0 α 1 β The following statements hold 1) If T is α,β)--normal, then λt is α,β)--normal for λ C 2) If T is α,β)--normal, then T + λ for λ C is α,β)--normal, if one of the following conditions holds: i) µ 1 λt) 0 ii) µ 1 λt) < 0, λ 2 +2 λ T µ 1 λt) > 0 Proof 1) Since N) is invariant subspace for T we observe that TP R) = P R) T and P R) = P R) = Let T be α,β)--normal then β 2 T T TT α 2 T T β 2 λ 2 T T λ 2 TT λ 2 α 2 T T Therefore λt is α, β)--normal operator λt λt λtλt α 2 λt λt P R) λt λt P R) λtλt α 2 P R) λt λt β 2 λt) λt) λt)λt) α 2 λt) λt) β 2 λt) λt) λt)λt) α 2 λt) λt) 2) ssume that T is α,β)--normal and the condition i) holds We need to prove that T ) α 2 ) T ) ) +λ T +λ u u +λ T +λ u u T ) ) T ) +λ T +λ u u β 2 ) 21) +λ T +λ u u In order To verify 21) we have T ) α 2 ) { T +λ T +λ u u = α 2 Tu u + λt u u λp + R) Tu u } + λ P 2 R) u u { T = α 2 Tu u +2Re λtu u } + λ 2 u 2 α 2 T Tu u {2Re +α2 λtu u } + λ 2 u 2 The condition i) implies that 2Re λtu u 0 and it follows that T ) α 2 ) { TT +λ T +λ u u u u +2Re λtu u } + λ 2 u 2 T ) ) = +λ T +λ u u { TT = u u +2Re λtu u } + λ 2 u 2 T ) β 2 ) +λ T +λ u u

9 bdelkader Benali and Ould hmed Mahmoud Sid hmed 9 and hence T + λ is α,β)--normal On the other hand if the condition ii) is satisfied then we have for λ 0 λ 2 +2 λ T µ 1 λt) ) { }) Re λtu u = λ 2 +2 λ sup Tu inf, u = 1, Tu / N 1 2 ) u =1 λ Tu λ 2 +2 inf u =1Re λtu u λ 2 +2Re λtu u similar argument used as above shows that T +λ is α,β)--normal Corollary 22 Let T B H) be an α,β)--normal operator The following statement hold 1) If µ 1 T) 0 then T +λ is α,β)--normal for every λ > 0 1) If µ 2 T) 0 then T +λ is α,β)--normal for every λ < 0 Proof 1) For every λ > 0 we have µ 1 λt) = µ1 λt) = µ1 T) 0By using Theorem 22 i) we have that T +λ is an α,β)--normal 2) For every λ < 0 we have µ 1 λt) = µ2 T) 0By using Theorem 22 ii) we have that T +λ is an α,β)--normal Lemma 21 [18], Lemma 21 ) Let T,S BH) such that T S and let R B H) Then the following properties hold 1) R TR R SR 2) RTR RSR 3) If R is -selfadjoint then RTR RSR Proposition 23 Let T,V B H) such that N) is invariant subspace for both T and VIf T is an α,β)--normal 0 α 1 β) and V is an -isometry, then VTV is an α, β)--normal operator Proof ssume that β 2 T T TT α 2 T V and V V = P R) This implies β 2 VTV ) VTV ) = β 2 V ) T V VTV ) = β 2 P R) VP R) T P R) TV ) = β 2 VP R) T T VP R) ) ) VP R) TT VP R) ) by Lemma 21) VTV ) VTV )

10 bdelkader Benali and Ould hmed Mahmoud Sid hmed 10 Similarly, we have VTV ) VTV ) = VP R) TT VP R) ) α 2 VP R) T T VP R) ) by Lemma 21) α 2 VTV ) VTV ) The conclusion holds Proposition24 LetT,S B H) suchthat T is α,β)--normalands is-selfadjoint If T S = ST then TS is α,β)--normal Proof Since T is α,β)--normal we have for u H α TSu T Su β TSu On the other hand T Su 2 = T Su T Su = ST u ST u = TS) u TS) u = TS) u 2 This implies α TSu TS) u β TSu Proposition 25 Let T,S B H) such that T is α,β)--normal and S is -unitary If TS = ST and N) is invariant subspace for T then TS is α,β)--normal Proof Since N) is invariant subspace for T we observe that TP R) = P R) T and T P R) = P R) T Let S be -unitary then S S = SS = P R) Now it is easy to see that ) ) ) ) β TS) 2 TS) = β 2 T S ST = β 2 T P R) T = β P 2 R) T TP R) By using the fact that T is α, β)--normal,it follows immediately from Lemma 21 that ) ) ) β TS) 2 TS) P R) TT P R) α 2 P R) T TP R) } {{ } } {{ } 1) 2) Notice that 1) gives and similarly 2) gives So P R) TT P R) = TP R) T = TSS T = TSTS) P R) T TP R) = T P R) T = T S ST = TS) TS) Hence T S is α, β)--normal operator β 2 TS) TS) TSTS) α 2 TS) TS)

11 bdelkader Benali and Ould hmed Mahmoud Sid hmed 11 The following example proves that even if T and S are α, β)- normal operators, their product T S is not in general α, β)--normal operator Example 23 1) Consider T = and S = which are α,β)-i 3 -normal and their product is α,β)-i 3 -normal ) ) ) Consider T = and S = which are α,β)-i normal whereas ) 1 0 their product TS = is not α,β)-i normal Theorem 23 Let T,S B H) such that T is α,β)--normal 0 α 1 β) and S is α,β ) --normal 0 α 1 β ) Then the following statements hold: 1) If T S = ST,then TS is αα,ββ )--normal operator 2) If S T = TS,then ST is αα,ββ )--normal operator Proof 1) Since T is α,β)--normal and S is α,β )- -normal, it follows that for all u H It follows that αα TSu α T Su = α ST u S T u = TS) u β ST u = β T Su ββ TSu αα STu TS) u ββ TSu The proof ofthe second assertion is completed inmuch the same way as the first assertion The following example shows that the power of α, β)--normal operator not necessarily an α, β)--normal ) ) Example 24 Let = and T = BC ) By Example 21,T is 1, 10)--normalHowever by direct computation one can show that T 2 is is nei- 6 ther 1 6, 10)--normal nor 1 6,10)--normal But it is 1 36,100)--normal ie, T2 is 1 6 ) 2 2, 10) 2 2 ) --normal Question If T B H) which is α,β)--normal operator, is that true T n is α n2,β n2 )- -normal operator?

12 bdelkader Benali and Ould hmed Mahmoud Sid hmed 12 Remark 23 Let T B H),then 1) If T is -normal, then r T) = T see,[21,corollary 32]) 2) If T is -hyponormal, then r T) = T see,[18,theorem 26]) The following theorem presents a generalization of these results to α, β)--normal Our inspiration cames from [12, Theorem 25] Theorem 24 Let T B H) be an α,β)--normal such that T 2n is α,β)--normal for every n N,too Then, we have 1 β T r T) T Proof It is we know that if T B H) then T T = TT = T 2 and if T is -selfadjoint then T 2 = T 2 From the definition of α, β)--normal operator and Lemma 211 we deduce that and so β 2 T )2 T 2 T T )2 α 2 T )2 T 2 T sup )2 T 2 u u 1 T sup T )2 u u u =1 β 2 u =1 Hence ) T 2 T T T ) 2 2 = 1 β 2 β 2 T 4 Now using a mathematical induction, we observe that for every positive integer number n, We have T ) 2 n T 2n r T) 2 = r T )r T) = lim sup 1 β 2n+1 2 T 2n+1 n T ) 2 n 1 lim ) T 2 n n lim n 1 β 2 T 2 2 n T ) 2 n T 2n lim sup T 2 n 1 n ) 1 T 2 n ) 1 2 n 2 n 2 n Therefore, we get This completes the proof 1 β T r T) T

13 bdelkader Benali and Ould hmed Mahmoud Sid hmed 13 Let H H denote the completion, endowed with a reasonable uniform crose-norm, of the algebraic tensor product H H of H with H Given non-zero T,S BH), let T S BH H) denote the tensor product on the Hilbert space H H, when T S is defined as follows T Sξ 1 η 1 ) ξ 2 η 2 ) = Tξ 1 ξ 2 Sη 1 η 2 The operation of taking tensor products T S preserves many properties of T,S BH), but by no means all of them Thus, whereas T S is normal if and only if T and S are 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 BH),T S is p-hyponormal if and only if T and S are p-hyponormal Thus result was extended to p-quasi-hyponormal operators in [16] Recall that for T B H) and S B B H), T S is α,β)- B)-normal operator with 0 α 1 β, if β 2 T S ) T S ) B T S ) T S ) B α 2 T S ) T S ) or equivalently α T S ) u v ) B for all u,v H T S ) u v ) B β T S ) u v ) B, Lemma 22 [18], Lemma 31 ) Let T k,s k BH), k = 1,2 and Let,B BH) +, such that T 1 T 2 0 and S 1 B S 2 B 0, then T1 S 1 ) B T2 S 2 ) B 0 Proposition 26 [18], Proposition 32 ) Let T 1,T 2,S 1,S 2 BH) and let,b BH) + such that T k is - positive and S k is B-positive for k = 1,2 If T 1 0 and S 1 0,then the following conditions are equivalents 1) T 2 S 2 B T 1 S 1 2) there exists d > 0 such that dt 2 T 1 and d 1 S 2 B S 1 The following theorem gives a necessary and sufficient condition for T S to be α,β)- - B-normal operator when T and S are both nonzero operators Proposition 27 Let T B H) and let S B B H) with T 0 and S 0 Let α,β) R 2 and α,β ) R 2 such that 0 α, α 1 and 1 β, 1 β The following properties hold: 1) If T is an α,β)- normal and S is an α,β )-B-normal,then T S is a αα,ββ )- B-normal operator 2) If T S is an α,β)- B-normal, then there exist two constants d > 0 and d 0 > 0 such that T is d 1 0 α, dβ ) --normal and S is 1 ) d 0, -B-normal operator d

14 bdelkader Benali and Ould hmed Mahmoud Sid hmed 14 Proof ssumethatt isanα,β)-normalands isanα,β )-B-normalByassumptions we have β 2 T T TT α 2 T T and β 2 S S B SS B α 2 S S It follows from the inequalities above and Lemma 22 that and so β 2 β 2 T T S S B TT SS B α 2 α 2 T T S S ββ ) 2 T S ) T S ) B T S ) T S ) B αα ) 2 T S ) T S ) Hence,T S is a αα,ββ )- B-normal operator Conversely assume that T S is a α,β)- B-normal operator We have So and β 2 T T S S B TT SS B α 2 T T S S β 2 T T S S B TT SS 22) TT SS B α 2 T T S S 23) We deduce from inequality 22) and Proposition 26 that there exists a constant d > 0 such that dβ 2 T T TT and d 1 S S B SS and so dβ 2 sup T Tu u sup TT u u u =1 u =1 dβ 2 T T TT Thus, dβ 2 1 Similarly, we obtain d 1 1 On the other had by inequality 23) and we can find a constant d 0 > 0 satisfies d 0 TT α 2 T T and It easily to see that d 1 0 SS B S S d 1 0 α 1 and d 0 1

15 bdelkader Benali and Ould hmed Mahmoud Sid hmed 15 Consequently we have and dβ ) 2 T T TT d 1 0 α)2 T T 1 d ) 2S S B SS B d0 ) 2S S This proof is completes Theorem 25 Let T,S B H) such that Tis α,β)--normal and S is α,β )- - normal operators with 0 α 1 β and 0 α 1 β The following statements hold: 1) If T S = ST, then TS T is α 2 α,β 2 β )- )-normal operator and TS S is αα 2,ββ 2 )- -normal operator 2) If S T = TS,then ST T is α α 2,β β 2 )- )-normal operator and ST S is α 2 α,β 2 β)- -normal operator Proof The proof is an immediate consequence of Theorem 23 and Proposition 27 3 INEQULITIES INVOLVING -OPERTOR NORMS ND -NUMERICL RDIUS OF α, β)--norml OPERTORS 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 -operator norm and -numerical radius ω of α,β)--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] Lemma 31 Let r R and u,v H such that u v and u,v / N),then the following inequalities hold u 2r Re u v + v 2r 2 u r v r u v r 2 u 2r 2 u v 2 if r 1 and v 2r 2 u v 2 if r < 1 Theorem 31 T B H) be an α,β)--normal operator Then 2β r ω T 2 )+β 2r 2 βt T 2 if r 1 ) α 2r +β 2r T 2 and 31) 32) 2β r ω T 2 )+ βt T 2 if r < 1

16 bdelkader Benali and Ould hmed Mahmoud Sid hmed 16 Proof Firstly, assume that r 1 and let u H with u = 1 Since T is α,β)-normal α 2 Tu 2 T u 2 β2 Tu 2 we have α 2r +β 2r ) Tu 2r β2r Tu 2r + T u 2r pplying Lemma 31 with the choices u 0 = β Tu and v 0 = T u we get βtu 2r + T u 2r 2 βtu r 1 From which, it follows that α 2r +β 2r ) Tu 2r T u r 1 Re βtu T u r2 βtu 2r 2 2 βtu r 1 T u r 1 +r 2 βtu 2r 2 βtu T u 2 33) βt 2 u u βtu T u 2 34) Taking the supremum in 34) over u H, u = 1 and using the fact that we get α 2r +β 2r ) T 2r sup T 2 u u = ω T 2 ) u =1 2βr T 2r 2 ω T 2 )+r 2 β 2r 2 T 2r 2 βt T 2 So α 2r +β 2r ) T 2 2β2r ω T 2 )+r 2 β 2r 2 βt T 2 which is the first inequality in 32) By employing a similar argument to that used in the first inequality in 31), gives the second inequality of 32) Theorem 32 Let T B H) be an αβ)-normal operator Then ω T) 2 1 ) β T 2 2 +ω T 2 ) 35) Proof Since for all u,v and e H u v u e e v u e e v u v we have by applying the inequalities reproduced from [11] u v u v u e e v + u e e v u v

17 bdelkader Benali and Ould hmed Mahmoud Sid hmed 17 that u e e v 1 2 ) u v + u v 36) for all u,v,e H with e = 1 Let x H with x = 1 and choosing in 36) u = Tx, v = T x and e = x we get Tx x x T x 1 Tx T 2 x + Tx T x ) 37) Since T is α, β)--normal, it follows that Tx x 2 1 β Tx T 2 x x ) 38) Tanking the supremum over x H x = 1, we get the desired inequality in 35) Theorem 33 Let T B H) be an α,β)--normal operator and λ C Then α T 2 ω T 2 )+ 2β T λt 2 1+ λ α ) 2 39) Proof For λ = 0, the inequality 39) is obvious ssume that λ 0From the following inequality [13] ) 1 u u + v 2 u v v u v, ; u,v H {0} which is well known in the literature as the Dunkl-Williams inequality, it follows that 1 ) u + v u 2 v u v u v for all u,v H / u,v / N) simple computation shows that which shows that u v u v 2 = 2 2 Re u v u v 4 u v 2 u + v ) 2 u v u v u v 2 u v 2 u + v ) 2, for all u,v H / u,v / N) and so u v u v + 2 u v u + v ) 2 u v 2

18 bdelkader Benali and Ould hmed Mahmoud Sid hmed 18 Let x H with x = 1 and consider u = Tx and v = λt x with x / N 1 2T) = N 1 2T ) we obtain Tx λt x Tx λt x 2 Tx λt + x ) Tx λt Tx + λt 2 x 2 x Since T being α, β)--normal operator, we get α Tx 2 T 2 x x 2β Tx 2 + ) Tx λt 2 x 2 Tx +α λ Tx Tanking the supremum over x H; x = 1, we get the desired inequality in 39) Theorem 34 Let T B H) be an α,β)--normal operator and λ C Then [ α 2 ] 1 λ +β) 2 T 4 ω T 2 ) 310) Proof We apply the following inequality inspired from [8] for all u,v H and λ C,λ 0 0 u 2 v 2 u v 1 λ 2 u 2 u v 2 311) Let x H and set u = Tx and v = T x in 311) we get α 2 Tx 4 T 2 x x ) 2 λ 2 Tx 2 1+ λ β Tx 2 312) Tanking the supremum over x H x = 1, we get the desired inequality in 310) References [1] T ndo, Operators with a norm condition, cta Sci MathSzeged) ), [2] ML rias, G Corach, MC Gonzalez, Partial isometries in semi-hilbertian spaces, Linear lgebra ppl 428 7) 2008) [3] ML rias, G Corach, MC Gonzalez,Metric properties of projections in semi- Hilbertian spaces,integral Equations Operator Theory 62 1) 2008) [4] M L rias, G Corach, M C Gonzalez, Lifting properties in operator ranges,cta Sci Math Szeged) 75: ), [5] JB Conway, Course in Functional nalysis, Second Edition, Springer Verlag, Berlin - Heildelberg - New York 1990

19 bdelkader Benali and Ould hmed Mahmoud Sid hmed 19 [6] RG Douglas, On majorization, factorization and range inclusion of operators in Hilbert space, Proc mer Math Soc ) [7] S S Dragomir and M S Moslehian, Some inequalities for α, β)-normal operators in Hilbert spaces, Facta Universitatis, vol 23,2008) pp 3947 [8] S S Dragomir,Inequalities for normal operators in Hilbert spaces, pplicable nalysis and Discrete Mathematics, ), [9] BPDuggal,Tensor products of operators-strong stability and p-hyponormalityglasgow Math Journal ), [10] P Fillmore, JPWilliams, On operator ranges, dv Math ) [11] CF Dunkl and KS Williams, simple norm inequality, mer Math Monthly, 711) 1964), 4344 [12] R Eskandari, F Mirzapour, and Morassaei, More on α, β)-normal Operators in Hilbert Spaces,bstract and pplied nalysis Volume 2012, rticle ID , 11 pages [13] Goldstein, JV Ryff and LE Clarke, Problem 5473, mer Math Monthly, 753) 1968), 309 [14] Gupta and PSharma,α, β)-normal Composition Operators, Thai Journal of Mathematics Volume ) Number 1 : [15] JC Hou, On the tensor products of operators, cta Math Sinica NS) ), no 2, [16] IH Kim,On p; k)-quasihyponormal operators Mathematical inequalities and pplications Vol 7,Number ), [17] OMahmoud Sid hmed and Saddi, -m-isomertic operators in semi-hilbertian spaces,linear lgebra and its pplications ) [18] OMahmoud Sid hmed and Benali, Hyponormal and k-quasi-hyponormal operators on Semi-Hilbertian spaces The ustralian Journal of Mathematical nalysis and pplicationsvolume 13, Issue 1, rticle 7, 2016), pp 1 22 [19] MS Moslehian, On α, β)-normal operators in Hilbert spaces, IMGE, ) Problem 39 4 [20] CR Putnam, On normal operators in Hilbert space, mer J Math ), [21] Saddi, -Normal operators in Semi-Hilbertian spacesthe ustralian Journal of Mathematical nalysis and pplicationsvolume 9, Issue 1, rticle 5, 2012), pp 1 12

20 bdelkader Benali and Ould hmed Mahmoud Sid hmed 20 [22] D Senthilkumar,On p-α, β)-normal Operators, pplied Mathematical Sciences, Vol 8, 2014, no 41,