PAijpam.eu CLASS OF (A, n)-power QUASI-NORMAL OPERATORS IN SEMI-HILBERTIAN SPACES

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1 International Journal of Pure and Applied Mathematics Volume 93 No , ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpam.eu CLASS OF (A, n)-power QUASI-NORMAL OPERAORS IN SEMI-HILBERIAN SPACES Sidi Hamidou Jah Department of Mathematics College of Science Qassim University P.O. Box 6640 Buraydah 51452, SAUDI ARABIA Abstract: In this paper, the concept of n-power quasi-normal operators on a Hilbert space defined by Sid Ahmed in [14] is generalized when an additional semi-inner product is considered. his new concept is described by means of oblique projections. For a Hilbert space operator B(H) is (A, n)-power quasi-normal operators for some positive operator A and for some positive integer n if, [ n A n] = 0, n = 1,2,... AMS Subject Classification: 47B20, 47B99 Key Words: operator, quasi-normal, n-normal, reducing subspace, Hilbert space 1. Introduction and erminologies A bounded linear operator on a complex Hilbert space is quasi-normal if and commute. he class of quasi-normal operators was first introduced and studied by A.Brown [5] in From the definition, it is easily seen that this class contains normal operators and isometries. In [14], the author introduced the class of n-power quasi-normal operators as a generalization of the class of quasi-normal operators and study some properties of such class for Received: November 4, 2013 c 2014 Academic Publications, Ltd. url:

2 62 S.H. Jah different values of the parameter n, in particular for n = 2 and n = 3, (see [14] and [15]). he purposeof this paper is to study the class of (A,n)-power quasi-normal operators in semi-hilbertian spaces. Along this work, H denotes a complex Hilbert space with inner product, B(H) is the algebra of all bounded linear operators on H, B(H) + is the cone of positive (semi-definite) operators of B(H) i.e., B(H) + := { B : (H) ξ ξ 0, ξ H } and B(H) cr is the subset of B(H) of all operators with closed range. For every B(H), we denote by N(),R() and R() respectively, the null space, the range and the closure of the range of. Also the adjoint operator of is denoted by and stands for the Moore-Penrose inverse of. In addition, if 1, 2 B(H) then 1 2 means that 1 2 B(H) +. For a closed subspace S of H, P S denotes the orthogonal projection onto S. Note that for A B(H) +, the functional A : H H C, ξ η A = Aξ η is a semi-inner product on H. By. A, we denote the semi-norm induced by A, i.e., ξ A = ξ ξ 1 2 A. Observe that ξ A = 0 if and only if ξ N(A). hen. A is a norm if and only if A is an injective operator, and the semi-normed space (H,. A ) is complete if and only if R(A) is closed. Moreover, A induces a semi-norm on a certain subspace of B(H), namely, on the subspace { } B(H)/ c > 0 : ξ A c ξ A, ξ H. In such case, it holds A = sup ξ R(A) ξ 0 { = sup = inf ξ A = sup ξ A ξ A ξ A 1 } ξ A : ξ A =1 { c > 0 : ξ A c ξ A, ξ H } <. Moreover, { } A = sup ξ η A ; ξ,η H, : ξ 1, η 1.

3 CLASS OF (A, n)-power QUASI-NORMAL For ξ,η H, we say that ξ and η are A-orthogonal if ξ η A = 0. Define { B 1 A2 (H) := B(H) : ξ A c ξ A for every ξ H ItiseasytoseethatB A 1 2 (H)isasubspaceofB(H).Ingeneralif B A 1 2 (H), / B A 1 2 (H). From now on, A denotes a positive operator on H, that is A B(H) +. }. Definition 1.1. ([1]) For B(H), an operator S B(H) is called an A-adjoint of if for every ξ,η H ξ η A = ξ Sη A, i.e., AS = A or equivalently S is a solution of the equation AX = A. We say that is A-selfadjoint if A = A. Remark 1.1. he existence of an A- adjoint operator is not guaranteed. Observe that admits an A-adjoint if and only if the equation AX = A has solution. his hind of equation can be studied applying the next theorem due to Douglas (for its proof see [6],[7]). heorem 1.1. Let C, B B(H). he following conditions are equivalents. (1) R(B) R(C). (2) here exists a positive number λ such that BB λcc. (3) here exists S B(H) such that CS = B. If one of these conditions holds, then there exists a unique operator D B(H) such that CD = B and R(D) R(C ) and N(D) = N(B). Moreover } D = inf {λ > 0 /BB λcc. his solution will be called a reduced solution of the equation CX = B. If we denote by B A (H) the subalgebra of B(H) of all bounded operators which admit an A-adjoint operator, then { } B A (H) = B(H) : R(A) R(A).

4 64 S.H. Jah Furthermore, applying Douglas theorem, we can see that { } B 1 A2 (H) = B(H) : R(A ) R(A 2 ) { } = B(H) : R(A 1 2 A 1 2) R(A). he relationship between the above sets is proved in [13]. Proposition 1.1. he following inclusion B A (H) B A 1 2 (H) is satisfying. If an operator equation BX = C has a solution, then it is easy to see that thedistinguishedsolution ofdouglas theoremisgiven byb C. herefore, given B A (H), if we denote by A, the unique A-adjoint operator of whose range is included in R(A), then In view of heorem 1.1, A = A A. A A = A, R( A ) R(A) and N( A ) = N( A). Note that if S is an A-adjoint of then S = A +Z, with Z B(H) such that R(Z) N(A). Remark 1.2. Observe that if is A-selfadjoint, it does not mean in general that = A. In fact = A if and only if is A-selfadjoint and R() R(A). It is also clear that has a unique A-adjoint (namely A) if and only if A is injective. If this is the case, then we get the equality ( A) A =. In the following proposition we collect some properties of A. For its proof, see [1], [2] and [3]. Proposition 1.2. Let B A (H). hen the following statements hold (1) A B A (H),( A) A=P R(A) P R(A) and (( A) A) A= A. (2) If S B A (H), then S B A (H) and (S) A = S A A. (3) A and A are A-selfadjoint. (4) A = A A = A 1 2 A = A 1 2 A.

5 CLASS OF (A, n)-power QUASI-NORMAL (5) S A = A A for every S B(H) which is an A-adjoint of. (6) If S B A (H) then S A = S A. Nevertheless, A is not in general the unique A-adjoint of that realizes the minimal norm. In the following definition we collect the notions of some classes of A- operators ([1],[4],[16],[19]). Definition 1.2. Any operator B(H) is 1. A-contraction if ξ A ξ A for every ξ H, or equivalently if A A. 2. A-isometry if A = A ξ A = ξ A, ξ H. 3. A-normal if A = A ξ A = ξ A, ξ H. 4. A-partial isometry if ξ A = ξ A, ξ N(A) A. 5. A-unitary if for any ξ H, then A = A = A ξ A = ξ A = ξ A. 6. A-hyponormal if A A ξ A ξ A, ξ H. 7. A-quasi-isometry if and only if, A = 2 A 2 A = 2 A. 8. (A,m)-isometry if for every ξ H, then m m ( 1) k ( m k ) m k A m k =0 ( 1) k ( m k ) m k ξ 2 A=0. k=0 k=0 Remark 1.3. If B A (H), then m ( m ( is an (A,m) isometry ( 1) k) k ) A m k m k =0. k=0

6 66 S.H. Jah For m = 2, we get 2 A 2 2 A+A=0 ( A ) A +P R(A) =0 if Proposition 1.3. ([1]) If B A (H), then is an A-isometry if and only A = P R(A). Proposition 1.4. ([4] heorem 2.5) Let B A (H) with closed range. hen the following statements are equivalent 1. A =. 2. A is a projection (i.e., is an A-partial isometry). 2. (A, n)-power Quasi-Normal Operators As an extension of the classes of A-quasi-normal operators and A-normal operators ([16],[17]), the following definition describes the class of operators that we will study in this paper. Definition 2.1. An operator B A (H) is said to be (A,n)-power quasinormal operator for a positive integer n, if n A = A n. We denote the set of all (A,n)-power Quasi-normal operators by [nqn] A. his class includes the class of A-quasi-normal operator and A-normal operator. Remark 2.1. Clearly if n = 1, then (A,1)- power quasi-normal operator is precisely A- quasi-normal operator. In the following theorem, we collect some properties of the class [nqn] A. heorem 2.1. Let B A (H). If [nqn] A, then 1. is in the class [2nQN] A. 2. If has a dense range in H, then n A = A n. 3. If S is in theclass [nqn] A such that [, S] = [, S A] = [ A,S] = 0, then S is in the class [nqn] A.

7 CLASS OF (A, n)-power QUASI-NORMAL If S is in the class [nqn] A such that S = S = AS = S A = S A = S A = 0, then S + is in the class [nqn] A. Proof. 1. Since [nqn] A, then Multiplying (2.1) from left by n, we obtain hus is in the class [2nQN] A. n A = A n. (2.1) 2n A = A 2n. 2. Let be in the class [nqn] A, and η R() : η = ξ, ξ H. hen ( n A n )η = ( n A n )ξ = ( n A n+1 )ξ = 0. hus, Hence, ( n A n ) = 0 onr(). n A = A n onh. 3. (S) n (S) A S = n S n S A A S Hence, S is in the class [nqn]. = n A S n S A S = A n+1 S A S n+1 = (S) A (S) n (+S) n (+S) A (+S) = ( n +S n )( A +S A S) = n A +S n S A S = A n+1 +S A S n+1 = (+S) A (+S) n+1. Which implies that +S is in the class [nqn] A.

8 68 S.H. Jah Remark 2.2. It is clear that a (A, 2)-power quasi-normal operator is a (A, 2k)-power quasi-normal operator and a (A, 3)-power quasi-normal operator is a (A, 3k)-power quasi-normal operator. Lemma 2.1. Let B(H) such that A = A. hen is an (A,n)- power quasi-normal operator if and only if is a n-power quasi-normal. Proof. Note first that the conditions imposed on A and on imply that R(A) = H and that R(A) R(A). So A exists. Moreover hus, the assertion follows. A = A A = A A = P R(A) =. Proposition 2.1. Let B A (H), B = n + A and C = n. hen, we have 1. [nqn] A if and only if B commutes with C. 2. if [nqn] A, then M = n A commutes with B and C. 3. Assume that N(A) is an invariant subspace for. hen we have λ := λi [nqn] A, λ C is A normal. Proof. 1. We have BC=CB ( n + A )( n ) = ( n )( n + A ) 2n n A + A n ( A ) 2 = 2n + n A n ( A ) 2 n A = A n.

9 CLASS OF (A, n)-power QUASI-NORMAL Assume that is in the class [nqn]. hen MB = n A ( n + A ) = n A n + n ( A ) 2 = n n A + A n A = ( n + A ) n A = BM. he same steps can prove that M and C commute. 3. Assume that ( λi) is in the class [nqn] A for every λ C. hen for every λ C, we have ( λi) n ( λi) A ( λi) = ( λi) A ( λ)( λi) n. Hence, if we put a k = ( 1) k ( n k ), we obtain n a k λ k n k ( A λ A ) k=0 = ( A λ A ) n a k λ k n k. k=0 So, hus n 1 a k λ k ( n k A n k ) a k λ k+1 ( n k A n k ) = 0. n 1 n 1 a k λ k ( n k A n k ) n 2 a k λ k+1 ( n k A n k )

10 70 S.H. Jah ( 1) n nλ n ( A A )=0. Put λ = re iθ,0 θ 2π, r > 0, we get So, n 1 a k (re iθ ) k ( n k A n k ) n 2 a k (re iθ ) k+1 ( n k A n k ) ( 1) n n(re iθ ) n ( A A )=0. ( A A ) = ( 1)n n 1 n(re iθ ) n ( 1) k ( n k )(reiθ ) k ( n k A n k ) ( 1)n n 2 n(re iθ ) n ( 1) k ( n k )(reiθ ) k+1 ( n k A n k ). Letting r, we get A A = 0. Hence, is A-normal. Conversely it is known that an A- normal operators have translation invariant propertyi.e., if isa-normaloperator,then( λi)isa-normal for all λ C and hence ( λi) is in the class [nqn] A. Proposition 2.2. Let B A (H) with closed range. If [nqn] A such that is A-partial isometry, then [(n+1)qn] A. Proof. Since is A- partial isometry with closed range, then A = ( as in Proposition 1.4 ). (2.2) Multiplying (2.2) from left by A n+1 and using the fact that is in the class [nnq] A, we get A n+2 = A n+2 A = n A. A = n+1 A, which implies that is in the class [(n+1)qn] A.

11 CLASS OF (A, n)-power QUASI-NORMAL heorem 2.2. Let B A (H). 1. If [2QN] A [3QN] A, then [nqn] A, n If is in the class [nqn] A and in the class [(n + 1)QN] A, then it is in the class [(n+2)qn] A, that is [nqn] A [(n+1)qn] A [(n+2)qn] A. 3. If [nqn] A [(n+1)qn] A such that is injective, then is A-quasinormal. Proof. 1. We shall prove the assertion by induction. he case n = 4 is a consequence of heorem 2.1. Let us prove the assertion for n = 5. Since [2QN] A, then So by multiplying (2.3) from left by 3, we get hus, since [3QN] A, we have 2 A = A 3. (2.3) 5 A = 3 A 3. 5 A = 3 A 3 = A 4 2 = A 6. Now, assume that the result is true for an integer n 5, that is hen, n A = A n. n+1 A = A n+1 hus, is in the class [(n+1)qn] A. = A 3 n 2 = 3 A n 2 = A 4 (n 2) = A n+2.

12 72 S.H. Jah 2. Let [nqn] A [(n+1)qn] A. Since is in the class [nqn] A, then n A = A n. So, n+1 A = A n+1. Since is in the class [(n+1)qn] A, then A n+2 = n+2 A. Hence, is in the class [(n+2)qn] A. 3. We have n ( A 2 ) = 0. Since is injective, then A A 2 = 0. Hence, is A- quasi-normal. Proposition 2.3. Let B A (H)suchthatN(A) isaninvariant subspace under the action of. hen, we have the following properties: 1. If is in the class [2QN] A and is an (A,2)-isometry, then 2 is in the class [nqn] A for all positive integer n If [2QN] A such that is an (A,2)-isometry, then is an A-isometry. 3. If is in the class [2QN] A [3QN] A and is an (A,n)-isometry, then is an A- isometry. Proof. 1. From heorem 2.2, it suffices to prove that 2 is in the class [2QN] A and 2 is in the class [3QN] A. Since is in the class [2QN] A and is an (A,2)-isometry, we have 4 ( A2 2 ) = 4 (2 A P) Hence, 2 is in the class [2QN] A. = 2 4 A 4 P = 2 2 ( A 3 ) 4 P = 2 A 5 P 4 = (2 A P) 4 = A2 6.

13 CLASS OF (A, n)-power QUASI-NORMAL On the other hand, hus, 2 is in the class [3QN] A. 6 ( A2 2 ) = 6 (2 A P) = 2 6 A P 6 2. By the definition of (A, 2)-isometry, = 2 4 ( A 3 ) P 6 = 2 A 7 P 6 = (2 A P) 6 = ( A2 2 ) 6. ( A2 2 )( A ) 2( A ) 2 +P A = 0. Since is in the class [2QN] A, then A2 ( A ) 2 2( A ) 2 +P A = 0, that is A3 3 2( A ) 2 + A P = 0. (2.4) Also, we have A [ A2 2 2 A +P ] = 0 i.e. A3 3 2 A2 2 + A P = 0. (2.5) From (2.4) and (2.5) A2 2 = ( A) 2 and so, ( A ) 2 2( A )+P = A2 2 2 A +P = ( A P ) 2 = 0, or A = P. Hence, is an A-isometry by Proposition 1.3.

14 74 S.H. Jah 3. By the definition of (A, n)-isometry, An n A ( n 1) A n 1 n 1 A ( 1) n 2( n n 2 ) A 2 2 A + ( 1) n 1 ( n n 1 ) A A + ( 1) n P A = 0. (2.6) Since is in [2QN] A [3QN] A, we have by heorem 2.2 Also, we have that is An+1 n+1 ( n 1) A n n ( 1) n 2( n n 2 ) A ( 1) n( n n 1 ) ( A ) 2 A [ An n ( n 1) A n 1 n ( 1) n A P =0. (2.7) + ( 1) n 1( n n 1 ) A +( 1) n P ] = 0, An+1 n+1 ( n 1 ) An n ( 1) n 1( n n 1 ) A2 2 From (2.6) and (2.7), we get 2 A 2 = ( A) 2. Consequently + ( 1) n A P = 0. (2.8) hen, ( A ) k k =( A ) k, k N. ( A ) n ( n 1) ( A ) n ( 1) n 1( n n 1 ) ( A ) + ( 1) n P = 0 = (P ) n. his completes the proof.

15 CLASS OF (A, n)-power QUASI-NORMAL heorem 2.3. Let B A (H) such that N( A) N() and R(A) R(). hen the following properties hold: 1. If [nqn] A, then A n = n A and n is A normal. 2. If [2QN] A [3QN] A, then is A-normal. Proof. 1. Since [nqn] A, we have n A = A n. We deduce that ( n A n) = 0. It follows that By hypothesis, we obtain n A n = 0 on R(). n A n = 0 on R(A). On the other hand, since N( A) N(), we have n A n = 0 on N( A) = N ( (A) ). hen, the result follows from the identity 2. Since [2QN] A [3QN] A, then H = N ( (A) ) N ( (A) ). 2 A = A 2 and 3 A = A 3, so ( A ) 3 = 0. It follows that ( A ) 2 = 0 on R() and hence ( A A ) 2 = 0 on R(A). Since N( A) N(), we get ( A A ) 2 = 0 on N( A). Hence, ( A ) 2 = 0 on H. By repeating this process, we obtain A = 0.

16 76 S.H. Jah Proposition 2.4. Let B A (H)suchthatN(A) isaninvariant subspace under the action of, R(A) = R() and N(A( ki)) = N( ki) for k = 0,1. So, if and I are in the class class [2QN] A, then is A-normal. Proof. It is easy to see that the condition on I implies that 2 ( A ) 2 A 2( A )+2 A = ( A ) 2 2 2( A ) +2 A. Since is in the class [2QN] A, we have 2 A 2( A )+2 A = 2 2( A ) +2 A. As A, A are A-self-adjoint operators, R( A) R(A) and R( A) R(A), we have It follows that ( A ) A = A and ( A ) A = A. Let us now show that (2.9) implies A2 2( A ) A +2 A = 2 2 A ( A )+2 A (2.9) We suppose that Aξ = 0. hen from (2.9), we get So, N( A ) N(). (2.10) 3 A2 ξ +2 A ξ = 0. (2.11) 3 A3 ξ +2 A2 ξ = 0. herefore, as is in the class [2QN] A, then 3 A A2 ξ +2 A2 ξ = 0 and hence, 2 A2 ξ = 0.

17 CLASS OF (A, n)-power QUASI-NORMAL Consequently, (2.11) gives 2 Aξ = 0 or Aξ = 0, i.e., N( A) N(A) N(). hisproves (2.10). heorem 2.3 implies that 2 is A- normal. his together with (2.9) gives or ( A )+ A = ( A ) + A A ( A A ) = A A. (2.12) If N( A I) = {0}, then (2.12) implies that is A-normal. Now, assume that N( A I) is non trivial. Let Since A2 = A2, we get herefore, A2 ξ ξ = A ξ ξ. A ξ = ξ. ξ 2 A =< A ξ ξ > A =< ξ ξ > A =< ξ A ξ > A = ξ 2 A. Hence, ξ ξ 2 A = ξ 2 A + ξ 2 A 2Re < ξ ξ > A = ξ 2 A ξ 2 A = 0. But Aξ = Aξ. hen N( A I) N(A A) N( I). his together with (2.11), yields to and so, or ( A A ) = A A ( A A ) = ( A A ) A 2 2 A = A 2 A. Since 2 A = A 2 and 3 A = A, we deduce that A 2 = A. hus isa-quasi-normal i.e., [QN] A. Fromheorem2.3, thea-normality of follows. In attempt to extend the above result for operators in the class [nqn] A, we state the following result.

18 78 S.H. Jah heorem 2.4. Let B A (H) such that N(A) is an invariant subspace under the action of, R(A()) = R() and N(A( ki)) = N( ki) for k = 0,1. If is in the class [2QN] A [3QN] A such that I is in the class [nqn] A, then is A- normal. or Proof. Since I is in the class [nqn] A, we have n a k k A = A n a k k A n a k k n a k k, a k = ( 1) n k ( n k ). Under the condition on, we have by heorem 2.3 ( n a 1 ( A ) a k k) ( n A =a 1 ( A ) a k k) a 1 ( A ) A n a k Ak = a 1 A ( A ) ( which implies that N( A) N(). In fact, let Aξ = 0. From (2.13), we have n a k Ak ), (2.13) ( n a 1 A2 ξ a k k) ξ = 0. Since is in [2QN] A [3QN] A, we deduce that a 1 A2 ξ a 1 A ξ a 2 A2 ξ = 0. (2.14) hen Hence, a 1 A3 ξ a 1 A2 a 2 A3 ξ = 0. a 1 A2 ξ = 0.

19 CLASS OF (A, n)-power QUASI-NORMAL Consequently (2.14) gives Aξ = 0, which implies that Aξ = 0. It follows by heorem 2.3 that k is a A-normal for k = 2,3,...,n and hence, ( A ) A =( A ) or, Hence, A ( A ) = A. ( A I)( A ) = 0. A similar argument as in the proof of Proposition 2.4 gives the desired result. heorem 2.5. Let B A (H) such that ( [2QN] A [3QN] A ), 2 [2QN] A and N( 2 ) N( A2 ). If A is injective, then 2 is A-quasinormal. Proof. he condition 2 is in the class [2QN] A gives which implies that But is in the class [3QN] A, so Hence, A4 ( A2 2 ) = ( A2 2 ) A4, A5 ( A ) = ( A2 2 ) A4. A2 ( A ) A3 = ( A2 A 2) A4. A2 ( A ) 2 A2 = ( A2 2 ) A4 ( [2QN]A ). hen the facts that [2QN] A and ( A) A = give and Since N( 2 ) N( A2 ), then ( A ) 2 A4 = ( A2 2 ) A4 4 (( A ) ) = 0. A2 2 (( A ) ) = 0

20 80 S.H. Jah or Hence, 2 [( A ) )] = 0. (2.15) A2 [(( A ) )] = 0, ( N( 2 ) N( A2 ) ). Or [(( A ) )] 2 = 0. (2.16) Since is in the class [2QN] A, 2 commutes with ( A) 2, then the desired conclusion follows from (2.15) and (2.16). Hence, [(( A ) )] 2 = 0. (2.17) Proposition 2.5. ake an arbitrary nonnegative integer n. Let L A (H) such that is in the class (A,1)- power quasi-normal. hen (1) ( A) n = ( A) n. (2) ( A) n = An n. (3) ( An n ) = ( An n ). Proof. (1) Since A = A 2, then n ( A) n = ( A) n n. (2) ( A) n = A A... A }{{ } = ( A) n n. k times (3) ( A ) n = ( A ) n 1 ( A ) n = ( A ) n 1 n ( A ) = ( A ) n 2 n 1 ( A ) 2 2 = ( A ) n 3 n 2 ( A ) 3 3 =... = ( A ) n n. 3. ensor Product of (A, n)-power Quasi-Normal Operators Let H H denote the completion, endowed with a reasonable uniform crosenorm, of the algebraic tensor product H H of H with itself. Given non-zero

21 CLASS OF (A, n)-power QUASI-NORMAL operators,s B(H), let S B(H H) denote the tensor product defined on the Hilbert space H H as follows S(ξ 1 η 1 ) (ξ 2 η 2 ) = ξ 1 ξ 2 Sη 1 η 2. he operation of taking tensor products S preserves many properties of,s B(H), but by no means all of them. Whereas S is normal if and only if and S are normal, there exist paranormal operators and S such that S is not paranormal. It is proved that for non-zero,s B(H), S is p- hyponormalifandonlyif ands arep-hyponormal. heseresultsareextended to p-quasihyponormal operators for more details see [9],[10],[11],[12],[18] and the references therein. In the following proposition we will prove the stability of the class of (A,n)- power quasinormal operators under the direct sum and tensor product. Recall that ( S) ( S) = S S and by the uniqueness of positive square roots, S r = r S r for any positive rational number r. From the density of the rational set in the real set, we obtain S p = p S p for every positive real number p. Observe also that A B = (A I)(I B) = (I B)(A I). Proposition 3.1. Let A i L + (H) and i L Ai (H) such that i is in the class [nqn] Ai for i = 1,2,...,p, then we have the following properties p is in the class [nqn] ( A 1 A 2... A p ) p is in the class [nqn] ( A 1 A 2.. A p ). Proof. 1. ( ) ( 1... p ) n A ( 1... p ) 1... Ap ( 1... p ) = 1 n A n A p n Ap p = A n A n... Ap p p p n ( ) A = ( 1... p ) 1... Ap ( 1... p )( 1... p ) n. Hence, 1... p is in the class [nqn] ( A 1 A 2... A p ). p

22 82 S.H. Jah 2. Let x 1,x 2,...,x p H. hen ( ) ( 1... p ) n A ( 1... p ) 1... Ap ( 1... p )(x 1... x p ) = 1 n A x 1... p n Ap p p x p = A n x 1... Ap p p p n x p ( ) A = ( 1... p ) 1... Ap ( 1... p ) n+1 (x 1... x p ). References [1] M.L. Arias, G. Corach, M.C. Gonzalez, Partial isometries in semi- Hilbertian spaces, Linear Algebra Appl, 428, (7), (2008), DOI: /j.laa [2] M.L. Arias, G. Corach, M.C. Gonzalez, Metric properties of projections in semi- Hilbertian spaces, Integral Equations Operator heory, 62 (1), (2008). DOI: /s [3] M. L. Arias, G. Corach, M. C. Gonzalez, Lifting properties in operator ranges, Acta Sci. Math., (Szeged) 75:3-4, (2009), [4] M. L. Arias and M.Mbekhta, A-partial isometries and generalized inverses, Linear Algebra and its Applications, 439, (2013) [5] A. Brown, On a class of operators, Proc. Amer. Math. Soc, 4, (1953), [6] R.G. Douglas, On majorization, factorization and range inclusion of operators in Hilbert space, Proc. Amer. Math. Soc, 17, (1966), [7] P. A. Fillmore, J. P. Williams, On operator ranges, Adv. Math., 7, (1971), [8] M.C. Gonzalez, Operator norm inequalities in semi-hilbertian spaces, Linear Algebra and its Applications, 434, (2011), doi: /j.laa [9] J. C. Hou, On the tensor products of operators, Acta Math. Sinica (N.S.), 9(2), (1993),

23 CLASS OF (A, n)-power QUASI-NORMAL [10] I. H. Kim, On (p, k)-quasihyponormal operators, Math. Ineq. and Appl., 7, (2004), [11] C. S. Kubrusly, ensor product of proper contractions, stable and posinormal operators, Publicationes Mathematicae Debrecen, 71, (2007), [12] C. S. Kubrusly and N. Levan, Preservation of tensor sum and tensor product, Acta Math Univ. Comenianae, Vol. LXXX, 1, (2011), [13] W. Majdak, N.-A. Secelean and L. Suciu, Ergodic properties of operators, Linear and Multilinear Algebra, 61(2), 2013, DOI: / [14] Ould Ahmed Mahmoud Sid Ahmed, On the class of n-power quasi-normal operators on Hilbert spaces, Bull. Math. Anal. Appl., 3(2), (2011), [15] Ould Ahmed Mahmoud Sid Ahmed, On Some Normality-Like Properties and Bishops Property (β) for a Class of Operators on Hilbert Spaces Spaces, International Journal of Mathematics and Mathematical Sciences, (2012). doi: /2012/ [16] Ould Ahmed Mahmoud Sid Ahmed and A. Saddi, A-m-Isomertic operators in semi-hilbertian spaces, Linear Algebra and its Applications, 436, (2012), doi: /j.laa [17] S. Panayappan and N. Sivamani, A-Quasi Normal Operators in Semi Hilbertian Spaces, Gen. Math. Notes, 10(2), (2012), [18] J. Stochel, Semi-normality of operators from their tensor product, Proc. Amer. Math., [19] L. Suciu, Quasi-isometries in semi-hilbertian spaces, Linear Algebra and its Applications, 430, (2009) doi.org/ /j.laa

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