Norm Estimates for Norm-Attainable Elementary Operators

Transcription

1 International Journal of Mathematical Analysis Vol. 12, 2018, no. 3, HIKARI Ltd, Norm Estimates for Norm-Attainable Elementary Operators J. N. Kinyanjui Jaramogi Oginga Odinga University of Science and Technology P.O. Box Bondo, Kenya N. B. Okelo Jaramogi Oginga Odinga University of Science and Technology P.O. Box Bondo, Kenya O. Ongati Jaramogi Oginga Odinga University of Science and Technology P.O. Box Bondo, Kenya S. W. Musundi Chuka University P.O. Box Chuka, Kenya Abstract In this paper we estimates the norms of norm-attainable elementary operators: inner derivations, generalized derivation, basic elementary operator and Jordan elementary operator. Keywords: Norm, norm-attainability, nonseparable Hilbert spaces 1 Introduction Let H be an infinite dimensional complex nonseparable Hilbert space and EN A(H) be the set of all norm-attainable elementary operators. Let T : NA(H) NA(H) be defined by T (X) = n i=1 P i XQ i for all X NA(H)

2 138 J. N. Kinyanjui, N. B. Okelo, O. Ongati and S. W. Musundi where P i, Q i are fixed in NA(H). We have the following examples of elementary operators. (i) the inner derivations δ P = P X XP (ii)the generalized derivation δ P,Q = P X XQ (iii)the basic elementary operator M P,Q = P XQ (iv)the Jordan elementary operator U P,Q = P XQ + QXP. An operator A B(H) is said to be norm-attainable if there exists a unit vector x H such that Ax = A. Definition 1.1 A normed space in which every cauchy sequence is convergent is called a complete normed space or Banach space. Definition 1.2 A norm or length function on a vector space X is a nonnegative real valued functions : X R(real number) satisfying the following axioms: (i). x 0 x X. (ii). x = 0 if and only if x = 0 x X. (iii). λx = λ x x X and λ is scalar. (iv). x + y x + y x, y X. Definition 1.3 A normed space in which every cauchy sequence is convergent is called a complete normed space or Banach space. Theorem 1.4 A Banach space is a Hilbert space if and only if its norm satisfies the parallelogram law. Definition 1.5 An operator A B(H) is said to be norm-attainable if there exists a unit vector x H such that Ax = A. Lemma 1.6 For an operator T B(H), the operator T is norm-attainable if and only if its adjoint T is norm-attainable. Corollary 1.7 If the elementary operator T Ai B i is norm-attainable then there exists an isometry or co-isometry V 0 such that T Ai B i (X) = n i=1 A i V 0 B i. Definition 1.8 For any vectors y, z H, the rank one operator y z B(H), is defined by (y z)x = x, z y, for all x H. Definition 1.9 Let φ H and ξ H. We define φ ξ B(H) by (φ ξ)η = φ(η)ξ, for all η H. Definition 1.10 A generalized derivation δ A,B on a C -algebra Ω, is said to be norm-attainable if there exists a function ϕ H such that δ A,B ϕ = δ A,B.

3 Norm estimates for norm-attainable elementary operators Results and Discussion Lemma 2.1 Let H be an infinite dimensional complex nonseparable Hilbert space and E[N A(H)] be the set of all norm-attainable elementary operators. Let δ P E[NA(H)] and X H be defined by δ P = P X XP then, δ P (X) E[NA(H)] = 2 P. Let {x n } be a unit sequence in H such that x n = 1 then it implies that P x n P. Set P x n = α n x n + β n y n, where x n, y n = 0 and y n = 1. Set W n x n = x n and W n y n = y n. Then (P W n W n P )x n 2 = P W n x n W n P x n 2 = P x n W n (α n x n + β n y n ) 2 This means that = P x n W n α n x n W n β n y n 2 = P x n α n x n + β n y n 2 = β n y n + β n y n 2 = 4 β n y n 2 (P W n W n P )x n 2 = 4 β n y n 2 (1) Taking the positive square root of Equation 1 on both sides we get This means that From the Equation 2 we have (P V n V n P )x n = 2 β n y n. (P V n V n P )x n = 2 β n y n = 2 P xn α n x n (2) (P V n V n P )x n 2( P x n α n x n ) Allowing α n 0 as n we get, which implies that Also from the Equation 2 we have (P V n V n P )x n 2 P. δ P (X) E[NA(H)] 2 P. (3) (P V n V n P )x n 2( P x n + α n x n ) Allowing α n 0 as n we get, which implies that (P V n V n P )x n 2 P. δ P (X) E[NA(H)] 2 P. (4) From the Inequality 3 and Inequality 4 we have, δ P (X) E[NA(H)] = 2 P

4 140 J. N. Kinyanjui, N. B. Okelo, O. Ongati and S. W. Musundi Theorem 2.2 Let H be an infinite dimensional complex nonseparable Hilbert space and E[NA(H)] be the set of all norm-attainable operators. Let δ P,Q E[NA(H)] and X H be defined by δ P,Q = P X XQ then, δ P,Q (X) E[NA(H)] = P Q. Let x n be a unit sequence in H such that x n = 1 then it implies that P x n P and Qx n Q. Let Qx n = α n x n + β n y n, where x n, y n = 0 and x n = 1. Also let W n x n = Qx n and W n y n = P y n. Then (P W n W n Q)x n 2 = P W n x n W n Qx n 2 = P Qx n W n (α n x n + β n y n ) 2 This means that = P Qx n W n α n x n W n β n y n 2 = P Qx n α n Qx n + P β n y n 2 = (P Qβ n + P β n y n ) α n Qx n 2. (P W n W n Q)x n 2 = (P Qβ n + P β n y n ) α n Qx n 2. (5) Thus from the Equation 5 (P W n W n Q)x n 2 (P Qβ n + P β n y n ) 2 α n 2 Qx n 2 Allowing α n 0 and β n 0 as n we get, (P V n V n Q)x n 2 P Qx n 2 P 2 Qx n 2 P 2 Q 2 Taking the positive square root on both sides we get, which implies that, (P V n V n Q)x n P Q, Also from the Equation 5, we have that, δ P,Q (X) E[NA(H)] P Q (6) (P W n W n Q)X n 2 (P Qβ n + P β n y n ) 2 + α n Qx n 2 (P Qβ n + P β n y n ) 2 + α n 2 Qx n 2 Allowing α n 0 and β n 0 as n we get, (P V n V n Q)x n 2 P Qx n 2 P 2 Qx n 2 P 2 Q 2 Taking the positive square root on both sides we get, (P V n V n Q)x n P Q

5 Norm estimates for norm-attainable elementary operators 141 which implies that, δ P,Q (X) E[NA(H)] P Q (7) From the Inequality 6 and Inequality 7 we have, δ P,Q (X) E[NA(H)] = P Q. Theorem 2.3 Let H be an infinite dimensional complex nonseparable Hilbert space and E[NA(H)] be the set of all norm-attainable operators. Let M P,Q E[NA(H)] and X H be defined by M P,Q = P XQ then, M P,Q (X) = P Q. By the definition of Supremum norm we have, { } P XQ M P,Q E[NA(H)] = sup : X NA(H), X = 1 X { } P X Q sup : X NA(H), X = 1 X sup{ P Q } P Q This means that, M P,Q ENA(H) P Q. (8) Also by the definition of the Supremum norm where M P,Q E[NA(H)] = sup{ M P Q (R) E[NA(H)] : R E[NA(H)], R = 1}, we have M P,Q E[NA(H)] M P,Q (R) E[NA(H)] : R E[NA(H)], R = 1 (9) By Equation 9 and so letting P = φ ξ 1, ξ 1 H and ξ 1 = 1, and we have Q = ϕ ξ 2, ξ 2 H and ξ 2 = 1, M P,Q E[NA(H)] M P,Q (R) E[NA(H)], R E[NA(H)], R = 1 P RQ (φ ξ 1 Rϕ ξ 2 )η φ ξ 1 Rϕ(η)ξ 2 φ ξ 1 ϕ(η)rξ 2 ϕ(η) φ ξ 1 Rξ 2 ϕ(η) φr(ξ 2 )ξ 1 ϕ(η) φr(ξ 2 ) ξ 1 ϕ(η) φr(ξ 2 ) P Q

6 142 J. N. Kinyanjui, N. B. Okelo, O. Ongati and S. W. Musundi and thus we have M P,Q E[NA(H)] P Q. (10) From the Inequality 8 and Inequality 10 we get M P,Q E[NA(H)] = P Q. Theorem 2.4 Let H be an infinite dimensional complex nonseparable Hilbert space and EN A(H) be the set of all norm-attainable elementary operators. Let U P,Q = P XQ + QXP be Jordan elementary operator where P, Q ENA(H) then, U P,Q E[NA(H)] 2 P Q. Let {x n } and {y n } be two unit vectors in H such that, also lim P Qx n, x n = Q P and n lim P n Q y n, y n = Q P and lim Qx n = Q, n lim n Q y n = Q. But by Cauchy- Schwarz-Buniakowski inequality and the fact that P and Q are self adjoint, we have P Qx n, x n = Qx n, P x n Qy n P x n, also P Q y n, y n = Q y n, P y n Q y n P y n. This implies that lim n P x n = P and lim n P x n = P. Then for n 1 we have, U P,Q (x n y n )Q y n E[NA(H)] 2 = (P (x n y n )Q + Q(x n y n )P )Q 2 = P (x n y n )QQ y n + Q(x n y n )P Q y n 2 = QQ y n, y n P x n + P Q y n, y n Qx n 2 = Q y n, Q y n P x n + P Q y n, y n Qx n 2 = ( Q y n 2 P x n + P Q y n, y n Qx n ) 2 = Q y n 2 P x n + P Q y n, y n Qx n, Q y n 2 P x n + P Q y n, y n Qx n = Q y n 2 P x n, Q y n 2 P x n + Q y n 2 P x n, P Q y n, y n Qx n + P Q y n, y n Qx n, Q y n 2 P x n + P Q y n, y n Qx n, P Q y n, y n Qxn = Q y n 2 P x n 2 + Q y n 2 P x n, P Q y n, y n Qx n + Q y n 2 P Q y n, y n Qx n, P x n + P Q y n, y n 2 Q y n 2 = Q y n 4 P x n 2 + Q y n 2 2Re P Q y n, y n Qx n, P x n + P Q y n, y n 2 Q y n 2

7 Norm estimates for norm-attainable elementary operators 143 = Q y n 4 P x n 2 + Q y n 2 2Re P Q y n, y n P Qx n, x n + P Q y n, y n 2 Q y n 2 = Q y n 4 P x n 2 + Q y n 2 2Re P Q y n, y n P Qx n, x n + P Q y n, y n 2 Q y n 2 But we have U P,Q (x n y n )Q y n E[NA(H)] U P,Q (x n y n ) E[NA(H)] Q y n This implies that (11) U P,Q E[NA(H)] (x n y n ) Q y n U P,Q E[NA(H)] Q y n. U P,Q (x n y n )Q y n E[NA(H)] 2 U P,Q 2 Q y n E[NA(H)] 2 and therefore substituting in the Equation 11 we get U P,Q E[NA(H)] 2 Q y n 2 Q y n 4 P x n 2 + Q y n 2 2Re P Q y n, y n P Qx n, x n Taking limits as n we get, + P Q y n, y n 2 Q y n 2 U P,Q E[NA(H)] 2 lim n Q y n 2 lim n Q y n 4 lim n 2 + lim n n Q y n 2 2Re lim n Q y n, y n lim Qx n, x n n and therefore we get + lim P n Q y n, y n 2 lim n Q y n 2. U P,Q E[NA(H)] 2 Q 2 Q 4 P 2 + Q 2 (2 Q P )(2 Q P ) we get, + (2 Q P ) 2 Q 2 Q 4 P Q 4 P 2 + P 2 Q 4 4 Q 4 P 2. U P,Q E[NA(H)] 2 4 Q 2 P 2. (12) Getting the positive square root of Inequality 12 on both sides we get, U P,Q E[NA(H)] 2 Q P. Acknowledgements. The first author s appreciations go to National Research Fund (NRF ) for financial support towards this research.

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