GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS

Transcription

1 International Journal of Analysis Alications ISSN Volume 5, Number (04), -9 htt:// GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS ILYAS ALI, HU YANG, ABDUL SHAKOOR Abstract. In this article, we generalize some norms inequalities for sums, differences, roducts of absolute value oerators. Our results based on Minkowski tye inequalities generalized forms of the Cauchy-Schwarz inequality. Some other related inequalities are also discussed.. Introduction In this article, notations are same as in [3], for reader convenience we recall that let H be a comlex searable Hilbert sace B(H) denote the C -algebra of all bounded linear oerators on H. Let A denote the absolute value of A B(H), is defined as A = (A A), where A is the adjoint oerator of A. If A is comact oerator on comlex searable Hilbert sace H, then the singular values of A enumerated as s (A) s (A)... which are the eigenvalues of ositive oerator A. A norm. st for untarily invariant norm i.e., a norm with the roerty that UAV = A for all A for all unitary oerators U, V in B(H). Oerator norm Schatten -norms are denoted as.. P resectively. Excet the oerator norm, which is defined on all of B(H), each unitarily invariant norm is defined on an ideal in B(H). When we use the symbol A it is imlicit understood that oerator A is in this ideal. For 0 < <, a norm. defines a quasi-norm. For this norm it is well-known that (.) A + B ( A + B ). By the definition of the Schatten -norm, we have (.) A r A r, where r, are real numbers. Also, since the singular values of A r A r are same, so (.3) A r = A r. The unitarily invariant norms for differences of the absolute values of Hilbert sace oerators have attracted the attention of several mathematicians. It has been 000 Mathematics Subject Classification. 47A30; 47A63; 47B0. Key words hrases. Unitarily invariant norm; Schatten -norm; Cauchy-Schwarz inequality; Minkowski inequality; Absolute value oerators. c 04 Authors retain the coyrights of their aers, all oen access articles are distributed under the terms of the Creative Commons Attribution License.

2 ALI, YANG AND SHAKOOR roved by K. Shebrawi H. Albadawi in [3] that if A i, B i, X i (i =,,..., n) be oerators in B(H) such that X i is self adjoint oerator 0 < r. Then (.4) n r (A i X i A i ) r (B i X i B i ) r, which leads to the following inequality (.5) (.6) A r B r r A + B r A B r. Inequality (.5) generalize the result resented by Bhatia in [5] as follows: A B A + B A B. K. Shebrawi H. Albadawi also roved in [3] that if A, B, X be oerators in B(H) such that X is self adjoint oerator 0 < r,, then (.7) A XB + B XA r this leads to the following inequality r+ (A X A) r (B X B) r, (.8) A r B r r+ A + B r A B r. Inequality (.8) generalize the following result in [5] (.9) A B A + B A B, where. This article we have organized as: In Section, we generalize the inequality (.5) also we discuss some other related results. In Section 3, we resent some Schatten -norms inequalities, one of which generalize the inequality (.8).. Generalized unitarily invariant norms inequalities for absolute value oerators In this section, we generalize some unitarily invariant norms inequalities for absolute value oerators. Our results based on several lemmas. First two lemmas contain norm inequalities of Minkowski tye generalized forms of the Cauchy- Schwarz inequality, see [4] [] resectively. Lemma.. Let A i, B i B(H), i =,,..., n. Then n n ) n r A i + B i r (.) r r ( A i r r + B i r r for 0 < r, n n (.) n r A i + B i r r A r i r + B i r r

3 GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 3 for r, n ( )/r A i + B i r r A r i r + B i r r (.3) for, r <. Lemma.. For A, B, X B(H), for all unitarily invariant norms for all ositive real numbers, r such that + =, we have (.4) A XB r AA X r XBB r, also, if f g are nonnegative continuous functions on [0, ) satisfying f(t)g(t) = A XB r ( A f ( X )A ) r ( B g ( X )B ) r (.5) For following two lemmas see [] [6,. 93, 94]. Lemma.3. Let A be a ositive oerator in B(H). Then for every normalized unitarily invariant norm (i.e., diag(, 0, 0,..., 0) = ), we have. (.6) for 0 r (.7) for r. A r A r A r A r Lemma.4. Let A B be a ositive oerator in B(H). Then (.8) for 0 r (.9) for r. A r B r A B r A B r A r B r Last lemma is a consequence of the concavity (convexity) of the function f(t) = t r, 0 r (r ). Lemma.5. Let a b be two ositive real numbers (.0) for 0 r (.) for r. (a + b) r a r + b r (a + b) r r (a r + b r ) Theorem.. Let A i, B i, X i (i =,,..., n) be oerators in B(H) such that X i is self adjoint oerator if, are ositive real numbers, such that + =

4 4 ALI, YANG AND SHAKOOR 0 < r. Then (.) n r A i A i X i r X i B i B i r, also, if f g are nonnegative continuous functions on [0, ) satisfying f(t)g(t) = (.3) n r ( A i f ( X i )A i ) r ( B i g ( X i )B i ) r. Proof. By alying (.), the triangle inequality, (.3) (.4), resectively, we obtain r r n r ( n ) A i X i B i r r + Bi X i A i r r ( n r n r A i X i B i r r n r r n r ( n A i X i B i r ( n ) r ) r ( n + Bi X i A i r ) A i A i X i r X i B i Bi r r. ) r The roof is comleted. By alying (.5) the roof of the first art of Theorem., we can obtain (.3). Relace A i, B i by A i + B i, A i B i resectively also take f(t) = g(t) = t in Theorem., then, we obtain the following result. Corollary.. Let A i, B i, X i (i =,,..., n) be oerators in B(H) such that X i is self adjoint oerator if, are ositive real numbers, such that + = 0 < r. Then r n r A i X i A i Bi X i B i r (A i + B i )(A i + B i ) X i r X i (A i B i )(A i B i ) r,

5 GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 5 r n r A i X i A i Bi X i B i r ((A i + B i ) X i (A i + B i )) r ((A i B i ) X i (A i B i )) r. By similar way alying to the roof of theorem., based on the inequality (.), we can obtain the following result. Theorem.. Let A i, B i, X i (i =,,..., n) be oerators in B(H) such that X i is self adjoint oerator if, are ositive real numbers, such that + = r. Then (.4) r n r n A i A i X i r X i B i B i r, also, if f g are nonnegative continuous functions on [0, ) satisfying f(t)g(t) = (.5) A i X i B i + Bi X i A i r r n r n ( A i f ( X i )A i ) r ( B i g ( X i )B i ) r. Corollary.. Let A i, B i, X i (i =,,..., n) be oerators in B(H) such that X i is self adjoint oerator if, are ositive real numbers, such that + = r. Then n r A i X i A i Bi X i B i r (A i + B i )(A i + B i ) X i r X i (A i B i )(A i B i ) r, n r A i X i A i Bi X i B i r ((A i + B i ) X i (A i + B i )) r ((A i B i ) X i (A i B i )) r. Remark.. If we take f(t) = t α g(t) = t ( α) for α [0, ], then from the inequality (.3) we can obtain imortant secial case. Also, if we take f(t) =

6 6 ALI, YANG AND SHAKOOR g(t) = t then from (.3) we have n r (A i X i A i ) r (B i X i B i ) r, which is the more general form of the inequality (.4). Similar remark we can give for the inequality (.5), which is more general form of the inequality (.9) in [3]. Our following result contains a romised generalization of (.5). Corollary.3. Let A B be oerators in B(H), are ositive real numbers such that + = then A r B r r A + B r A B r (.6) for 0 < r, (.7) A A B B r A + B r A B r for r. Proof. By (.8) from second result of Corollary (.), we have A r B r A B r r A + B r A B r, for 0 < r. inequality (.7) is a secial case of the first result of Corollary (.). Remark.. Secial cases of second results in Corollary.. resectively are: Let A, B, X be oerators in B(H) such that X is self adjoint oerator if, are ositive real numbers such that + =, then A XA B XB r r ((A + B) X (A + B)) r ((A B) X (A B)) r. for 0 < r, for r. A XA B XB r ((A + B) X (A + B)) r ((A B) X (A B)) r. Corollary.4. Let A be an oerator in B(H) if, are ositive real numbers such that + =, then A A AA r +r ReA r ImA r, for 0 < r, for r. A A AA r r ReA r ImA r,

7 GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 7 3. Generalized norm inequalities for the Schatten -norm Schatten -norm for absolute value oerators are discussed in this section. Our these results refine some of the results in Section also, our first result leads to a generalization of (.8). Theorem 3.. Let A, B, X be oerators in B(H) such that X is self adjoint oerator if, are ositive real numbers such that + =, then (3.) A XB + B XA r r+ AA X r XBB r, also, if f g are nonnegative continuous functions on [0, ) satisfying f(t)g(t) = (3.) A XB + B XA r r+ ( A f ( X )A ) r ( B g ( X )B ) r. for 0 < r. Proof. By alying (.), (.), (.0), (.3) (.4) resectively, we obtain A XB + B XA r = A XB + B XA r r ( r ( A XB r + B XA r ) ) r r ( A XB r r + B XA r ) r r ( A XB r + B XA r ) = r+ A XB r r+ AA X r XBB r. The roof is comleted. By alying (.5) the roof of the first art of Theorem 3., we can obtain (3.). Corollary 3.. Let A, B, X be oerators in B(H) such that X is self adjoint oerator if, are ositive real numbers such that + =, then A XA B XB r r+ (A + B)(A + B) X r X(A B)(A B) r, A XA B XB r r+ ((A + B) X (A + B)) r for 0 < r. ((A B) X (A B)) r, Remark 3.. By using (.8) from second inequality in Corollary (3.), we can obtain A r + B r A B r r+ A + B r A B r,

8 8 ALI, YANG AND SHAKOOR which is the generalized form of the inequality (.8). Similarly to the roof of Theorem 3., based on (.), we can obtain the following result. Theorem 3.. Let A, B, X be oerators in B(H) such that X is self adjoint oerator if, are ositive real numbers such that + =, then (3.3) A XB + B XA r r AA X r XBB r, also, if f g are nonnegative continuous functions on [0, ) satisfying f(t)g(t) = (3.4) for r. A XB + B XA r r ( A f ( X )A ) r ( B g ( X )B ) r, Corollary 3.. Let A, B, X be oerators in B(H) such that X is self adjoint oerator if, are ositive real numbers such that + =, then A XA B XB r (A + B)(A + B) X r X(A B)(A B) r, for r A XA B XB r ((A + B) X (A + B)) r. ((A B) X (A B)) r, Similarly to the roof of Theorem 3., based on (.3), we can also obtain the following result. Theorem 3.3. Let A i, B i, X i (i =,,..., n) be oerators in B(H) such that X i is self adjoint oerator if, are ositive real numbers such that + =, Then A i X i B i + Bi X i A i r (3.5) r n A i A i X i r X i B i B i r, also, if f g are nonnegative continuous functions on [0, ) satisfying f(t)g(t) = A i X i B i + Bi X i A i r (3.6) r n ( A i f ) ( X i )A r i ( B i g ( X i )B i ) r,

9 GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS 9 for r,. Corollary 3.3. Let A i, B i, X i (i =,,..., n) be oerators in B(H) such that X i is self adjoint oerator if, are ositive real numbers such that + =, then n A i X i A i Bi X i B i r (A i + B i )(A i + B i ) X i r n A i X i A i Bi X i B i r ((A i + B i ) X i (A i + B i )) r for r,. X i (A i B i )(A i B i ) r, ((A i B i ) X i (A i B i )) r, Remark 3.. For r r or r (4 r), the results in Corollary 3.3 refine the results in Corollary. resectively. Acknowledgements The authors thank the referees for their careful reading of the manuscrit. This work was suorted by the National Natural Science Foundation of China (No. 736). References [] Hiai, F, Zhan, X: Inequalities involving unitarily invariant norms oerator monotone functions. Linear Algebra Al. 34, 5-69 (00). [] Albadawi, H: Holder-tye inequalities involving unitarily invariant norms. Positivity. 6, (0). [3] Shebrawi, K, Albadawi, H: Norm inequalities for the absolute value of Hilbert sace oerators. Linear Multilinear Algebra. 58, (00). [4] Shebrawi, K, Albadawi, H: Oerator norm inequalities of Minkowski tye. J. Ineq. Pure Al. Math. 9, Issue, Article 6, (008). [5] Bhatia, R: Perturbation inequalities for the absolute value ma in norm ideals of oerators. J. Oer. Theory. 9, 9-36 (988). [6] Bhatia, R: Matrix Analysis, Sringer-Verlag, New York, 997. College of Mathematics Statistics, Chongqing University, Chongqing 4033, P.R.China Corresonding author