A Diaz-Metcalf type inequality for positive linear maps and its applications

Transcription

1 Electronic Journal of Linear Algebra Volume Volume 0) Article 0 A Diaz-Metcalf type inequality for positive linear maps and its applications Mohammad Sal Moslehian Ritsuo Nakamoto Yuki Seo Follow this and additional works at: Recommended Citation Moslehian, Mohammad Sal; Nakamoto, Ritsuo; and Seo, Yuki 0), "A Diaz-Metcalf type inequality for positive linear maps and its applications", Electronic Journal of Linear Algebra, Volume DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository For more information, please contact scholcom@uwyoedu

2 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 A DIAZ METCALF TYPE INEQUALITY FOR POSITIVE LINEAR MAPS AND ITS APPLICATIONS MOHAMMAD SAL MOSLEHIAN, RITSUO NAKAMOTO, AND YUKI SEO Abstract We present a Diaz Metcalf type operator inequality as a reverse Cauchy Schwarz inequality and then apply it to get some operator versions of Pólya Szegö s, Greub Rheinboldt s, Kantorovich s, Shisha Mond s, Schweitzer s, Cassels and Klamkin McLenaghan s inequalities via a unified approach We also give some operator Grüss type inequalities and an operator Ozeki Izumino Mori Seo type inequality Several applications are included as well Key words Diaz Metcalf type inequality, Reverse Cauchy Schwarz inequality, Positive map, Ozeki Izumino Mori Seo inequality, Operator inequality AMS subject classifications 46L08, 6D5, 46L05, 47A30, 47A63 Introduction The Cauchy Schwarz inequality plays an essential role in mathematical analysis and its applications In a semi-inner product space H,, ) the Cauchy Schwarz inequality reads as follows x,y x,x / y,y / x,y H ) There are interesting generalizations of the Cauchy Schwarz inequality in various frameworks, eg, finite sums, integrals, isotone functionals, inner product spaces, C - algebras and Hilbert C -modules; see [5, 6, 7, 9,, 3, 7, 0] and references therein There are several reverses of the Cauchy Schwarz inequality in the literature: Diaz Metcalf s, Pólya Szegö s, Greub Rheinboldt s, Kantorovich s, Shisha Mond s, Ozeki Izumino Mori Seo s, Schweitzer s, Cassels and Klamkin McLenaghan s inequalities Inspired by the work of JB Diaz and FT Metcalf [4], we present several reverse Cauchy Schwarz type inequalities for positive linear maps We give a unified treatment of some reverse inequalities of the classical Cauchy Schwarz type for positive Received by the editors on September 9, 00 Accepted for publication on February 9, 0 Handling Editor: Bit-Shun Tam Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures CEAAS), Ferdowsi University of Mashhad, PO Box 59, Mashhad 9775, Iran moslehian@ ferdowsiumacir, moslehian@amsorg) Supported by a grant from Ferdowsi University of Mashhad No MP8944MOS) Faculty of Engineering, Ibaraki University, Hitachi, Ibaraki , Japan r-naka@net jwaynejp) Faculty of Engineering, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitamacity, Saitama , Japan yukis@sicshibaura-itacjp) 79

3 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 80 MS Moslehian, R Nakamoto, and Y Seo linear maps Throughout the paper BH ) stands for the algebra of all bounded linear operators acting on a Hilbert space H We simply denote by α the scalar multiple αi of the identity operator I BH ) For self-adjoint operators A, B the partially ordered relation B A means that Bξ,ξ Aξ,ξ for all ξ H In particular, if 0 A, then A is called positive If A is a positive invertible operator, then we write 0 < A A linear map Φ : A B between C -algebras is said to be positive if ΦA) is positive whenever A is We say that Φ is unital if Φ preserves the identity The reader is referred to [9, 9] for undefined notations and terminologies Operator Diaz Metcalf type inequality We start this section with our main result Recall that the geometric operator mean A B for positive operators A,B BH ) is defined by if 0 < A A B = A ) A BA A Theorem Let A, B BH ) be positive invertible operators and Φ : BH ) BK ) be a positive linear map i) If m A B M A for some positive real numbers m < M, then the following inequalities hold: Operator Diaz Metcalf inequality of first type Operator Cassels inequality MmΦA) + ΦB) M + m)φa B); ΦA) ΦB) M + m Mm ΦA B); Operator Klamkin McLenaghan inequality ΦA B) ΦB)ΦA B) ΦA B) ΦA) ΦA B) M m) ; Operator Kantorovich inequality ΦA) ΦA ) M + m Mm ii) If m A M and m B M for some positive real numbers m < M and m < M, then the following inequalities hold:

4 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 A Diaz Metcalf Type Inequality for Positive Linear Maps 8 Operator Diaz Metcalf inequality of second type M m M m ΦA) + ΦB) M m + m M ) ΦA B); Operator Pólya Szegö inequality ΦA) ΦB) ) M M m m + ΦA B); m m M M Operator Shisha Mond inequality ΦA B) ΦB)ΦA B) ΦA B) ΦA) ΦA B) M m Operator Grüss type inequality ΦA) ΦB) ΦA B) M M M M m m ) m m min m M ) ; { M, M } m m Proof i) If m A B M A for some positive real numbers m < M, then m A BA M ii) If m A M and m B M for some positive real numbers m < M and m < M, then m = m M A BA M m = M ) In any case we then have ) ) M A / ) ) BA A / BA m 0, whence Hence ) Mm + A BA M + m) A BA ) MmA + B M + m)a / A BA A / = M + m)a B ) Since Φ is a positive linear map, ) yields the operator Diaz Metcalf inequality of first type as follows: MmΦA) + ΦB) M + m)φa B) 3)

5 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 8 MS Moslehian, R Nakamoto, and Y Seo In the case when ii) holds we get the following, which is called the operator Diaz Metcalf inequality of second type: M m M ΦA) + ΦB) + m ) ΦA B) M m m M Following the strategy of [], we apply the operator geometric arithmetic inequality to MmΦA) and ΦB) to get: MmΦA) ΦB)) = MmΦA)) ΦB) MmΦA) + ΦB)) 4) It follows from 3) and 4) that ΦA) ΦB) M + m Mm ΦA B), which is said to be the operator Cassels inequality under the assumption i); see also [6] Under the case ii) we can represent it as the following inequality being called the operator Pólya Szegö inequality or the operator Greub Rheinboldt inequality: ΦA) ΦB) ) M M m m + ΦA B) 5) m m M M It follows from 5) that ) ) M M m m ΦA) ΦB) ΦA B) + ΦA B) m m M M M M ) m m It follows from ) that = m m M M ΦA B) 6) so m ) A A / A / BA A / M A, M m Now, 6) and 7) yield that ΦA) ΦB) ΦA B) m m M A B M M m 7) M M m m ) m m M M M M m An easy symmetric argument then follows that M M M M ) m m ΦA) ΦB) ΦA B) m m min { M, M }, m m

6 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 presenting a Grüss type inequality A Diaz Metcalf Type Inequality for Positive Linear Maps 83 If A is invertible and Φ is unital and m = m A M = M, then by putting m = /M B = A /m = M in 5) we get the following operator Kantorovich inequality: It follows from 3) that ΦA) ΦA ) M + m Mm ΦA B) ΦB)ΦA B) ΦA B) ΦA) ΦA B) M + m MmΦA B) ΦA)ΦA B) ΦA B) ΦA) ΦA B) M + m ) Mm Mm ΦA B) / ΦA)ΦA B) ) / ) ΦA B) ΦA) ΦA B) M m), 8) that is, an operator Klakmin Mclenaghan inequality when i) holds Under ii), we get the following operator Shisha Szegö inequality from 8): ) ΦA B) ΦB)ΦA B) ΦA B) ΦA) ΦA B) M m m M 3 Applications If a,,a n ) and b,,b n ) are n-tuples of real numbers with 0 < m a i M i n),0 < m b i M i n), we can consider the positive linear map ΦT) = Tx,x on BC n ) = M n C) and let A = diaga,,a n), B = diagb,,b n) and x =,,) t in the operator inequalities above to get the following classical inequalities: Diaz Metcalf inequality [4] k= b k + m M m M Pólya Szegö inequality [3] k= a k k= b k a k k= k= a kb k ) 4 Shisha Mond inequality [4] k= a k k= a k= a kb k n kb k k= b k M + m ) a k b k m M k= ) M M m m + ; m m M M M m m M ) ;

7 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 84 MS Moslehian, R Nakamoto, and Y Seo A Grüss type inequality n k= a k ) / / bk) k= a k b k k= M M M M m m ) m m min { M, M } m m Using the same argument with a positive n-tuple a,,a n ) of real numbers with 0 < m a i M i n), x = n,,) t, we get from Kantorovich inequality that Schweitzer inequality [] n i= a i ) n i= a i ) M + m ) 4M m If a,,a n ) and b,,b n ) are n-tuples of real numbers with 0 < m a i /b i M i n), we can consider the positive linear map ΦT) = Tx,x on BC n ) = M n C) and let A = diaga,,a n), B = diagb,,b n) and x = w,, w n ) t based on the weight w = w,,w n ), in the operator inequalities above to get the following classical inequalities: Cassels inequality [5] k= w ka k k= w kb k k= w ka k b k ) Klamkin McLenaghan inequality [4] M + m) 4mM ; w k a k k= n ) w k b ) k w k a k b k M m w k a k b k k= k= k= n k= w k a k Using the same argument, we obtain a weighted form of the Pólya Szegö inequality as follows: Grueb Rheinboldt inequality [0] k= w ka k k= w kb k k= w ka k b k ) M M + m m ) 4m m M M One can assert the integral versions of discrete results above by considering L X,µ), where X,µ) is a probability space, as a Hilbert space via h,h = X h h dµ, multiplication operators A,B BL X,µ))) defined by Ah) = f h

8 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 A Diaz Metcalf Type Inequality for Positive Linear Maps 85 and Bh) = g h for bounded f,g L X,µ) and a positive linear map Φ by ΦT) = X T)dµ on BL X,µ))) For instance, let us state integral versions of the Cassels and Klamkin McLenaghan inequalities These two inequalities are obtained, first for bounded positive functions f,g L X,µ) and next for general positive functions f,g L X,µ) as the limits of sequences of bounded positive functions Corollary 3 Let X,µ) be a probability space and f,g L X,µ) with 0 mg f Mg for some scalars 0 < m < M Then X ) f dµ g M + m) dµ fgdµ X 4Mm X and X f dµ g ) dµ fgdµ) M m fgdµ f dµ X X X X Considering the positive linear functional ΦR) = i= Rξ i,ξ i on BH ), where ξ,,ξ n H, we get the following versions of the Diaz Metcalf and Pólya Szegö inequalities in a Hilbert space Corollary 3 Let H be a Hilbert space, let ξ,,ξ n H and let T,S BH ) be positive operators satisfying 0 < m T M and 0 < m S M Then and M m M m Tξ i + i= Sξ i i= ) / n ) / Tξ i Sξ i i= i= M + m ) T S ) / ξ i m M i= ) M M m m + T S ) / ξ i m m M M i= 4 A Grüss type inequality In this section we obtain another Grüss type inequality, see also [8] Let A be a C -algebra and let B be a C -subalgebra of A Following [], a positive linear map Φ : A B is called a left multiplier if ΦXY ) = ΦX)Y for every X A, Y B The following lemma is interesting on its own right

9 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 86 MS Moslehian, R Nakamoto, and Y Seo Lemma 4 Let Φ be a unital positive linear map on A, A A and M,m be complex numbers such that Then ReM A) A m)) 0 4) Φ A ) ΦA) 4 M m Proof For any complex number c C, we have Φ A ) ΦA) = Φ A c ) ΦA c) 4) Since for any T A the operator equality 4 M m T M + m holds, the condition 4) implies that Therefore, it follows from 4) and 43) that = Re M T)T m) ) Φ A M + m ) 4 M m 43) Φ A ) ΦA) Φ A M + m ) 4 M m Remark 4 If i) Φ is a unital positive linear map and A is a normal operator or ii) Φ is a -positive linear map and A is an arbitrary operator, then it follows from [3] that 0 Φ A ) ΦA) 44) Condition 44) is stronger than positivity and weaker than -positivity; see [8] Another class of positive linear maps satisfying 44) are left multipliers, cf [, Corollary 4] Lemma 43 Let a positive linear map Φ : A B be a unital left multiplier Then ΦA B) ΦA) ΦB) Φ A ) ΦA) Φ B ) ΦB) ) 45)

10 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 A Diaz Metcalf Type Inequality for Positive Linear Maps 87 Proof If we put [X,Y ] := ΦX Y ) ΦX) ΦY ), then A is a right pre-inner product C -module over B, since ΦX Y ) is a right pre-inner product B-module, see [, Corollary 4] It follows from the Cauchy Schwarz inequality in pre-inner product C -modules see [5, Proposition ]) that ΦA B) ΦA) ΦB) = [B,A][A,B] and hence 45) holds [A,A] [B,B] = ΦA A) ΦA) ΦA) ΦB B) ΦB) ΦB)) Theorem 44 Let a positive linear map Φ : A B be a unital left multiplier If M,m,M,m C and A,B A satisfy the following conditions: then ReM A) A m ) 0 and ReM B) B m ) 0, ΦA B) ΦA) ΦB) 4 M m M m Proof By Löwner Heinz theorem, we have ΦA B) ΦA) ΦB) Φ A ) ΦA) Φ B ) ΦB) ) by Lemma 43) 4 M m M m by Lemma 4) 5 Ozeki Izumino Mori Seo type inequality Let a = a,,a n ) and b = b,,b n ) be n-tuples of real numbers satisfying 0 m a i M and 0 m b i M i =,,n) Then Ozeki Izumino Mori Seo inequality [, ] asserts that i= a i n ) b i a i b i n 3 M M m m ) 5) i= i= In [] they also showed the following operator version of 5): If A and B are positive operators in BH ) such that 0 < m A M and 0 < m B M for some scalars m M and m M, then A x,x)b x,x) A B x,x) 4γ M M m m ) 5)

11 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 88 MS Moslehian, R Nakamoto, and Y Seo for every unit vector x H, where γ = max{ m M, m M } Based on the Kantorovich inequality for the difference, we present an extension of Ozeki Izumino Mori Seo inequality 5) as follows Theorem 5 Suppose that Φ : BH ) BK ) is a positive linear map such that ΦI) is invertible and ΦI) I Assume that A,B BH ) are positive invertible operators such that 0 < m A M and 0 < m B M for some scalars m M and m M Then ΦB ) ΦA )ΦB ) ΦB ) ΦA B )ΦB ) M M m m ) and ΦA ) ΦB )ΦA ) ΦA ) ΦA B )ΦA ) M M m m ) 4 4 M m 53) M m 54) Proof Define a normalized positive linear map Ψ by ΨX) := ΦA) ΦA XA )ΦA) By using the Kantorovich inequality for the difference, it follows that ΨX ) ΨX) M m) 4 for all 0 < m X M with some scalars m M As a matter of fact, we have ΨX ) ΨX) ΨM + m)x Mm) ΨX) = ΨX) M + m ) M m) + 4 M m) 4 If we put X = A BA ), then due to m M 0 < m =) X = M) M m we deduce from 55) that ΦA) ΦB)ΦA) ΦA) ΦA B)ΦA) ) M M m m ) 4M m 55)

12 Electronic Journal of Linear Algebra ISSN Volume, pp 79-90, March 0 A Diaz Metcalf Type Inequality for Positive Linear Maps 89 Pre- and post-multiplying both sides by ΦA), we obtain ΦA) ΦB)ΦA) ΦA) ΦA B)ΦA) M M m m ) 4M m ΦA) M M m m ) 4 M m, since 0 ΦA) M Replacing A and B by A and B respectively, we have the desired inequality 54) Similarly, one can obtain 53) Remark 5 If Φ is a vector state in 53) and 54), then we get Ozeki Izumino Mori Seo inequality 5) Acknowledgement The authors would like to sincerely thank the anonymous referee for carefully reading the paper and for valuable comments REFERENCES [] Lj Arambašić, D Bakić and MS Moslehian A treatment of the Cauchy Schwarz inequality in C -modules J Math Anal Appl, to appear, doi:006/jjmaa0006 [] PS Bullen A Dictionary of Inequalities Pitman Monographs and Surveys in Pure and Applied Mathematics, 97, Longman, Harlow, 998 [3] M-D Choi Some assorted inequalities for positive linear maps on C -algebras J Operator Theory, 4:7 85, 980 [4] JB Diaz and FT Metcalf Stronger forms of a class of inequalities of G Pólya-G Szegö and LV Kantorovich Bull Amer Math Soc, 69:45 48, 963 [5] SS Dragomir A survey on Cauchy Bunyakovsky Schwarz type discrete inequalities JIPAM J Inequal Pure Appl Math, 4, 003, Article 63 [6] SS Dragomir A counterpart of Schwarz inequality in inner product spaces RGMIA Res Rep Coll, 6, 003, Article 8 [7] SS Dragomir Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces Nova Science Publishers, Inc, New York, 007 [8] DE Evans and JT Lewis, Dilations of Irreversible Evolutions in Algebraic Quantum Theory Comm Dublin Inst Adv Studies Ser A, no 4, 977 [9] T Furuta, JM Hot, JE Pečarić and Y Seo Mond Pečarić Method in Operator Inequalities Inequalities for Bounded Selfadjoint Operators on a Hilbert Space Monographs in Inequalities, Element, Zagreb, 005 [0] W Greub and W Rheinboldt On a generalization of an inequality of LV Kantorovich Proc Amer Math Soc, 0:407 45, 959 [] D Ilišević and S Varošanec On the Cauchy-Schwarz inequality and its reverse in semi-inner product C -modules Banach J Math Anal, :78 84, 007 [] S Izumino, H Mori and Y Seo On Ozeki s inequality J Inequal Appl, :35 53, 998 [3] M Joiţa On the Cauchy Schwarz inequality in C -algebras Math Rep Bucur), 353):43 46, 00 [4] MS Klamkin and RG Mclenaghan An ellipse inequality Math Mag, 50:6 63, 977 [5] EC Lance Hilbert C -Modules London Math Soc Lecture Note Series 0, Cambridge Univ Press, 995

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