Norm inequalities related to the Heinz means
|
|
- Griselda Goodwin
- 5 years ago
- Views:
Transcription
1 Gao and Ma Journal of Inequalities and Applications (08) 08:303 R E S E A R C H Open Access Norm inequalities related to the Heinz means Fugen Gao * and Xuedi Ma * Correspondence: gaofugen08@6.com College of Mathematics and Information Science, Henan Normal University, Xinxiang, China Abstract Let (I, ) be a two-sided ideal of operators equipped with a unitarily invariant norm. We generalize the results of Kapil s, using a new contractive map in I to obtain a norm inequality. And we give a new inequality, which is a comparison between the Heinz means and other related inequalities; moreover, we will obtain some correlative conclusions. Keywords: Norm inequalities; Contractive maps; Unitarily invariant norm Introduction and preliminaries In this paper, let B(H) denote the algebra of all bounded linear operators on a complex separable Hilbert space, B(H) + denote the cone of positive operators and K(H)denotethe ideal of compact operators in B(H). And (I, ) is a two-sided ideal of B(H) equipped with a unitarily invariant norm. We shall denote this by I instead of (I, )for convenience. For any compact operator A K(H), let S (A), S (A),... be the eigenvalues of A = (A A) arranged in decreasing order. If A M n, which is the algebra of all n n matrices over C, wetakes k (A) =0fork > n. A unitarily invariant norm in K(H) isamap : K(H) [0, ] givenby A = g(s(a)), A K(H), where g is a symmetric gauge function, cf. [6, 9]. The Schatten p-norms A p =( Sp (A)) p for p aresignificant examples of the unitarily invariant norm, and is a special unitarily invariant norm which is the Hilbert Schmidt norm defined, for A M n, as follows: A = i, a i = tr ( A A ). It is well known that the arithmetic geometric mean inequality ab a + b for positive numbers a and b has been generalized in various directions. Bhatia et al. [] haveobtainedtheresultthatifa, B, andx are n n matrices with A and B are positive definite, then A XB AX + XB. The Author(s) 08. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Gao and MaJournal of Inequalities and Applications (08) 08:303 Page of 7 The Heinz means H v (a, b)= a v b v + a v b v is an interpolation between the arithmetic and geometric means for a, b 0andv [0, ]. It is easy to see that H v is symmetric and convex function for v [0, ] and attains its minimum at v = ;thuswehave ab Hv (a, b) a + b. Thematrixversionhasbeenprovedin[]: if A, B and X are positive definite, then for every unitarily invariant norm the function g(v)= A v XB v + A v XB v is convex on [0, ], and attains its minimum at v =.Thuswehave A XB A v XB v + A v XB v AX + XB. The Heron means is defined by F α (a, b)=( α) ab + α a + b for 0 α, see []. This family is the linear interpolation between the geometric and the arithmetic mean. Clearly, F α F β whenever α β. Bhatia and Davis have proved that the inequality ( ) ( ) ( α)a XB AX + XB + α ( β)a XB AX + XB + β is always true for 0 α β, β,andthisrestrictiononβ is necessary in []. The Heinz means and the Heron means satisfy the inequality H v (a, b) F α(v) (a, b) for 0 v. And α(v) = 4(v v ), this is a convex function, its minimum value is α( ) = 0, and its maximum value is α(0) = α() =. In [0] the authors have presented the result that A v XB v + A v XB v ( ) ( α)a XB AX + XB + α for 4 v 3 4 and α [, ] and they used the properties of contractive map on I to prove it. A different version of the Heinz inequality, A α XB α A α XB α α AX XB, for α [0, ] was proved by Bhatia and Davis [3] in 995.
3 Gao and MaJournal of Inequalities and Applications (08) 08:303 Page 3 of 7 A further generalization, namely +t A α XB α + A α XB α AX + XB + ta XB, was proved for t [, ], and α [ 4, 3 ]. For more details refer to [3]and[4]. 4 Contractive maps on I play a key role to prove the inequalities, more details refer to [4, 5, 7, 8, ]. This in turn proves the corresponding Schur multiplier maps to be contractive. We use the ideals to establish the fact that some special maps on I are contractive. Our paper consists of three parts. In the second part, we will give a new norm inequality whichisprovedbythepropertiesofcontractivemapsoni.inthethirdpart,wewillpresent a inequality related to the Heinz means and we notice that this inequality is a comparison between the Heinz means and the geometric mean. Our results are extensions of some previous conclusions about norm inequalities. Norm inequalities with contractive maps Let L X, R Y denote the left and right multiplication maps on B(H), respectively, that is, L X (T)=XT and R Y (T)=TY and we have e L X+R Y (T)=e X Te Y. () Neeb []hasprovedthat(l X ±R Y ) sin(l X ±R Y )isanexpansivemaponi by using the Weierstrass factorization theorem. We use X and Y to denote two selfadoint operators in B(H)andD = L X R Y. The following proposition is a result for a contractive map in I; for more details refer to [0]. Proposition. ([0, Proposition.4]) Let 0 r sors r 0, then each of the following operator maps is a contraction on I. () (+t) cosh( r D) t+cosh(sd) () t+cosh(rd) t+cosh(sd) for t. for t. (3) cosh(rd) cosh(sd). In particular r (4) s sinh(rd) r sinh(sd) cosh(rd) cosh(sd) dr = sd. The case r =0would become sinh(rd) rd cosh(sd) sinh(sd). is a contraction. Corollary. ([0, Corollary.5]) The following maps are contractive on I. (t+) cosh((v )D) () for t+cosh D 4 v 3, and t. 4 cosh(rd) () cosh(s D)+cosh(s D) for 0 r max{ s +s, s s } or min{ s +s, s s } r 0. (s (3) +s ) sinh(rd) r(sinh(s D)+sinh(s D)) for 0 r s +s or s +s r 0. Corollary.3 Let 0 v and t. Then the map (t+) cosh((v )D) t+cosh(d) is contractive on I. Proof This map is a special case of () of Proposition. when s =. From the condition 0 v, we can obtain v. Letting r =v,wegetthedesiredresult. With these above results about contractive maps in I, we obtain a new norm inequality which derives from the Heinz means and other related inequalities. Let R be an invertible operator in B(H) +, then there exists a selfadoint operator S B(H) suchthatr = e S.To avoid repetitions, we denote the two invertible operators A and B in B(H) + by e X and
4 Gao and MaJournal of Inequalities and Applications (08) 08:303 Page 4 of 7 e Y,respectively,whereX and Y in B(H) are selfadoint. The corresponding operator map L X R Y is denoted by D. Theorem.4 Let v [0, ] and α [, ). Supposed that A and B are any two invertible operators in B(H) +, X I, then A v XB v + A v XB v ( α)a XB + α A 3 XB + A XB 3. () Proof Put A XB = T and use () we can notice that A v XB v + A v XB v = A v A XB B v + A v A XB B v = A v TB v + A v TB v = e(v )(L X R Y ) T + e (v )(L X R Y ) T = cosh ( (v )D ) T. (3) And we also have ( α)a XB + α A 3 XB + A XB 3 =( α)t + α AA XB B + A A XB B =( α)t + α ATB + A TB =( α)t + α e(l X R Y ) T + e (L X R Y ) T = [ ( α)+αcosh(d) ] T. (4) Setting = α in (4) and applying Corollary.3, we can obtain ()from(3)and(4). +t 3 Norm inequality related to the Heinz means According to the above results, we set H v (a, b)= av b v + a v b v, G v (a, b)=( v)a b + v a 3 b + a b 3 Then we have the following result, which is a interpolation of H v and G v..
5 Gao and MaJournal of Inequalities and Applications (08) 08:303 Page 5 of 7 Theorem 3. Let a, b >0and v [,].Then ( H (a, b) ) v H v (a, b) G v (a, b). (5) H (a, b) Proof Without loss of generality, we can suppose that a =. Then the inequality (5) can be simplified to b v + b v To prove this inequality, let ( ) b v [ ( v) b + v b + b 3 ]. +b f (v)=log ( b v + b v) ( v) log b +b log[ ( v) b + v ( b + b 3 )]. Calculations show that f (v)= 4bv b v log b (b + b 3 b ) + (b v + b v ) [( v)b + v(b + b 3 )]. We can notice that f (v) 0for v, thus f is convex on [,].Thenwehavef (v) max{f ( ), f ()}. By calculation, we have f ( ) ( b 4 ( + b) ) = log b + b + b 0 3 because of b 4 (+b) b + b + b 3 (b >0),whichequals b 4 ( + b) b + b + b 3,that is, 8( + b)b 3 6b + b 4 +4b 3 +4b +. Setting b = x (x >0),thuswehavex 8 +4x 6 8x 5 +6x 4 8x 3 +4x + 0, that is, (x ) (x 6 +x 5 +7x 4 +4x 3 +7x +x +) 0, which is always true when x >0,thenwe get the desired results. In the same way, ( ) +b f () = log 0 b + b 3 +b because of (b >0),whichequals+b b b +b 3 + b 3,thatis(b )(b 3 ) 0, whichisalwaystruewhenb >0.Hencethevaluesoff ( )andf() are both less than 0, that is, f (v) 0. Then we complete the proof. Remark 3. The inequality which we obtained from Theorem 3. can also be written as H v (a, b) ( ) ab v G v, a + b and we notice that it is a new comparison between the Heinz means and the geometric means.
6 Gao and MaJournal of Inequalities and Applications (08) 08:303 Page 6 of 7 Corollary 3.3 Let A, B M n +, X M n and v. If there are two positive numbers m, Msuchthatm A, B M, then A v XB v + A v XB v ( ) m + M v ( v)a XB + v A 3 XB + A XB 3 mm. Proof Let A = U diag(λ i )U and B = V diag(μ )V be the spectral decompositions of A and B, respectively. Letting U XV = Y,wehave A v XB v + A v XB v = U diag(λv i )Y diag(μ v )+diag(λ v i )Y diag(μ v ) = U [λv i μ v + λ v i μ v ] [y i] V, (6) where denotes the Schur product. Applying (5) we have A v XB v + A v XB v = ( λ v i μ v + λ v i i, i, ( λi + μ λ i λ μ v ) y i ) (v ) ( ( v)λ i μ V + v λ 3 i μ + λ i μ 3 ) y i ( ) m + M (v ) ( v)a XB + v A 3 XB + A XB 3 mm, (7) wherewehaveusedthefactthatm λ i, μ M to obtain the last inequality. Funding This research is supported by the National Natural Science Foundation of China (7054), the Natural Science Foundation of the Department of Education, Henan Province (6A0003), (9A000), and the Program for Graduate Innovative Research of Henan Normal University (No.YL070). Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed to each part of this work equally, and they all read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to urisdictional claims in published maps and institutional affiliations. Received: 9 July 08 Accepted: 8 October 08 References. Bhatia, R.: Interpolating the arithmetic geometric mean inequality and its operator version. Linear Algebra Appl. 43, (006). Bhatia, R., Davis, C.: More matrix forms of the arithmetic geometric mean inequality. SIAM J. Matrix Anal. Appl. 4, 3 36 (993) 3. Bhatia, R., Davis, C.: A Cauchy Schwarz inequality for operators with applications. Linear Algebra Appl. 3/4, 9 9 (995)
7 Gao and MaJournal of Inequalities and Applications (08) 08:303 Page 7 of 7 4. Bhatia, R., Parthasarathy, K.R.: Positive definite function and operator inequalities. Bull. Lond. Math. Soc. 3, 4 8 (000) 5. Drissi, D.: Sharp inequalities for some operator means. SIAM J. Matrix Anal. Appl. 8, 8 88 (006) 6. Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear nonselfadoint opeators. Transl. Math. Monogr, vol. 8. Am. Math. Soc., Providence (969) 7. Hiai, F., Kosaki, H.: Comparison of various means for operators. J. Funct. Anal. 63, (999) 8. Hiai, F., Kosaki, H.: Means of matrices and comparison of their norm. Indiana Univ. Math. J. 48(3), (999) 9. Kapil, Y., Conde, C., Moslehian, M.S., Singh, M., Sababheh, M.: Norm inequalities related to the Heron and Heinz means. Mediterr. J. Math. 4(5), 3 (07) 0. Kapil, Y., Singh, M.: Contractive maps on operator ideals and norm inequalities. Linear Algebra Appl. 459, (04). Kosaki, H.: Arithmetic geometric mean and related inequalities for operators. J. Funct. Anal. 56, (998). Neeb, K.H.: A Cartan Hadamard theorem for Banach Finsler manifolds. Geom. Dedic. 95, 5 56 (00) 3. Singh, M., Aula, J.S., Vasudeva, H.L.: Inequalities for Hadamard product and unitarily invariant norm of matrices. Linear Multilinear Algebra 48, 47 6 (00) 4. Zhan, X.: Inequalities for unitarily invariant norms. SIAM J. Matrix Anal. Appl. 0(), (999)
Extensions of interpolation between the arithmetic-geometric mean inequality for matrices
Bakherad et al. Journal of Inequalities and Applications 017) 017:09 DOI 10.1186/s13660-017-1485-x R E S E A R C H Open Access Extensions of interpolation between the arithmetic-geometric mean inequality
More informationInterpolating the arithmetic geometric mean inequality and its operator version
Linear Algebra and its Applications 413 (006) 355 363 www.elsevier.com/locate/laa Interpolating the arithmetic geometric mean inequality and its operator version Rajendra Bhatia Indian Statistical Institute,
More informationSome inequalities for unitarily invariant norms of matrices
Wang et al Journal of Inequalities and Applications 011, 011:10 http://wwwjournalofinequalitiesandapplicationscom/content/011/1/10 RESEARCH Open Access Some inequalities for unitarily invariant norms of
More informationThe Improved Arithmetic-Geometric Mean Inequalities for Matrix Norms
Applied Mathematical Sciences, Vol 7, 03, no 9, 439-446 HIKARI Ltd, wwwm-hikaricom The Improved Arithmetic-Geometric Mean Inequalities for Matrix Norms I Halil Gumus Adıyaman University, Faculty of Arts
More informationThe matrix arithmetic-geometric mean inequality revisited
isid/ms/007/11 November 1 007 http://wwwisidacin/ statmath/eprints The matrix arithmetic-geometric mean inequality revisited Rajendra Bhatia Fuad Kittaneh Indian Statistical Institute Delhi Centre 7 SJSS
More informationarxiv: v1 [math.fa] 19 Aug 2017
EXTENSIONS OF INTERPOLATION BETWEEN THE ARITHMETIC-GEOMETRIC MEAN INEQUALITY FOR MATRICES M. BAKHERAD 1, R. LASHKARIPOUR AND M. HAJMOHAMADI 3 arxiv:1708.0586v1 [math.fa] 19 Aug 017 Abstract. In this paper,
More informationNorm inequalities related to the matrix geometric mean
isid/ms/2012/07 April 20, 2012 http://www.isid.ac.in/ statmath/eprints Norm inequalities related to the matrix geometric mean RAJENDRA BHATIA PRIYANKA GROVER Indian Statistical Institute, Delhi Centre
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 6 (2012), no. 1, 139 146 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/bjma/ AN EXTENSION OF KY FAN S DOMINANCE THEOREM RAHIM ALIZADEH
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationarxiv: v1 [math.fa] 1 Oct 2015
SOME RESULTS ON SINGULAR VALUE INEQUALITIES OF COMPACT OPERATORS IN HILBERT SPACE arxiv:1510.00114v1 math.fa 1 Oct 2015 A. TAGHAVI, V. DARVISH, H. M. NAZARI, S. S. DRAGOMIR Abstract. We prove several singular
More informationNORMS ON SPACE OF MATRICES
NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system
More informationSINGULAR VALUE INEQUALITIES FOR COMPACT OPERATORS
SINGULAR VALUE INEQUALITIES FOR OMPAT OPERATORS WASIM AUDEH AND FUAD KITTANEH Abstract. A singular value inequality due to hatia and Kittaneh says that if A and are compact operators on a complex separable
More informationGENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS
International Journal of Analysis Alications ISSN 9-8639 Volume 5, Number (04), -9 htt://www.etamaths.com GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS ILYAS ALI, HU YANG, ABDUL SHAKOOR Abstract.
More informationNotes on matrix arithmetic geometric mean inequalities
Linear Algebra and its Applications 308 (000) 03 11 www.elsevier.com/locate/laa Notes on matrix arithmetic geometric mean inequalities Rajendra Bhatia a,, Fuad Kittaneh b a Indian Statistical Institute,
More informationOn (T,f )-connections of matrices and generalized inverses of linear operators
Electronic Journal of Linear Algebra Volume 30 Volume 30 (2015) Article 33 2015 On (T,f )-connections of matrices and generalized inverses of linear operators Marek Niezgoda University of Life Sciences
More informationSTRUCTURAL AND SPECTRAL PROPERTIES OF k-quasi- -PARANORMAL OPERATORS. Fei Zuo and Hongliang Zuo
Korean J Math (015), No, pp 49 57 http://dxdoiorg/1011568/kjm01549 STRUCTURAL AND SPECTRAL PROPERTIES OF k-quasi- -PARANORMAL OPERATORS Fei Zuo and Hongliang Zuo Abstract For a positive integer k, an operator
More informationSome Hermite-Hadamard type integral inequalities for operator AG-preinvex functions
Acta Univ. Sapientiae, Mathematica, 8, (16 31 33 DOI: 1.1515/ausm-16-1 Some Hermite-Hadamard type integral inequalities or operator AG-preinvex unctions Ali Taghavi Department o Mathematics, Faculty o
More informationarxiv: v1 [math.fa] 1 Sep 2014
SOME GENERALIZED NUMERICAL RADIUS INEQUALITIES FOR HILBERT SPACE OPERATORS MOSTAFA SATTARI 1, MOHAMMAD SAL MOSLEHIAN 1 AND TAKEAKI YAMAZAKI arxiv:1409.031v1 [math.fa] 1 Sep 014 Abstract. We generalize
More informationInequalities involving eigenvalues for difference of operator means
Electronic Journal of Linear Algebra Volume 7 Article 5 014 Inequalities involving eigenvalues for difference of operator means Mandeep Singh msrawla@yahoo.com Follow this and additional works at: http://repository.uwyo.edu/ela
More informationarxiv: v1 [math.fa] 12 Oct 2016
UNITARILY INVARIANT NORM INEQUALITIES FOR ELEMENTARY OPERATORS INVOLVING G 1 OPERATORS FUAD KITTANEH, MOHAMMAD SAL MOSLEHIAN, AND MOHAMMAD SABABHEH arxiv:161.3869v1 [math.fa] 1 Oct 16 Abstract. In this
More informationGeneralized Numerical Radius Inequalities for Operator Matrices
International Mathematical Forum, Vol. 6, 011, no. 48, 379-385 Generalized Numerical Radius Inequalities for Operator Matrices Wathiq Bani-Domi Department of Mathematics Yarmouk University, Irbed, Jordan
More informationSalah Mecheri. Communicated by S. L. Troyanski
Serdica Math J 26 (2000), 119-126 ON MINIMIZING S (AX XB) p p Salah Mecheri Communicated by S L Troyanski Abstract In this paper, we minimize the map F p (X)= S (AX XB) p p, where the pair (A, B) has the
More informationWavelets and Linear Algebra
Wavelets and Linear Algebra () (05) 49-54 Wavelets and Linear Algebra http://wala.vru.ac.ir Vali-e-Asr University of Rafsanjan Schur multiplier norm of product of matrices M. Khosravia,, A. Sheikhhosseinia
More informationSingular Value and Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices
Electronic Journal of Linear Algebra Volume 32 Volume 32 (2017) Article 8 2017 Singular Value Norm Inequalities Associated with 2 x 2 Positive Semidefinite Block Matrices Aliaa Burqan Zarqa University,
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J Math Anal (009), no, 64 76 Banach Journal of Mathematical Analysis ISSN: 75-8787 (electronic) http://wwwmath-analysisorg ON A GEOMETRIC PROPERTY OF POSITIVE DEFINITE MATRICES CONE MASATOSHI ITO,
More informationSome Range-Kernel Orthogonality Results for Generalized Derivation
International Journal of Contemporary Mathematical Sciences Vol. 13, 2018, no. 3, 125-131 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2018.8412 Some Range-Kernel Orthogonality Results for
More informationOn the Generalized Reid Inequality and the Numerical Radii
Applied Mathematical Sciences, Vol. 5, 2011, no. 9, 441-445 On the Generalized Reid Inequality and the Numerical Radii J. O. Bonyo 1, D. O. Adicka 2, J. O. Agure 3 1,3 Department of Mathematics and Applied
More informationAnn. Funct. Anal. 5 (2014), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:www.emis.
Ann. Funct. Anal. 5 (2014), no. 2, 147 157 A nnals of F unctional A nalysis ISSN: 2008-8752 (electronic) URL:www.emis.de/journals/AFA/ ON f-connections OF POSITIVE DEFINITE MATRICES MAREK NIEZGODA This
More informationAbstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization relations.
HIROSHIMA S THEOREM AND MATRIX NORM INEQUALITIES MINGHUA LIN AND HENRY WOLKOWICZ Abstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization
More informationEigenvalue inequalities for convex and log-convex functions
Linear Algebra and its Applications 44 (007) 5 35 www.elsevier.com/locate/laa Eigenvalue inequalities for convex and log-convex functions Jaspal Singh Aujla a,, Jean-Christophe Bourin b a Department of
More informationAdditive functional inequalities in Banach spaces
Lu and Park Journal o Inequalities and Applications 01, 01:94 http://www.journaloinequalitiesandapplications.com/content/01/1/94 R E S E A R C H Open Access Additive unctional inequalities in Banach spaces
More informationSome Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces
Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results
More informationPositive definite solutions of some matrix equations
Available online at www.sciencedirect.com Linear Algebra and its Applications 49 (008) 40 44 www.elsevier.com/locate/laa Positive definite solutions of some matrix equations Aleksandar S. Cvetković, Gradimir
More informationThe operator equation n
isid/ms/29/2 February 21, 29 http://www.isid.ac.in/ statmath/eprints The operator equation n i= A n i XB i = Y Rajendra Bhatia Mitsuru Uchiyama Indian Statistical Institute, Delhi Centre 7, SJSS Marg,
More informationarxiv: v1 [math.fa] 6 Nov 2015
CARTESIAN DECOMPOSITION AND NUMERICAL RADIUS INEQUALITIES FUAD KITTANEH, MOHAMMAD SAL MOSLEHIAN AND TAKEAKI YAMAZAKI 3 arxiv:5.0094v [math.fa] 6 Nov 05 Abstract. We show that if T = H + ik is the Cartesian
More informationTrace inequalities for positive semidefinite matrices with centrosymmetric structure
Zhao et al Journal of Inequalities pplications 1, 1:6 http://wwwjournalofinequalitiesapplicationscom/content/1/1/6 RESERCH Trace inequalities for positive semidefinite matrices with centrosymmetric structure
More informationj jf, S K cf = j K c j jf, f H.
DOI 10.1186/s40064-016-2731-2 RESEARCH New double inequalities for g frames in Hilbert C modules Open Access Zhong Qi Xiang * *Correspondence: lxsy20110927@163.com College of Mathematics and Computer Science,
More informationMoore-Penrose Inverse and Operator Inequalities
E extracta mathematicae Vol. 30, Núm. 1, 9 39 (015) Moore-Penrose Inverse and Operator Inequalities Ameur eddik Department of Mathematics, Faculty of cience, Hadj Lakhdar University, Batna, Algeria seddikameur@hotmail.com
More informationJENSEN S OPERATOR INEQUALITY AND ITS CONVERSES
MAH. SCAND. 100 (007, 61 73 JENSEN S OPERAOR INEQUALIY AND IS CONVERSES FRANK HANSEN, JOSIP PEČARIĆ and IVAN PERIĆ (Dedicated to the memory of Gert K. Pedersen Abstract We give a general formulation of
More informationMatrix Inequalities by Means of Block Matrices 1
Mathematical Inequalities & Applications, Vol. 4, No. 4, 200, pp. 48-490. Matrix Inequalities by Means of Block Matrices Fuzhen Zhang 2 Department of Math, Science and Technology Nova Southeastern University,
More informationarxiv: v1 [math.ca] 7 Jan 2015
Inertia of Loewner Matrices Rajendra Bhatia 1, Shmuel Friedland 2, Tanvi Jain 3 arxiv:1501.01505v1 [math.ca 7 Jan 2015 1 Indian Statistical Institute, New Delhi 110016, India rbh@isid.ac.in 2 Department
More informationPOSITIVE DEFINITE FUNCTIONS AND OPERATOR INEQUALITIES
POSITIVE DEFINITE FUNCTIONS AND OPERATOR INEQUALITIES RAJENDRA BHATIA and K. R. PARTHASARATHY Abstract We construct several examples of positive definite functions, and use the positive definite matrices
More informationClarkson Inequalities With Several Operators
isid/ms/2003/23 August 14, 2003 http://www.isid.ac.in/ statmath/eprints Clarkson Inequalities With Several Operators Rajendra Bhatia Fuad Kittaneh Indian Statistical Institute, Delhi Centre 7, SJSS Marg,
More informationSOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES. S. S. Dragomir
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: 011), 151 16 DOI: 10.98/FIL110151D SOME INEQUALITIES FOR COMMUTATORS OF BOUNDED LINEAR
More information2. reverse inequalities via specht ratio To study the Golden-Thompson inequality, Ando-Hiai in [1] developed the following log-majorizationes:
ON REVERSES OF THE GOLDEN-THOMPSON TYPE INEQUALITIES MOHAMMAD BAGHER GHAEMI, VENUS KALEIBARY AND SHIGERU FURUICHI arxiv:1708.05951v1 [math.fa] 20 Aug 2017 Abstract. In this paper we present some reverses
More informationSOME OPERATOR INEQUALITIES FOR UNITARILY INVARIANT NORMS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Volumen 46, Número 1, 2005, Páginas 53 66 SOME OPERATOR INEQUALITIES FOR UNITARILY INVARIANT NORMS CRISTINA CANO, IRENE MOSCONI AND DEMETRIO STOJANOFF Abstract.
More informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More informationPRODUCT OF OPERATORS AND NUMERICAL RANGE
PRODUCT OF OPERATORS AND NUMERICAL RANGE MAO-TING CHIEN 1, HWA-LONG GAU 2, CHI-KWONG LI 3, MING-CHENG TSAI 4, KUO-ZHONG WANG 5 Abstract. We show that a bounded linear operator A B(H) is a multiple of a
More informationOn some Hermite Hadamard type inequalities for (s, QC) convex functions
Wu and Qi SpringerPlus 65:49 DOI.86/s464-6-676-9 RESEARCH Open Access On some Hermite Hadamard type ineualities for s, QC convex functions Ying Wu and Feng Qi,3* *Correspondence: ifeng68@gmail.com; ifeng68@hotmail.com
More informationPERTURBATION ANAYSIS FOR THE MATRIX EQUATION X = I A X 1 A + B X 1 B. Hosoo Lee
Korean J. Math. 22 (214), No. 1, pp. 123 131 http://dx.doi.org/1.11568/jm.214.22.1.123 PERTRBATION ANAYSIS FOR THE MATRIX EQATION X = I A X 1 A + B X 1 B Hosoo Lee Abstract. The purpose of this paper is
More informationHERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GEOMETRICALLY CONVEX FUNCTIONS
HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR GEOMETRICALLY CONVEX FUNCTIONS A TAGHAVI, V DARVISH, H M NAZARI, S S DRAGOMIR Abstract In this paper, we introduce the concept of operator geometrically
More informationarxiv: v1 [math.ra] 8 Apr 2016
ON A DETERMINANTAL INEQUALITY ARISING FROM DIFFUSION TENSOR IMAGING MINGHUA LIN arxiv:1604.04141v1 [math.ra] 8 Apr 2016 Abstract. In comparing geodesics induced by different metrics, Audenaert formulated
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationSubstrictly Cyclic Operators
Substrictly Cyclic Operators Ben Mathes dbmathes@colby.edu April 29, 2008 Dedicated to Don Hadwin Abstract We initiate the study of substrictly cyclic operators and algebras. As an application of this
More informationFixed points of monotone mappings and application to integral equations
Bachar and Khamsi Fixed Point Theory and pplications (15) 15:11 DOI 1.1186/s13663-15-36-x RESERCH Open ccess Fixed points of monotone mappings and application to integral equations Mostafa Bachar1* and
More informationProperties for the Perron complement of three known subclasses of H-matrices
Wang et al Journal of Inequalities and Applications 2015) 2015:9 DOI 101186/s13660-014-0531-1 R E S E A R C H Open Access Properties for the Perron complement of three known subclasses of H-matrices Leilei
More informationThe iterative methods for solving nonlinear matrix equation X + A X 1 A + B X 1 B = Q
Vaezzadeh et al. Advances in Difference Equations 2013, 2013:229 R E S E A R C H Open Access The iterative methods for solving nonlinear matrix equation X + A X A + B X B = Q Sarah Vaezzadeh 1, Seyyed
More informationSEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES
SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. The main aim of this paper is to establish some connections that exist between the numerical
More informationThe reflexive and anti-reflexive solutions of the matrix equation A H XB =C
Journal of Computational and Applied Mathematics 200 (2007) 749 760 www.elsevier.com/locate/cam The reflexive and anti-reflexive solutions of the matrix equation A H XB =C Xiang-yang Peng a,b,, Xi-yan
More informationIMPROVED ARITHMETIC-GEOMETRIC AND HEINZ MEANS INEQUALITIES FOR HILBERT SPACE OPERATORS
IMPROVED ARITHMETI-GEOMETRI AND HEINZ MEANS INEQUALITIES FOR HILBERT SPAE OPERATORS FUAD KITTANEH, MARIO KRNIĆ, NEDA LOVRIČEVIĆ, AND JOSIP PEČARIĆ Abstract. The main objective of this paper is an improvement
More informationWirtinger inequality using Bessel functions
Mirković Advances in Difference Equations 18) 18:6 https://doiorg/11186/s166-18-164-7 R E S E A R C H Open Access Wirtinger inequality using Bessel functions Tatjana Z Mirković 1* * Correspondence: tatjanamirkovic@visokaskolaedurs
More informationCOMMUTATIVITY OF OPERATORS
COMMUTATIVITY OF OPERATORS MASARU NAGISA, MAKOTO UEDA, AND SHUHEI WADA Abstract. For two bounded positive linear operators a, b on a Hilbert space, we give conditions which imply the commutativity of a,
More informationA fixed point problem under two constraint inequalities
Jleli and Samet Fixed Point Theory and Applications 2016) 2016:18 DOI 10.1186/s13663-016-0504-9 RESEARCH Open Access A fixed point problem under two constraint inequalities Mohamed Jleli and Bessem Samet*
More informationOptimal bounds for the Neuman-Sándor mean in terms of the first Seiffert and quadratic means
Gong et al. Journal of Inequalities and Applications 01, 01:55 http://www.journalofinequalitiesandapplications.com/content/01/1/55 R E S E A R C H Open Access Optimal bounds for the Neuman-Sándor mean
More informationExistence and multiplicity of positive solutions of a one-dimensional mean curvature equation in Minkowski space
Pei et al. Boundary Value Problems (28) 28:43 https://doi.org/.86/s366-8-963-5 R E S E A R C H Open Access Existence and multiplicity of positive solutions of a one-dimensional mean curvature equation
More informationarxiv: v1 [math.fa] 30 Oct 2011
AROUND OPERATOR MONOTONE FUNCTIONS MOHAMMAD SAL MOSLEHIAN AND HAMED NAJAFI arxiv:111.6594v1 [math.fa] 3 Oct 11 Abstract. We show that the symmetrized product AB + BA of two positive operators A and B is
More informationMATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
More informationAn Iterative Procedure for Solving the Riccati Equation A 2 R RA 1 = A 3 + RA 4 R. M.THAMBAN NAIR (I.I.T. Madras)
An Iterative Procedure for Solving the Riccati Equation A 2 R RA 1 = A 3 + RA 4 R M.THAMBAN NAIR (I.I.T. Madras) Abstract Let X 1 and X 2 be complex Banach spaces, and let A 1 BL(X 1 ), A 2 BL(X 2 ), A
More informationInequalities for Modules and Unitary Invariant Norms
Int. J. Contemp. Math. Sciences Vol. 7 202 no. 36 77-783 Inequalities for Modules and Unitary Invariant Norms Loredana Ciurdariu Department of Mathematics Politehnica University of Timisoara P-ta. Victoriei
More informationFixed point theorems of nondecreasing order-ćirić-lipschitz mappings in normed vector spaces without normalities of cones
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 18 26 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Fixed point theorems of nondecreasing
More informationApproximations to inverse tangent function
Qiao Chen Journal of Inequalities Applications (2018 2018:141 https://doiorg/101186/s13660-018-1734-7 R E S E A R C H Open Access Approimations to inverse tangent function Quan-Xi Qiao 1 Chao-Ping Chen
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More informationNumerical Ranges and Dilations. Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement
Numerical Ranges and Dilations Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement Man-Duen Choi Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3.
More informationA CLASS OF SCHUR MULTIPLIERS ON SOME QUASI-BANACH SPACES OF INFINITE MATRICES
A CLASS OF SCHUR MULTIPLIERS ON SOME QUASI-BANACH SPACES OF INFINITE MATRICES NICOLAE POPA Abstract In this paper we characterize the Schur multipliers of scalar type (see definition below) acting on scattered
More informationTrace Inequalities for a Block Hadamard Product
Filomat 32:1 2018), 285 292 https://doiorg/102298/fil1801285p Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Trace Inequalities for
More informationarxiv: v3 [math.ra] 22 Aug 2014
arxiv:1407.0331v3 [math.ra] 22 Aug 2014 Positivity of Partitioned Hermitian Matrices with Unitarily Invariant Norms Abstract Chi-Kwong Li a, Fuzhen Zhang b a Department of Mathematics, College of William
More informationExistence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions
International Journal of Mathematical Analysis Vol. 2, 208, no., 505-55 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ijma.208.886 Existence of Solutions for a Class of p(x)-biharmonic Problems without
More informationSome inequalities for sum and product of positive semide nite matrices
Linear Algebra and its Applications 293 (1999) 39±49 www.elsevier.com/locate/laa Some inequalities for sum and product of positive semide nite matrices Bo-Ying Wang a,1,2, Bo-Yan Xi a, Fuzhen Zhang b,
More informationA Spectral Characterization of Closed Range Operators 1
A Spectral Characterization of Closed Range Operators 1 M.THAMBAN NAIR (IIT Madras) 1 Closed Range Operators Operator equations of the form T x = y, where T : X Y is a linear operator between normed linear
More informationA generalized Schwarz lemma for two domains related to µ-synthesis
Complex Manifolds 018; 5: 1 8 Complex Manifolds Research Article Sourav Pal* and Samriddho Roy A generalized Schwarz lemma for two domains related to µ-synthesis https://doi.org/10.1515/coma-018-0001 Received
More informationSOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES
SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES SEVER S DRAGOMIR Abstract Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert
More informationA Note on Product Range of 3-by-3 Normal Matrices
International Mathematical Forum, Vol. 11, 2016, no. 18, 885-891 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6796 A Note on Product Range of 3-by-3 Normal Matrices Peng-Ruei Huang
More informationHigher rank numerical ranges of rectangular matrix polynomials
Journal of Linear and Topological Algebra Vol. 03, No. 03, 2014, 173-184 Higher rank numerical ranges of rectangular matrix polynomials Gh. Aghamollaei a, M. Zahraei b a Department of Mathematics, Shahid
More informationConvexity properties of Tr[(a a) n ]
Linear Algebra and its Applications 315 (2) 25 38 www.elsevier.com/locate/laa Convexity properties of Tr[(a a) n ] Luis E. Mata-Lorenzo, Lázaro Recht Departamento de Matemáticas Puras y Aplicadas, Universidad
More informationResearch Article Constrained Solutions of a System of Matrix Equations
Journal of Applied Mathematics Volume 2012, Article ID 471573, 19 pages doi:10.1155/2012/471573 Research Article Constrained Solutions of a System of Matrix Equations Qing-Wen Wang 1 and Juan Yu 1, 2 1
More informationLinear Operators Preserving the Numerical Range (Radius) on Triangular Matrices
Linear Operators Preserving the Numerical Range (Radius) on Triangular Matrices Chi-Kwong Li Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, USA. E-mail:
More informationEigenvalue analysis of constrained minimization problem for homogeneous polynomial
J Glob Optim 2016) 64:563 575 DOI 10.1007/s10898-015-0343-y Eigenvalue analysis of constrained minimization problem for homogeneous polynomial Yisheng Song 1,2 Liqun Qi 2 Received: 31 May 2014 / Accepted:
More informationHomogeneity of isosceles orthogonality and related inequalities
RESEARCH Open Access Homogeneity of isosceles orthogonality related inequalities Cuixia Hao 1* Senlin Wu 2 * Correspondence: haocuixia@hlju. edu.cn 1 Department of Mathematics, Heilongjiang University,
More informationAvailable online at J. Math. Comput. Sci. 4 (2014), No. 1, 1-9 ISSN:
Available online at http://scik.org J. Math. Comput. Sci. 4 (2014), No. 1, 1-9 ISSN: 1927-5307 ON -n-paranormal OPERATORS ON BANACH SPACES MUNEO CHŌ 1, KÔTARÔ TANAHASHI 2 1 Department of Mathematics, Kanagawa
More informationStrong convergence theorems for asymptotically nonexpansive nonself-mappings with applications
Guo et al. Fixed Point Theory and Applications (2015) 2015:212 DOI 10.1186/s13663-015-0463-6 R E S E A R C H Open Access Strong convergence theorems for asymptotically nonexpansive nonself-mappings with
More informationGEOMETRIC DISTANCE BETWEEN POSITIVE DEFINITE MATRICES OF DIFFERENT DIMENSIONS
GEOMETRIC DISTANCE BETWEEN POSITIVE DEFINITE MATRICES OF DIFFERENT DIMENSIONS LEK-HENG LIM, RODOLPHE SEPULCHRE, AND KE YE Abstract. We show how the Riemannian distance on S n ++, the cone of n n real symmetric
More informationELA ON A SCHUR COMPLEMENT INEQUALITY FOR THE HADAMARD PRODUCT OF CERTAIN TOTALLY NONNEGATIVE MATRICES
ON A SCHUR COMPLEMENT INEQUALITY FOR THE HADAMARD PRODUCT OF CERTAIN TOTALLY NONNEGATIVE MATRICES ZHONGPENG YANG AND XIAOXIA FENG Abstract. Under the entrywise dominance partial ordering, T.L. Markham
More informationOn a decomposition lemma for positive semi-definite block-matrices
On a decomposition lemma for positive semi-definite bloc-matrices arxiv:10.0473v1 [math.fa] Feb 01 Jean-Christophe ourin, Eun-Young Lee, Minghua Lin January, 01 Abstract This short note, in part of expository
More informationExtremal numbers of positive entries of imprimitive nonnegative matrices
Extremal numbers of positive entries of imprimitive nonnegative matrices Xingzhi Zhan East China Normal University Shanghai, China e-mail: zhan@math.ecnu.edu.cn A square nonnegative matrix A is said to
More informationBulletin of the Iranian Mathematical Society
ISSN: 7-6X (Print ISSN: 735-855 (Online Special Issue of the ulletin of the Iranian Mathematical Society in Honor of Professor Heydar Radjavi s 8th irthday Vol. 4 (5, No. 7, pp. 85 94. Title: Submajorization
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationOn non-expansivity of topical functions by a new pseudo-metric
https://doi.org/10.1007/s40065-018-0222-8 Arabian Journal of Mathematics H. Barsam H. Mohebi On non-expansivity of topical functions by a new pseudo-metric Received: 9 March 2018 / Accepted: 10 September
More informationOn Toader-Sándor Mean 1
International Mathematical Forum, Vol. 8, 213, no. 22, 157-167 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/imf.213.3344 On Toader-Sándor Mean 1 Yingqing Song College of Mathematics and Computation
More informationMoore-Penrose Inverses of Operators in Hilbert C -Modules
International Journal of Mathematical Analysis Vol. 11, 2017, no. 8, 389-396 HIKARI Ltd, www.m-hikari.com https//doi.org/10.12988/ijma.2017.7342 Moore-Penrose Inverses of Operators in Hilbert C -Modules
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More information