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Establishment and its application of operator inequalities by using computer

Research Project

Project/Area Number 06640273
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionScience University of Tokyo

Principal Investigator

FURUTA Takayuki  Science University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (40007612)

Project Period (FY) 1994 – 1996
Project Status Completed (Fiscal Year 1996)
Budget Amount *help
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1996: ¥100,000 (Direct Cost: ¥100,000)
Fiscal Year 1995: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1994: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsLowner-Heinz inequality / Furuta inequality / relative operator entropy / chaotic order / log majorization / order preserving inequality / log majorization / The Furuta inguality / Lowner-Heinz therkm / relative operatov entropy / positive operaton / trace inequality / monotone decreasing / The Furuta Jnequality / Lowner-Heing thecxem / relative operator entropy / trace norm inequality / positive operator / monotone clecreasing
Research Abstract

In what follows, a capital letter means a bounded linear operator on a complex Hilbert space H.The very famous Lowner-Heinz theorem (1934) asserts that if A<greater than or equal>B<greater than or equal>0 ensures A^<alpha><greater than or equal>B^<alpha> for any alpha <not a member of> [0,1]. But A<greater than or equal>B<greater than or equal>0 does not always hold A^p<greater than or equal>B^p for p>1 in general. As an extension of the Lowner-Heinz theorem, we established the Furuta Inequality (1987) which reads as follows. If A<greater than or equal>B<greater than or equal>0, then for each r<greater than or equal>0 (A^rA^pA^r)^<1/q><greater than or equal>(A^rB^pA^r)^<1/q> hold for p<greater than or equal>0 and q<greater than or equal>1 with (1+2r) q<greater than or equal>p+2r. The Furuta inequality yield
Recently we obtained a lot of applications by using Furuta inequality in the following three branches (a) operator inequalities, (b) norm inequalities and (c) operator equations. We cite some of them as follows : (alpha_1) Applications to relative operator entropy, (alpha_2) Applications to Ando-Hiai log majorization, (alpha_3) Generalized Aluthge transformation, (b_1) Heinz-Kato inequality, (b_2) Kosaki trace inequality, (c_1) Pedersen-Takesaki operator equation.

Report

(4 results)
  • 1996 Annual Research Report   Final Research Report Summary
  • 1995 Annual Research Report
  • 1994 Annual Research Report
  • Research Products

    (44 results)

All Other

All Publications (44 results)

  • [Publications] T.Furuta: "An extension of the Furuta inequality and Ando-Hiai log majorization" Linear Algebra and Its Applications. 219. 139-155 (1995)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Characterization of operators satisfying logA【greater than or equal】logB and its app " 15th Operator Theory Conference,IMAR. 1. 101-113 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Generalized Aluthge transformation on p-hyponormal operators" Proceedings of american Mathematical Society. 124. 3071-3075 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Generalization of Kosaki trace inequalities and related trace inequalities" Linear Algebra and Its Applications. 235. 153-161 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Characterizations of chaotic orden via generalized Furuta inegnality" Journal of Inequalities and Applications. 1. 11-24 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M.Fujii: "Norm inequalities in the Corach-Porta-Recht theory and operator means" Illinois Mathematical Journal. (to appear). (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Applications of order preserving operator inequalities" General Inequalities 7,Birkhauser. (to appear). (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M.Fujii: "Complements to the Furuta inequality,III" Mathematical Japon. (to appear). (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Parallelism selated to the inequality″A【greater than or equal】B【greater than or " Mathematical Japon. (to appear). (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] M.Fujii: "Operator means and covariance in probability theory" Mathematical Japon. (to appear). (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "An extension of the Furuta inequality and Ando-Hiai log majorization" Linear Alg.and Its Appl.219. 139-155 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Characterizations of operators satisfying logA<greater than or equal>logB and its app " Proc.of 15th Operator Theory Conference IMAR. 101-113 (1995)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Generalizations of Kosaki trace inequalities and related trace inequalities on chaotic order" Linear Alg.and Its Appl.235. 153-161 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Generalized Aluthge transformation on p-hyponormal operators" Proc.Amer.Math.Soc.124. 3071-3075 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Characterizations of chaotic order via generalized Furuta inequality" J.Inequalities and Applicatoins. 1. 11-24 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Applications of order preserving operator inequalities to a generalized relative operator entropy" General Inequalities 7, Birkhauser.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Complements to the Furuta inequality, III,(with M.Fujii and E.Kamei)" Math.Japonica. 45. 25-32 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Norm inequalities in the Corach-Porta-Recht theory and operator means, (with M.Fujii and R.Nakamoto)." Illinois J.Math.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Parallelism related to the inequality "A<greater than or equal>B<greater than or equal>0 ensures (A^<r/2>A^pA^<r/2>)^<(1+r)/><greater than or equal>(A^<r/2>B^pA^<r/2>)^<(1+r)/> for p<greater than or equal>1 and r<greater than or equal>0"" Math.Japonica.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "Operator inequalities and covariance in noncommutative probability, (with M.Fujii, R.Nakamoto and S.Takahasi)" Math.Japonica.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1996 Final Research Report Summary
  • [Publications] T.Furuta: "An extension of the Furuta inequality and Ando-Hiai log majorization." Linear Algebra and Its Applications. 219. 139-155 (1995)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Furuta: "Characterizatin of operatrvs satisfying logA【greater than or equal】logB and its app " 15th Poerator Theory Conference IMAR. 1. 101-113 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Furuta: "Generalized Aluthge transformation on p-hyponormal operators" Proceedings of American Mathematical Society. 124. 3071-3075 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Furuta: "Generaligization of Kosaki trace inequalities and related trace inequalities" Linear Algebra and Its Applications. 235. 153-161 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Furuta: "Characterizations of chaotic order via generalized Furuta inequality" Journal of Inequalities and Applications. 1. 11-24 (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Fujii: "Norm inequalities in the Corach-Lorta-Recht thery and operaton means" Illinois Mathematical Journal. (to appear). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Furuta: "Applications of order preserving operator inequalities" General Inequalities 7,Birkhauser. (to appear). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] M.Fujii: "Complements to the Furuta inequality,III" Mathematical Japan. (to appear). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T.Furuta: "Parallelism nelated to the inequality “A【greater than or equal】B【greater than or " Mathematical Japan. (to appear). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] M.Fujii: "Operator means and covariance in probability theory" Mathematical Japan. (to appear). (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] T. Furuta: "Extension of the Furuta inequality and Ando-Hiai log majorijation" Linear Algebra and Its Applications. 219. 139-155 (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] T. Furuta: "Generalizations of Kosaki tvace ivegualities and related tvace inzualities on chastic oeder" Linear Algebra and Its Applications. (掲載予定)(印刷中). (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] T. Furuta: "Characterizatin of operators satisfying logA【greater than or equal】logB and its app " Operator Jheory. (印刷中). (1995)

    • Related Report
      1995 Annual Research Report
  • [Publications] T. Furuta: "Generalized Aluthge transformation on p-hyponormal operators" Proceedings of American Mathematical Soiety. (掲載予定). (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] M. Fujii: "Complenents to the Furuta inequality, III" Mathematica Japonica. (掲載予定). (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] T. Furuta: "Parallelism related to the inequality “A【greater than or equal】B【greater than or equal】0 onsures(A^<v/2>A^pA^<v/2>)(1+v)/(p+v)【greater than or equal】(A^<v/2>B^pA^<v/2>)(1+v)/(p+v)"" Mathematica Japonica. (掲載予定). (1996)

    • Related Report
      1995 Annual Research Report
  • [Publications] T.Furuta: "A Note on the Arithmetic-Geometric Mean inequality for Every Unitary Jmasiant Matrix Norm" Linear Algebra and Its Applicatins. 208/209. 223-228 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Furuta: "An extension of the Heing-Kato Theovem" Pvoceedings of the American Mathematical Society. 120. 785-787 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Furuta: "Determinant tyne generalijation of the Heing-Kato Reovorm via the Furuta inequality" Proceedings of the American Mathematical Society. 120. 223-231 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Furuta: "Extension of the Furuta inequality and log majorigation by Ando-Hiai" Linesr Algebra and Its Applications. (掲載予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Furuta: "Characterigation of operators satisfying logA【greater than or equal】logBand its app " Operator Theory,Birlshaueer. (掲載予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] T.Furuta: "Generaligation of Kosaki tvace inequalities and nelated tvace inequalities on chaotic order" Linear Algebra and Its Applications. (掲載予定). (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] M.Fujii: "An application of the Furuta inequality to operator inequalities on chaotic order" Mathematica Japonica. 40. 317-321 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] M.Fujii: "Complenrento to the Furuta inequality" 70. 239-242 (1994)

    • Related Report
      1994 Annual Research Report

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Published: 1994-04-01   Modified: 2016-04-21  

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