1998 Fiscal Year Annual Research Report

ヒルベルト空間上の作用素不等式の開発とその応用

Research Project

Project/Area Number	09640225
Research Institution	Science University of Tokyo
Principal Investigator	古田孝之東京理科大学, 理学部, 教授 (40007612)
Keywords	Lowner-Heinz inequality / Furuta inequality / generalized Furuta inequality / log majorization / order preserving inequality / chaotic order / relative operator entropy / positive definite operator
Research Abstract	ヒルベルト空間上の有界線形作用素の順序保存に関する有名なLowner-Heinz(1934)の定理は次のことである。A≧B≧0ならばA^p≧B^pただし1≧p≧0、しかしA≧B≧0であってもp>1に対しては必ずしもA^p≧B^pとは限らない。このため応用の上で大変不便であったので、それを解消するために我々はFuruta inequality(1987)を次のように確立した。 If A【greater than or equal】B【greater than or equal】0,then for (*)(A^<r/2>A^pA^<r/2>)^<1/q>【greater than or equal】(A^<r/2>B^pA^<r/2>)^<1/q> holds for p【greater than or equal】0 and q【greater than or equal】1 with(1+r)q 最近このFuruta inequalityの応用が多方面において見つかっている。それは主に次の三分野においてである。(a)作用素不等式、(b)ノルム不等式、(c)作用素方程式。これらの応用のうち主なものを次に述べてみよう。(a_1)relative operator entoropyへの応用(a_2)Ando-Hiaiのlog majorizationへの応用(a_3)Generalized Aluthge transformation(b_1)Heinz-Kato inequalityの一般化(b_2)Kosaki trace inequalityの一般化(c_1)Pedersen-Takesakiの作用素方程式の一般化などである。最近Furuta inequalityの一般化が得られたが、これはAndo-Hiaiによるlog majorizationと同値な作用素不等式とFuruta inequalityを補間する作用素不等式である。このGeneralizedFuruta inequalityのたった1頁の証明が得られた。更にこれらに関連した作用素関数との同値性が示された。今後益々Furuta inequalityの応用や発展が期待される。

Research Products
(23 results)

All Other

All Publications (23 results)

[Publications] M.Fuji: "Complements to the Furuta irequality,III" Math.Japonica. 45. 25-32 (1997)
[Publications] T.Furuta: "Chavacterizations of chaotic order via generalized Furuta ineqvality" J.Ineqvalities & Applications. 1. 11-24 (1997)
[Publications] T.Furuta: "Parallelism related to the iveqvality“AZBZO ensures……"" Math.Japonica. 45. 203-209 (1997)
[Publications] T.Furuta: "Extensions of Holder McCarthy and Kantroich inequalities……" Roc.Japan Acad.73. 38-41 (1997)
[Publications] T.Furuta: "Applications of order presering operator inequalities to a generalzed…" General Inequalities 7,Birkhauser. 123. 65-76 (1997)
[Publications] T.Furuta: "Farther extensions of Aluthge transformation on p-hyronomal" Integral Equations and Operatoy Theory. 29. 122-125 (1997)
[Publications] M.Fujii: "Operater inequalities and covariance in noncommutative…" Math.Japonica. 46. 317-320 (1997)
[Publications] T.Furuta: "Equivalance relations among Reid,Lowner-Heinz and Heing-Kato…" Integral Equations and Operator Theory. 29. 1-9 (1997)
[Publications] T.Furuta: "Operator inequalities associated with Holder McCarthy and……" J.Inequalities & Applications. 2. 137-148 (1998)
[Publications] T.Furuta: "Operator functions inplying generalized Furuta inequality" Mathematical Inequalities & Applications. 1. 123-130 (1998)
[Publications] T.Furuta: "Goneralized means and convoxity of inversion for positive operators" Aner.Math.Monthly. 105. 258-259 (1998)
[Publications] J.I.Fujii: "An operater version of the Wilf-Diaz-Matcalf inequality" Nihonkai Math.J.9. 47-52 (1998)
[Publications] T.Furuta: "A decreasing operator function associted with the Furuta ioquality" Proc.Aner.Math.Soc. 126. 2427-2432 (1998)
[Publications] T.Furuta: "Order preserving operator inequalities via Furute inequality" Math.Japonica. 48. 471-476 (1998)
[Publications] T.Furuta: "Order preserving operater function via Furuta inequality" in press in Proc.d IWOTA 96.
[Publications] T.Furuta: "Parmetric operator function via Furuta inquality" Scientiae Mathematical. 1. 1-5 (1998)
[Publications] T.Furuta: "Operator functions involving order preserving inequalities" Scientiae Mathematical. 1. 1-7 (1998)
[Publications] T.Furuta: "Equiualance relation among Furuta-type inequalities with negative" Science Mathematical. 1. 223-229 (1998)
[Publications] T.Furuta: "Equivalence relation between generalized Furuta inequality and…" Scientiae Mathematical. 1. 257-259 (1998)
[Publications] T.Furuta: "On a conjecture related to Furuta type inequalities with …" Nihonkai Mash.J.9. 213-218 (1998)
[Publications] T.Furuta: "Sinplified proof of an order preserving operator inequality" Proc.Japan Acad.74. 114(1) (1998)
[Publications] T.Furuta: "A subclass of paranormal operators including class of…" Scientiae Mathematical. to appear.
[Publications] J.I.Fujii: "Simplified prirof of chaotic order via Specht‘s ratio" Scientiae Mathemratical. to appear.

1998 Fiscal Year Annual Research Report

ヒルベルト空間上の作用素不等式の開発とその応用

Principal Investigator

古田 孝之 東京理科大学, 理学部, 教授 (40007612)

Research Products

[Publications] M.Fuji: "Complements to the Furuta irequality,III" Math.Japonica. 45. 25-32 (1997)

[Publications] T.Furuta: "Chavacterizations of chaotic order via generalized Furuta ineqvality" J.Ineqvalities & Applications. 1. 11-24 (1997)

[Publications] T.Furuta: "Parallelism related to the iveqvality“AZBZO ensures……"" Math.Japonica. 45. 203-209 (1997)

[Publications] T.Furuta: "Extensions of Holder McCarthy and Kantroich inequalities……" Roc.Japan Acad.73. 38-41 (1997)

[Publications] T.Furuta: "Applications of order presering operator inequalities to a generalzed…" General Inequalities 7,Birkhauser. 123. 65-76 (1997)

[Publications] T.Furuta: "Farther extensions of Aluthge transformation on p-hyronomal" Integral Equations and Operatoy Theory. 29. 122-125 (1997)

[Publications] M.Fujii: "Operater inequalities and covariance in noncommutative…" Math.Japonica. 46. 317-320 (1997)

[Publications] T.Furuta: "Equivalance relations among Reid,Lowner-Heinz and Heing-Kato…" Integral Equations and Operator Theory. 29. 1-9 (1997)

[Publications] T.Furuta: "Operator inequalities associated with Holder McCarthy and……" J.Inequalities & Applications. 2. 137-148 (1998)

[Publications] T.Furuta: "Operator functions inplying generalized Furuta inequality" Mathematical Inequalities & Applications. 1. 123-130 (1998)

[Publications] T.Furuta: "Goneralized means and convoxity of inversion for positive operators" Aner.Math.Monthly. 105. 258-259 (1998)

[Publications] J.I.Fujii: "An operater version of the Wilf-Diaz-Matcalf inequality" Nihonkai Math.J.9. 47-52 (1998)

[Publications] T.Furuta: "A decreasing operator function associted with the Furuta ioquality" Proc.Aner.Math.Soc. 126. 2427-2432 (1998)

[Publications] T.Furuta: "Order preserving operator inequalities via Furute inequality" Math.Japonica. 48. 471-476 (1998)

[Publications] T.Furuta: "Order preserving operater function via Furuta inequality" in press in Proc.d IWOTA 96.

[Publications] T.Furuta: "Parmetric operator function via Furuta inquality" Scientiae Mathematical. 1. 1-5 (1998)

[Publications] T.Furuta: "Operator functions involving order preserving inequalities" Scientiae Mathematical. 1. 1-7 (1998)

[Publications] T.Furuta: "Equiualance relation among Furuta-type inequalities with negative" Science Mathematical. 1. 223-229 (1998)

[Publications] T.Furuta: "Equivalence relation between generalized Furuta inequality and…" Scientiae Mathematical. 1. 257-259 (1998)

[Publications] T.Furuta: "On a conjecture related to Furuta type inequalities with …" Nihonkai Mash.J.9. 213-218 (1998)

[Publications] T.Furuta: "Sinplified proof of an order preserving operator inequality" Proc.Japan Acad.74. 114(1) (1998)

[Publications] T.Furuta: "A subclass of paranormal operators including class of…" Scientiae Mathematical. to appear.

[Publications] J.I.Fujii: "Simplified prirof of chaotic order via Specht‘s ratio" Scientiae Mathemratical. to appear.

古田孝之東京理科大学, 理学部, 教授 (40007612)