コンピューターを活用した作用素不等式の開発とその応用

1996 Fiscal Year Annual Research Report

• Project/Area Number: 06640273
• 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 / Relative operator entropy / chaotic order / log majorization / order preserving inequality

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【greater than or equal】B【greater than or equal】0ならばA^α【greater than or equal】B^αただしα【not a member of】[0,1]、しかしA【greater than or equal】B【greater than or equal】0であってもp>1に対しては必ずしもA^α【greater than or equal】B^αとは限らない。この定理の不便さを解消するために我々はFuruta inequality(1987)を次のように確立した。A【greater than or equal】B【greater than or equal】0ならば(A^rA^pA^r)^<1/q>【greater than or equal】(A^rB^pA^r)^<1/q>がなりたつ、ただしr【greater than or equal】0,p【greater than or equal】0,q【greater than or equal】1かつ(1+2r)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 entropy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。
この他に通常の順序A【greater than or equal】B【greater than or equal】0とcaotic order log A【greater than or equal】logBとの間を連続的に結ぶ或るorderをFuruta inequalityを用いて図形的に説明することが可能であることなどが判明している。今後この作用素不等式の分野の発展にFuruta

Research Products

• [Publications] T.Furuta: "An extension of the Furuta inequality and Ando-Hiai log majorization." Linear Algebra and Its Applications. 219. 139-155 (1995)
• [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)
• [Publications] T.Furuta: "Generalized Aluthge transformation on p-hyponormal operators" Proceedings of American Mathematical Society. 124. 3071-3075 (1996)
• [Publications] T.Furuta: "Generaligization of Kosaki trace inequalities and related trace inequalities" Linear Algebra and Its Applications. 235. 153-161 (1996)
• [Publications] T.Furuta: "Characterizations of chaotic order via generalized Furuta inequality" Journal of Inequalities and Applications. 1. 11-24 (1997)
• [Publications] T.Fujii: "Norm inequalities in the Corach-Lorta-Recht thery and operaton means" Illinois Mathematical Journal. (to appear). (1997)
• [Publications] T.Furuta: "Applications of order preserving operator inequalities" General Inequalities 7,Birkhauser. (to appear). (1997)
• [Publications] M.Fujii: "Complements to the Furuta inequality,III" Mathematical Japan. (to appear). (1997)
• [Publications] T.Furuta: "Parallelism nelated to the inequality “A【greater than or equal】B【greater than or " Mathematical Japan. (to appear). (1997)
• [Publications] M.Fujii: "Operator means and covariance in probability theory" Mathematical Japan. (to appear). (1997)