Project/Area Number  06640273 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
解析学

Research Institution  Science University of Tokyo 
Principal Investigator 
FURUTA Takayuki Science University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (40007612)

Project Fiscal Year 
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)

Keywords  LownerHeinz inequality / Furuta inequality / relative operator entropy / chaotic order / log majorization / order preserving inequality / Relative operator entropy / log majorigation / The Furuta inguality / LownerHeinz therkm / relative operatov entropy / positive operaton / trace inequality / monotone decreasing / The Furuta Jnequality / LownerHeing thecxem / 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 LownerHeinz 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 LownerHeinz 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 AndoHiai log majorization, (alpha_3) Generalized Aluthge transformation, (b_1) HeinzKato inequality, (b_2) Kosaki trace inequality, (c_1) PedersenTakesaki operator equation. この他に通常の順序A【greater than or equal】B【greater than or equal】0とcaotic order logA【greater than or equal】log B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
