1996 Fiscal Year Final Research Report Summary
Establishment and its application of operator inequalities by using computer
| Project/Area Number |
06640273
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Science University of Tokyo |
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Principal Investigator |
FURUTA Takayuki Science University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (40007612)
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| Project Period (FY) |
1994 – 1996
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| Keywords | Lowner-Heinz inequality / Furuta inequality / relative operator entropy / chaotic order / log majorization / order preserving inequality |
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| 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.
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