1998 Fiscal Year Final Research Report Summary
Research of inequalities of matrix algebra using analytic method
| Project/Area Number |
09640143
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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 | HOKKAIDO UNIVERSITY OF EDUCATION |
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Principal Investigator |
OSADA Masayuki Sapporo College, Assistant Professor, 教育学部札幌校, 助教授 (10107229)
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| Co-Investigator(Kenkyū-buntansha) |
KOMURO Naoto Asahikawa College, Assistant Professor, 教育学部旭川校, 助教授 (30195862)
NISIMURA Jun-ichi Sapporo College, Assistant Professor, 教育学部札幌校, 助教授 (00025488)
HASEGAWA Izumi Sapporo College, Professor, 教育学部札幌校, 教授 (50002473)
SAKURADA Kuninori Sapporo College, Professor, 教育学部札幌校, 教授 (30002463)
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| Project Period (FY) |
1997 – 1998
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| Keywords | unitary rho dilation / operator radius / Schur product / Holder type inequality |
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| Research Abstract |
1 It is considered many (quasi-)norms on the space of all n*n complex matrices. The rho radius omega_<rho>(.) is one of them. In this research, we will characterize this norm in the case of 2 * 2 matrices by using the theory of complex functions, and apply it to calculation of the rho-radius for some matrices. The main result is the following : Let a, b *D : ={z*C||z|<less than or equal> 1}. Then for rho <greater than or equal> A=[a c 0 b] is a rho-contraction if a
|c|<@D12@>D1+|a-b|<@D12@>D1 <less than or equal> inf|<@D7{rho+(rho)azeta}{rho+(-rho)bzeta}-ab|zeta<@D12@>D1(/)5 pizet
2 Besides of usual mulpiplication, entrywise product (it is called Schur product) or matrices is considered. We had Holder type inequality for p-radius with respect to Schur product as following : Let 0<rho0<rho1<*. Then for any non-negative matrices A, B,
omega_<rho>(A^<(alpha)> o B^<(1-alpha>) <less than or equal> omega_<rho0>(A)^<alpha>_<rho1>(B)^<1-alpha> (0<alpha<1 ; alphazeta0+(1-alpha)zeta1).
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