| Project/Area Number |
11640146
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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 |
Basic analysis
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| Research Institution | HOKKAIDO UNIVERSITY OF EDUCATION |
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Principal Investigator |
OKUBO Kazuyoshi Hokkaido Univ.of Education, 教育学部・札幌校, 教授 (80113661)
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| Co-Investigator(Kenkyū-buntansha) |
NISHIMURA Junichi Hokkaido Univ.of Education, 教育学部・札幌校, 助教授 (00025488)
OSADA Masayuki Hokkaido Univ.of Education, 教育学部・札幌校, 教授 (10107229)
SAKURADA Kuninori Hokkaido Univ.of Education, 教育学部・札幌校, 教授 (30002463)
KOMURO Naoto Hokkaido Univ.of Education, 教育学部・旭川校, 助教授 (30195862)
HASEGAWA Izumi Hokkaido Univ.of Education, 教育学部・旭川校, 教授 (50002473)
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| Project Period (FY) |
1999 – 2000
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| Project Status |
Completed (Fiscal Year 2000)
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| Budget Amount *help |
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1999: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | Trace / Numerical range / Numerical radius / Spectrum / Square root of matrix / rank reducing / Approximant / Operator radius / Spectral norm / positive semidefinite / Majorization / Golden-Thompson inequality / numerical range |
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| Research Abstract |
Let M_n be the algebra of all n×n complex matrices. The spectral norm, which is the operator norm of linear operator on C^n, is the one of the important norms on C^n. Another important norm is the numerical radius in M_n, which is of measurement of magnitude of a particle in quantum mechanics. The operator radius ω_ρ ( )(ρ>0) were defined by J.A.R.Holbrook. These radius interpolate among the spectral norm, the numerical radius and the spectral radius. We give an explicit description of all matrices A∈M_2 such that ω_ρ(A)【less than or equal】1. This description leads to the formulas for ρ-radii when the eigenvalues of such matrices either have equal absolute values or (mod π) argument.
Trace inequalities for multiple products of powers of two matrices are discussed via the method of log majorization. For instance, the trace inequality
|Tr (A^<p1>B^<q1>A^<p2>B^<q2>…A^<pK>B^<qK>|【less than or equal】Tr (AB)
is obtained for positive semidefinite matrices A, B and p_i, q_i【greater than or equal】0 with p1+…+pK=q1+…+qK=1 under some additional condition.
For A∈M_n (C), let W (A) denote the numerical range of A.It is shown that if W (A)∩(-∞, 0)=φ, then A has a unique square root B∈M_n (C) with W (B) in the closed right half plane.
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