| Project/Area Number |
15540148
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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, Sapporo Campus, Professor, 教育学部・札幌校, 教授 (80113661)
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| Co-Investigator(Kenkyū-buntansha) |
SAKURADA Kuninori Hokkaido Univ.of Education, Sapporo Campus, Professor, 教育学部・札幌校, 教授 (30002463)
OSADA Masayuki Hokkaido Univ.of Education, Sapporo Campus, Assistant Professor, 教育学部・札幌校, 助教授 (10107229)
HASEGAWA Izumi Hokkaido Univ.of Education, Sapporo Campus, Professor, 教育学部・札幌校, 教授 (50002473)
KOMURO Naoto Hokkaido Univ.of Education, Asahikawa Campus, Assistant Professor, 教育学部・札幌校, 助教授 (30195862)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2004: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2003: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | Aluthge transformation / λ-Aluthge transformation / Operetor radius / Weakly unitarily invariant norm / Norm Inequality / Numerical radius / Absolute norm / Numerical range / Spectrum / Aluthge transform / operator radius / ρ-contraction |
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| Research Abstract |
Let T∈B(Η) and T=UP be a polar decomposition of T. For 0<λ<1, we define the λ-Aluthge transformation of T by P^λUP^<1-λ>. In particular, for λ=1/2, ^^〜T:=P^<1/2>UP^<1/2> is called the Aluthge transformation of T.
For ρ>0 a operator T∈β(Η) is called a ρ-contraction if there is a Hilbert space Κ⊃Η and a unitary operator V on Κ such that Τ^n=ρQV^n|_Η(n=1,2,・・・) where Q is the orthogonal projection from Κ to Η. Also ρ-radius ω_ρ(T) of T is defined by ω_ρ(T)=inf{r>0:1/rT is a ρ-contraction}. It is known that for 0<ρ<2, ω_ρ(・) is a weakly unitarily invariant norm on β(Η). Then we have
Theorem 1.Let T=UP be a polar decomposition of T∈β(Η) and f be a polynomial. Then for 0<ρ and 0【less than or equal】λ【less than or equal】1, we have ω_ρ(f(P^λUP^<1-λ>))【less than or equal】ω_ρ(f(T)).
Let|||・||| be a weakly unitarily invariant norm on β(Η), that is, |||・||| satisfies |||VXV^*|||=|||Χ|||(V∈β(Η):unitary, Χ∈β(Η)), then we have
Theorem 2.Let T=UP be a polar decomposition of invertible operator T∈β(Η) and satisfies TB=BT for some B∈β(Η). Then for weakly unitarily in variant norm |||・||| on β(Η) we have |||P^λBUP^<1-λ>||【less than or equal】||BT||| (0【less than or equal】λ【less than or equal】1).
By using this result we have the following consequence.
Theorem 3.Let T=UP be a polar decomposition of invertible operator T∈β(Η). Then for weakly unitarily in variant norm |||・||| on β(Η) and any polynomial f we have |||f(P^λUP^<1-λ>||【less than or equal】||f(T)||| (0【less than or equal】λ【less than or equal】1).
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