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Research of Matrix Inequalities and Norm Inequalities

矩阵不等式和范数不等式研究

基本信息

批准号：
15540148
负责人：
OKUBO Kazuyoshi
金额：
$ 2.18万
依托单位：
HOKKAIDO UNIVERSITY OF EDUCATION
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540148/
关键词：
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

项目摘要

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 haveTheorem 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 haveTheorem 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).

设T∈B（H），T=UP是T的一个极分解.当0<λ<1时，我们定义T的λ-λ-thge变换为P^λUP^<1-λ>。特别地，当λ=1/2时，称T为T的ρ-压缩变换。当ρ>0时，称算子T∈β（H）为ρ-压缩，如果存在Hilbert空间λ ∈ H和λ上的酉算子V，使得T ^n=ρQV^n|其中Q是从n到H的正交投影。T的ρ-半径ω_ρ（T）定义为：ω_ρ（T）=inf{r>0：1/rT是ρ-压缩集}.已知当0<p <2时，ω_p（·）是β（H）上的弱酉不变范数.定理1.设T=UP是T∈β（H）的极分解，f是多项式。则对于0<ρ和0[小于或等于]λ[小于或等于]1，我们有ω_ρ（f（P^λUP^<1-λ>））[小于或等于]ω_ρ（f（T））。令|||·||| 是β（H）上的弱酉不变范数，即， |||·||| 满足 ||| VXV^*||| =|||Χ|||（V∈β（H）：酉，τ ∈β（H）），则有定理2.设T=UP是可逆算子T∈β（H）的极分解，且对某个B∈β（H）满足TB=BT.则对于弱酉在变范 |||·||| 关于β（H），我们有 ||| P^λBUP^<1-λ>||[小于或等于]|| BT||| （0[小于或等于]λ[小于或等于]1）.利用这个结果，我们得到如下结果：定理3.设T=UP是可逆算子T∈β（H）的极分解.则对于弱酉在变范 |||·||| 在β（H）和任何多项式f上， ||| f（P^λUP^<1-λ>||[小于或等于]|| f（T）||| （0[小于或等于]λ[小于或等于]1）。