Research of Matrix Inequalities and Norm Inequalities

点击翻译按钮获取中文摘要

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).