Research of Matrix Inequalities and Norm Inequalities on Matrices Algebra

Let T∈B(H) 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>U P^<1/2> is called the Aluthge transformation of T (See [A]). the numerical rangeW(T) of T is defined by W(T) = {<x, Tx> || |x|| = 1}. RecentlyYamazaki and Wu showed that W(T^^~)⊂W(T)，then w（T^^~）【less than or equal】w(T) for the numerical radius w（·）. In this研究我们扩展这些results.我们给予following results as the parts of我们的作品.(i) On the generalizednumerical rangeLet T，C be n × n complex matrices. The C-numerical range of T is defined by w_c (T) = {tr(CU^*AU) | U；If C is a Hermitian matrix or a rank one matrix，then the following inclusion relation holds: w_c (f(T^^~))⊂w_c (f(T))for f(z) is a complexpolynomial.(ii) The inequality on semi-norms.Let A∈B（Η），and | | |和| | | be a semi-norm on b(Η). if | | |和| | | satisfy欧洲γ,|||X|||【less than or equal】γ ||X|| (X∈B(H))，||| s ^* xs |||【less than or equal】|| s ||^ 2·||| x ||| (x,S∈B（Η）). Then for 0【less than or equal】λ【less than or equal】1，| | | f (a _颗心)| | |【less than or equal】max {| | | f (a) | | |,|||U^*·f(A)·U + f(0)(I-U^*U)|||} for any polynomial f. From this fact，we can prove that for the operator radii w_ρ(ρ > 0)，0【less than or equal】λ【less than or equal】1,and polynomial fwe have w_ρ(f(a_λ))【less than or equal】w_ρ(f(A))。

Let T ∈ B(H) 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>U P^<1/2> is called the Aluthge transformation of T (See [A]). The numerical range W(T) of T is defined by W(T) := {<x, Tx> | ||x|| = 1}. Recently, Yamazaki and Wu showed that W(T^^~) ⊂ W(T), then w(T^^~) 【less than or equal】 w(T) for the numerical radius w(・). In this research we extended these results. We give the following results as the parts of our works.(i) On the generalized numerical rangeLet T, C be n × n complex matrices. The C-numerical range of T is defined by W_C(T) := {tr(CU^*AU) | U; unitary}.If C is a Hermitian matrix or a rank one matrix, then the following inclusion relation holds:W_C(f(T^^~)) ⊂ W_C(f(T))for f(z) is a complex polynomial.(ii) The inequality on semi-norms.Let A ∈ B(Η), and |||・||| be a semi-norm on B(Η). If |||・||| satisfy ∃γ, |||X||| 【less than or equal】 γ ||X|| (X ∈ B(H)), |||S^*XS||| 【less than or equal】 ||S||^2・|||X||| (X, S ∈ B(Η)). Then for 0 【less than or equal】 λ 【less than or equal】 1, |||f(A_λ)||| 【less than or equal】 max {|||f(A)|||, |||U^* ・f(A) ・ U + f(0)(I-U^*U)|||} for any polynomial f. From this fact, we can prove that for the operator radii w_ρ(・) (ρ > 0), 0 【less than or equal】 λ 【less than or equal】 1, and polynomial f, we have w_ρ(f(A_λ)) 【less than or equal】 w_ρ(f(A)).