| Co-Investigator(Kenkyū-buntansha) |
HASEGAWA Izumi Hokkaido Univ. of Education, 教育学部・旭川校, 教授 (50002473)
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)
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| Research Abstract |
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 range
Let 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)).
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