令A=U|A| 為一n乘n矩陣A的極分解。而D_t(A)=|A|^t U|A|^(1-t)對於t屬於[0,1], 稱之為A的廣義Aluthge轉換。本計畫的目的在於研究A的廣義Aluthge轉換的數值半徑其行為，特別的，對於以下的數值半徑不等式:(1) 2w(A) - ‖A‖ ≦ w(D_t(A)) ≦ w(A),(2) w(A) ≦ (‖A‖ + ‖A^2‖^(1/2))/2, and(3) w(A)^2 ≦ (‖A‖^2 + w(A^2))/2.我們將考慮這些不等式何時等式成立。對於每一個不等式，我們將給出等式成立的充分必要條件。 ;Let A=U|A| be the polar decomposition of an n-by-n matrix A. The generalized Aluthge transform is then given by D_t(A)=|A|^t U|A|^(1-t) for t in [0,1]. We investigate the behavior of the numerical radius w(D_t(A)). In particular, for the following numerical radius inequalities:(1) 2w(A) - ∥A∥ ≦ w(D_t(A)) ≦ w(A),(2) w(A) ≦ (∥A∥ + ∥A^2∥^(1/2))/2, and(3) w(A)^2 ≦ (∥A∥^2 + w(A^2))/2.We consider when these inequalities become equalities. We want to give complete characterizations of A for which the equality is attained.
