Mathematica Eterna
Open Access

Abstract

Generalized Dunkl-Williams inequality

Bahram Dastourian, Khodarahm Marzban and Nasiballah Mohammadi

Dunkl and Williams showed that u kuk − v kvk ≤ 4ku − vk kuk + kvk . for any nonzero elements u, v in a normed linear space X. Peˇcari´c and Raji´c gave a refinement and, moreover, a generalization to operators A, B belong to the algebra B(H) of all bounded linear operators on a separable complex Hilbert space H, such that |A|, |B| are invertible as follows: |A|A| −1 − B|B| −1 | 2 ≤ |A| −1 (r|A − B| 2 + s(|A| − |B|) 2 )|A| −1 , where r, s > 1 with 1 r + 1 s = 1. In this paper, we generalized this inequality in the framework of Hilbert C*-modules. As a consequence we investigate this inequality without assumption of the invertibility of the absolute value of operator B.