Operator inequalities related to p-angular distances

Abstract

For any nonzero elements x, y in a normed space X, the angular and skew-angular distance is respectively defined by α[x, y] = x x − y y and β[x, y] = x y − y x. Also inequality α ≤ β characterizes inner product spaces. Operator version of α p has been studied by Pečarić, Rajić, and Saito, Tominaga, and Zou et al.In this paper, we study the operator version of p-angular distance β p by using Douglas' lemma. We also prove that the operator version of inequality α p ≤ β p holds for normal and double commute operators. Some examples are presented to show essentiality of these conditions.