Operator inequalities related to angular distances

Abstract

For any nonzero elements $x,y$ in a normed space $X$, the angular and skew-angular distance is respectively defined by $\alpha[x,y]=\left\| \frac{x}{\|x\|}-\frac{y}{\|y\|}\right\|$ and $\beta[x,y]=\left\| \frac{x}{\|y\|}-\frac{y}{\|x\|}\right\|$. Also inequality $\alpha\leq \beta$ characterizes inner product spaces. Operator version of $\alpha$ has been studied by Pe\v{c}ari\'{c}, Raji\'{c}, and Saito, Tominaga, and Zou et al.\\
In this paper, we study the operator version of $\beta$ by using Douglas' lemma. We also prove that the operator version of inequality $\alpha\leq \beta$ holds for normal and double commute operators. Some examples are presented to show essentiality of these conditions.