Dunkl-Williams inequality for operators associated with $p$-angular distance

Abstract

We present several operator versions of the Dunkl--Williams inequality with respect to the $p$-angular distance for operators. More precisely, we show that if $A, B \in \mathbb{B}(\mathscr{H})$ such that $|A|$ and $|B|$ are invertible, $\frac{1}{r}+\frac{1}{s}=1$ $(r>1)$ and $p\in\mathbb{R}$, then

\begin{equation*} \left|\,A|A|^{p-1}-B|B|^{p-1}\,\right|^{2} \leq |A|^{p-1}\left(\,r|A-B|^{2}+s\left|\,|A|^{1-p}|B|^{p}-|B|\,\right|^2\,\right)|A|^{p-1}.%\nonumber \end{equation*}

In the case that $0 < p \leq 1$, we remove the invertibility assumption and show that if $A=U|A|$ and $B=V|B|$ are the polar decompositions of $A$ and $B$, respectively, $t>0$, then

$$\left|\,\left(U|A|^{p}-V|B|^{p}\right)|A|^{1-p}\,\right|^{2}\leq \left(1+t\strut\right)|A-B|^{2}+\left(1+\frac{1}{t}\right) \left| |B|^{p}|A|^{1-p}-|B| \right|^2 .$$

We obtain several equivalent conditions, when the case of equalities hold.