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 $A,B\in \mathbb{B}(\mathcal{H})$ such that $|A|$ and $|B|$ are invertible, $\frac{1}{r}+\frac{1}{s}=1(r>1)$ and $p\in \mathbb{R}$, then \begin{equation*} |A|A|^{p-1}-B|B|^{p-1}|^{2} \leq |A|^{p-1}(r|A-B|^{2}+s||A|^{1-p}|B|^{p}-|B||^2)|A|^{p-1}.%\nonumber \end{equation*} In the case that $0<p\le 1$, 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 $$|(U|A{|}^p-V|B{|}^p)|A{|}^{1-p}{|}^2\le (1+t)|A-B{|}^2+(1+\frac{1}{t})||B{|}^p|A{|}^{1-p}-|B|{|}^2.$$ We obtain several equivalent conditions, when the case of equalities hold.

11 pages, to appear in Nihonkai Math J

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