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

TL;DR

This paper develops operator versions of the Dunkl--Williams inequality related to the $p$-angular distance, providing new bounds and conditions for operators with invertible and non-invertible cases.

Contribution

It introduces novel operator inequalities involving $p$-angular distance, extending previous results to cases without invertibility assumptions and characterizing equality conditions.

Findings

Established inequalities for operators with invertible $|A|$ and $|B|$.

Extended inequalities to non-invertible cases using polar decompositions.

Derived conditions characterizing equality cases in the inequalities.

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*} |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 $0<p \leq 1$, 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 $$|(U|A|^{p}-V|B|^{p})|A|^{1-p}|^{2}\leq (1+t)|A-B|^{2}+(1+\frac{1}{t})||B|^{p}|A|^{1-p}-|B||^2 \,.$$ We obtain several equivalent conditions, when the case of equalities hold.