Inequalities for absolute value operators (Q1932618)

Let \(A, B \in B(H)\) with polar decompositions \(A = U|A|\), \(B = V|B|\), and let \(p, q > 1\) with \(1/p + 1/q = 1\). \textit{K.-S. Saito} and \textit{M. Tominaga} [Linear Algebra Appl. 432, No. 12, 3258--3264 (2010; Zbl 1195.26044)] proved that \(|(U -V)|A|\,|^2 - p|A - B|^2 + q(|A| - |B|)^2\). In the paper under review, the authors give refinements of the inequality above and prove some lower estimates for \(|(U - V)|A|\,|^2\). The presented inequalities are related to the Dunkl-Williams inequality; see \textit{M. S. Moslehian} et al. [Banach J. Math. Anal. 5, No. 2, 138--151 (2011; Zbl 1225.47022)].