Estimates of operator moduli of continuity (Q647594)

The operator modulus of continuity \(\Omega_f\) of a function \(f\) on \(\mathbb R\) is defined by \[ \Omega_f (\delta )=\sup \|f(A)-f(B)\|,\quad \delta >0, \] where the supremum is taken over all selfadjoint operators \(A,B\) such that \(\|A-B\|\leq \delta\). Estimates of \(\Omega_f\) imply estimates for differences of functions of selfadjoint operators. The authors improve some estimates of \(\Omega_f\) proved in their earlier paper [Adv. Math. 224, No. 3, 910--966 (2010; Zbl 1193.47017)]. This leads, in particular, to a sharp estimate for \(\|\,|S|-|T|\,\|\), where \(S,T\) are bounded operators, \(|S|=(S^*S)^{1/2}\), improving that by \textit{T. Kato} [Proc. Japan Acad. 49, 157--160 (1973; Zbl 0301.47006)]. They find a lower estimate for \(\Omega_f\) for a certain specific function \(f\), thus proving sharpness of some estimates obtained in earlier papers. Sharp estimates of \(\|f(A)-f(B)\|\) are found also in the case where the spectrum of \(A\) is a finite set.