A generalization of the Powers-Størmer-Ogata inequality

RESUMO

We show that for any positive operator monotone function $f$ the inequality $\int_{[0,\infty)} f(t) \, \mathrm{d} \left\lVert E^{\Delta_{\varphi,\psi}}(t) \xi_\psi \right\rVert^2 + f'_\infty \varphi(1-s(\psi)) \geq \frac{f(1)}{2} \Bigl( \varphi(1) + \psi(1) - \lVert \varphi - \psi \rVert \Bigr)$ holds, where $\varphi, \psi \in \mathcal{M}_{*,+}$ are two normal positive linear functionals on a von Neumann algebra $\mathcal{M}$, and $\Delta_{\varphi,\psi}$ is the associated relative modular operator. Choosing $f(t) = t^s$ with $s \in [0,1]$, the Powers-Størmer-Ogata inequality is recovered.