ABSTRACT

Let Φ be a unital positive linear map and let A be a positive invertible operator. We prove that there exist partial isometries U and V such that $|\mathrm{\Phi }\left(f\right(A\left)\right)\mathrm{\Phi }\left(A\right)\mathrm{\Phi }\left(g\right(A\left)\right)|\le U^{\ast }\mathrm{\Phi }\left(f\right(A\left)Ag\right(A\left)\right)U$ and $\left|\mathrm{\Phi }{\left(f\left(A\right)\right)}^{-r}\mathrm{\Phi }(A{)}^r\mathrm{\Phi }{\left(g\left(A\right)\right)}^{-r}\right|\le V^{\ast }\mathrm{\Phi }\left(f\left(A{)}^{-r}{A}^{r}g\right(A{)}^{-r}\right)V$ hold under some mild operator convex conditions and some positive numbers r. Further, we show that if $f^2$ is operator concave, then $|\mathrm{\Phi }\left(f\right(A\left)\right)\mathrm{\Phi }\left(A\right)|\le \mathrm{\Phi }\left(Af\right(A\left)\right).$ In addition, we give some counterparts to the asymmetric Choi–Davis inequality and asymmetric Kadison inequality. Our results extend some inequalities due to Bourin–Ricard and Furuta.

摘要

设 Φ 为单位正线性映射，设A为正可逆算子。我们证明存在部分等距U和V使得$|\mathrm{\Phi }\left(F\right(\mathrm{一个}\left)\right)\mathrm{\Phi }\left(\mathrm{一个}\right)\mathrm{\Phi }\left(G\right(\mathrm{一个}\left)\right)|\le {ü}^{*}\mathrm{\Phi }\left(F\right(\mathrm{一个}\left)\mathrm{一个}G\right(\mathrm{一个}\left)\right)ü$和$\left|\mathrm{\Phi }{\left(F\left(\mathrm{一个}\right)\right)}^{-r}\mathrm{\Phi }(\mathrm{一个}{)}^r\mathrm{\Phi }{\left(G\left(\mathrm{一个}\right)\right)}^{-r}\right|\le {五}^{*}\mathrm{\Phi }\left(F\left(\mathrm{一个}{)}^{-r}{\mathrm{一个}}^{r}G\right(\mathrm{一个}{)}^{-r}\right)五$在一些温和的算子凸条件和一些正数r下成立。此外，我们证明如果$F^2$是算子凹的，那么$|\mathrm{\Phi }\left(F\right(\mathrm{一个}\left)\right)\mathrm{\Phi }\left(\mathrm{一个}\right)|\le \mathrm{\Phi }\left(\mathrm{一个}F\right(\mathrm{一个}\left)\right).$此外，我们给出了非对称 Choi-Davis 不等式和非对称 Kadison 不等式的对应物。由于 Bourin-Ricard 和 Furuta，我们的结果扩展了一些不等式。