On relations between operator valued \(\alpha\)-divergence and relative operator entropies (Q2815616)

Let \(\mathcal{H}\) be a complex Hilbert space and \(B(\mathcal{H})_{+}\) be the set of all positive definite bounded linear operators on \(\mathcal{H}\). The authors give elementary properties of the operator-valued \(\alpha\)-divergence, defined byNEWLINENEWLINENEWLINE\[NEWLINE D_{\alpha}(A,B):= \frac{1}{\alpha(1-\alpha)}(A\nabla_{\alpha}B- A\sharp_{\alpha}B) NEWLINE\]NEWLINENEWLINENEWLINEfor \(\alpha\in (0,1)\), \(A,B\in B(\mathcal{H})_{+}\), where \(A\nabla_{\alpha}B=(1-\alpha)A+\alpha B\) and \(A\sharp_{\alpha}B=A^{\frac{1}{2}} (A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{\alpha} A^{\frac{1}{2}}\) for \(\alpha\in [0,1]\). Next, the authors define the noncommutative ratio as follows:NEWLINENEWLINENEWLINE\[NEWLINE \mathcal{R}(u,v;A,B):=(A\natural_{u+v}B) (A\natural_{u}B)^{-1} \quad (u,v\in \mathbb{R}), NEWLINE\]NEWLINENEWLINENEWLINEwhere \(A\natural_{u}B= A^{\frac{1}{2}} (A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{u} A^{\frac{1}{2}}\) for \(u\in \mathbb{R}\) and \(A,B\in B(\mathcal{H})_{+}\), and study some elementary properties of it from a geometrical viewpoint.