Operator versions of Shannon type inequality (Q2794827)

Let \(B(\mathcal{H})\) be the \(C^{*}\)-algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\). For positive definite \(A,B\in B(\mathcal{H})\) and \(q\in \mathbb{R}\), the generalized relative operator entropy is defined byNEWLINENEWLINENEWLINE\[NEWLINE S_{q}(A|B)=A^{\frac{1}{2}} (A^{\frac{-1}{2}}BA^{\frac{-1}{2}})^{q} (\log A^{\frac{-1}{2}}BA^{\frac{-1}{2}}) A^{\frac{1}{2}}. NEWLINE\]NEWLINENEWLINENEWLINEThe author gives upper and lower bounds of \(\sum_{j=1}^{n}S_{q}(A_{j}|B_{j})\) for positive definite \(A_1,\dots,A_n\), \(B_1,\dots,B_n\in B(\mathcal H)\) and \(0\leq q\leq 1\). This is a more precise estimation than in [\textit{T. Furuta}, Linear Algebra Appl. 381, 219--235 (2004; Zbl 1057.47022)].