Another proof of a result on the doubly superstochastic matrices (Q6593274)

Let \(H\in\mathbb{R}^{2n\times 2n}\) be symplectic, that is\N\[\NH^TJH=J,\quad J=\left( \begin{array}{cc} 0&I_n\\\N-I_n&0 \end{array} \right).\N\]\NThe author establishes a trace inequality for \(H\) and, as a consequence, gives a new proof of a theorem by \textit{R. Bhatia} and \textit{T. Jain} [J. Math. Phys. 56: 112201 (2015; Zbl 1329.15048)] on a doubly superstochastic matrix corresponding to \(H\). An entrywise nonnegative square matrix \(U\) is doubly superstochastic if there is a doubly stochastic matrix \(V\) such that \(U\ge V\) entrywise.