About the matrix function X->AX+XA

Abstract: Let K be an infinite field such that its characteristic is not 2. We show that, for every $A\in\mathcal{M}_n(K)$ such that $\mathrm{rank}(A)\geq n/2$, there exists $B\in\mathcal{M}_n(K)$ such that $B$ is similar to $A$ and $A+B$ is invertible. Let $K$ be a subfield of $\mathbb{R}$. We show that, if $n$ is even, then for every $X\in\mathcal{M}_n(K)$, $\det(AX+XA)\geq 0$ if and only if either $\mathrm{rank}(A)<n/2$ or there exists $\alpha\in K,\alpha\leq 0$, such that $A^2=\alpha I_n$.