I am trying to determine if an inequality holds for Hermitian matrices $A$,$B$, and $C$ of the same dimension. Let $A$ be positive definite, $B$ and $C$ be positive semi-definite, and $B-C$ positive semi-definite. Is it then true in general that the maximal eigenvalue for the matrix $BAB$ is at least as large as the maximal eigenvalue for $CAC$ (or equivalently that the maximal eigenvalue for $A^{0.5}BB A^{0.5}$ is at least as large as the maximal eigenvalue for $A^{0.5}CC A^{0.5}$ )?
If $A^{0.5}B$ and $A^{0.5}C$ are both Hermitian, then the inequality can be derived using some well known eigenvalue inequalities for sums and products of positive semi-definite matrices. However, this argument breaks down if at least one of the two is non-hermitian.