Necessary and sufficient conditions for the function $f(t) := \mbox{trace}(A^{-1}e^{-tB} A e^{-tB})$ to be monotone decreasing.

Question

Let $A$ and $B$ be a positive-definite $n \times n$ matrices. For any $t \ge 0$, define $f(t) := \mbox{trace}(A^{-1}e^{-tB} A e^{-tB}) = \|A^{-1/2}e^{-tB}A^{1/2}\|_F^2$.

Question. What are necessary and sufficient conditions for $f$ to be monotone decreasing.

A sufficient condition

If $A$ and $B$ commute, then $A$ and $e^{-tB}$ commute for any $t \ge 0$. Thus, $f(t)=\mbox{trace}(AA^{-1}e^{-2tB}) = \mbox{trace}(e^{-2tB})$, which is clearly monotone decreasing in $t$.

Answer 1

I doubt that you'll find a simple necessary and sufficient condition. You do have

$$ \dfrac{df(t)}{dt} = -\text{trace}\left(A^{-1} e^{-Bt} (BA + AB) e^{-Bt}\right) = -\text{trace}\left(A^{-1/2} e^{-Bt} (BA+AB) e^{-Bt} A^{-1/2}\right) $$

so a somewhat more general sufficient condition is that $BA + AB$ is positive definite.