Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}))^2)$ for $T$?

Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ and $\hat{O} \in SU(N)$. $\log$ is the principle matrix log. I'm only interested in positive $T$ solutions.

I have solved it in the case that $e^{iT\hat{H}_0}$ and $\hat{O}$ commute, the answer is obtained from the quadratic formula and is not an especially pleasing form.

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?

Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ and $\hat{O} \in SU(N)$. $\log$ is the principle matrix log. I'm only interested in positive $T$ solutions.

I have solved it in the case that $e^{iT\hat{H}_0}$ and $\hat{O}$ commute, the answer is obtained from the quadratic formula and is not an especially pleasing form.