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