Sorry, I wasn't very clear! I had intended G to be a function of t in general. So where G appears above I really mean G(t). So the most general case I'm interested in would be to write $E_t = e^{tG(t)}$, provided $t$ is sufficiently small but finite. Naturally, the first thing one might try to do is take logs, this seems to run into problems as discussed above. Thank you again for your help!

Ah, this answer is very helpful! Thank you! I'm particularly interested in the case where we can write $E_t = \sum_j p_j e^{-it ad_{h_j}}$ as a semi-group $E_t = e^{tG}$ for some generator $G$. I was hoping this could be proved by the existence of the matrix logarithm , but if $E_t$ is not generally invertible it seems this proof method must fail as the matrix log may not exist.

If I restrict to the invariant subspace of Hermitian operators, then it looks like this has a trivial kernel, right?

From your answer, I can see that if $n=2$, $A=((0,1)(0,0))$ then it is possible to find unitaries with 1 and -1 eigenvalues, then it looks like it has a non-trivial kernel.

Thanks for your answer! I'm not sure I understand your counter example. If $A\in \mathbb{R}$, then surely the linear mapping in your example is E(A) = (1/2)*(-1)A(-1) + (1/2)*(1)A(1) = A? But I feel like I've misunderstood you here?