I'm reading J. Schwinger's book "Quantum Kinematics and Dynamics" and I'm trying to make sense of his formulation of his famous quantum action principle. In essence, he starts from considering arbitrary variations of the the quantum amplitude $\langle a, t_1 |b, t_2 \rangle$, where $|a, t_1 \rangle$ is an eigenvector, corresponding to eigenvalue $a$, of some arbitrary but fixed Hermitian operator $A(t_1)$ (considered in the Heisenberg picture of QM), and $|b, t_2 \rangle$ is an eigenvector, corresponding to eigenvalue $b$, of another arbitrary but fixed Hermitian operator $B(t_2)$. An arbitrary variation of the amplitude reads, obviously, $$ \delta (\langle a, t_1 |b, t_2 \rangle)= (\delta\langle a, t_1 |)|b, t_2 \rangle + \langle a, t_1 |(\delta |b, t_2 \rangle) $$ where $\delta\langle a, t_1 |$ and $\delta |b, t_2 \rangle$ are arbitrary variations of the eigenbra and eigenket.
Now comes the puzzle. Schwinger writes $$\delta |b, t_2 \rangle = -iG_b(t_2)|b, t_2 \rangle$$$$\delta\langle a, t_1 | = iG_a(t_1)\langle a, t_1 |$$where he says $G_a(t_1)$ and $G_b(t_2)$ are infinitesimal Hermitian operators. I don't understand as to why an arbitrary infinitesimal variation $\delta |b, t_2 \rangle$ should be generated by a Hermitian operator, as if only unitary infinitesimal variations (instead of general ones) are considered, and not by a more general operator.
Could you, please, help clarify as to why the generators must be Hermitian for the most general variation $\delta |b, t_2 \rangle$?