The Schrödinger equation for states is known as $\frac{d}{dt} \psi(t)= - iH(t)\psi(t) $. The solution can be expressed via the time evolution operator $U(t)$ so that $\psi(t)=U(t)\psi(0)$ where now also $U(t)$ satisfies the Schrödinger equation \begin{align} \frac{d}{dt} U(t) = -i H(t) U(t)\end{align}
Applying a time-dependent unitary basis transformation $T(t)$, the Hamiltonian can be shown to transform as \begin{align} \breve{H} = THT^\dagger + i \dot{T}T^\dagger \end{align} yielding a transformed Schrödinger equation \begin{align} \frac{d}{dt} \breve{\psi} = -i\breve{H}(t)\breve{\psi}(t) \end{align} where $\breve{\psi}(t)=T\psi$
Question: Does the time evolution operator solving the transformed Schrödinger equation also satisfy the transformed Schrödinger equation? I.e. $\frac{d}{dt}\breve{U}=-i\breve{H}\breve{U}$?
Background: I am actually interested in introducing a time-dependent basis transformation of the operator basis that $U$ may be written in. Naively, such a basis transform would transform $U$ as $U\mapsto \breve{U} = T U T^\dagger$. However the equation of motion for this $\breve{U}$ takes the form \begin{align*} \frac{d}{dt} \breve{U} =& \dot{T}UT^{\dagger}+T\dot{U}T^{\dagger}+TU\dot{T^{\dagger}}\\ = & \dot{T}T^{\dagger}TUT^{\dagger}-iTHUT^{\dagger}+TUT^{\dagger}T\dot{T^{\dagger}}\\ = & \dot{T}T^{\dagger}\breve{U}-iTHUT^{\dagger}+\breve{U}T\dot{T^{\dagger}}\\ = & \dot{T}T^{\dagger}\breve{U}-iTHT^{\dagger}TUT^{\dagger}+\breve{U}T\dot{T^{\dagger}}\\ = & \dot{T}T^{\dagger}\breve{U}-iTHT^{\dagger}\breve{U}+\breve{U}T\dot{T^{\dagger}}\\ = & \dot{T}T^{\dagger}\breve{U}-iTHT^{\dagger}\breve{U}+\breve{U}T\dot{T^{\dagger}} \end{align*} and upon replacing $THT^{\dagger}=\breve{H}-i\dot{T}T^{\dagger}$ we find \begin{align*} \frac{d}{dt}\breve{U}= & \dot{T}T^{\dagger}\breve{U}-i\left(\breve{H}-i\dot{T}T^{\dagger}\right)\breve{U}+\breve{U}T\dot{T^{\dagger}}\\ = & \dot{T}T^{\dagger}\breve{U}-i\breve{H}\breve{U}-\dot{T}T^{\dagger}\breve{U}+\breve{U}T\dot{T^{\dagger}}\\ = & -i\breve{H}\breve{U}+\breve{U}T\dot{T^{\dagger}} \end{align*}
The final equation of motion for $\breve{U}$ almost looks like the Schrödinger equation with transformed Hamiltonian $\breve{H}$ if it were not for the addition term featuring $T \dot{T^\dagger}$ operated from the right on $\breve{U}$. How does one handle this equation? I am used to factoring out $U$ or $\psi$ in any equation of motion. This does not seem to be possible here.
I am wondering: is there any literature/ mathematical theories treating such a time-dependent transformation?
