Let us consider a system. In Schrodinger picture, its Hamiltonian $H$ is given by $H = H_0 + V(t)$, where $H_0$ is the unperturbed Hamiltonian and $V(t)$ is the time-dependent perturbation.
In interaction picture, the state ket $|\psi, t \rangle_I$ of this system is defined as $$|\psi, t \rangle_I = e^{iH_0(t-t_0)/\hbar} \, |\psi, t \rangle, $$ where $|\psi, t \rangle$ is the state ket of the system in Schrodinger picture.
In interaction picture, an operator $\hat{A}_I$ is defined as $$\hat{A}_I = e^{iH_0(t-t_0)/\hbar} \,\hat{A} \, e^{-iH_0(t-t_0)/\hbar},$$ where $\hat{A}$ is the operator in Schrodinger picture.
My questions are as follows.
What is the eigenvalue equation for $\hat{A}_I(t)$?
If the eigenkets are time dependent, then how do they evolve in time?
If an eigenket of $\hat{A}_I$ is $|i, t\rangle_I$, then how is it related to the eigenket $|i\rangle$ of $\hat{A}$ in Schrodinger picture?