Adiabatic theorems for general linear operators with time-independent domains

We establish adiabatic theorems with and without spectral gap condition for general -- typically dissipative -- linear operators $A(t): D(A(t)) \subset X \to X$ with time-independent domains $D(A(t)) = D$ in some Banach space $X$. Compared to the previously known adiabatic theorems -- especially those without spectral gap condition -- we do not require the considered spectral values $\lambda(t)$ of $A(t)$ to be (weakly) semisimple. We also impose only fairly weak regularity conditions. Applications are given to slowly time-varying open quantum systems and to adiabatic switching processes.