Infinite dimensional dynamical maps

Completely positive trace preserving maps are widely used in quantum information theory. These are mostly studied using the master equation perspective. A central part in this theory is to study whether a given system of dynamical maps $\{\Lambda_t: t \ge 0\}$ is Markovian or non-Markovian. We study the problem when the underlying Hilbert space is of infinite dimensional. We construct a sufficient condition for checking P (resp. CP) divisibility of dynamical maps. We construct several examples where the underlying Hilbert space may not be of finite dimensional. We also give a special emphasis on Gaussian dynamical maps and get a version of our result in it.