Sufficient conditions for a memory-kernel master equation

We derive sufficient conditions for the memory-kernel governing nonlocal master equation which guarantee a legitimate (completely positive and trace-preserving) dynamical map. It turns out that these conditions provide natural parametrizations of the dynamical map being a generalization of the Markovian semigroup. This parametrization is defined by the so-called legitimate pair---monotonic quantum operation and completely positive map---and it is shown that such a class of maps covers almost all known examples from the Markovian semigroup, the semi-Markov evolution, up to collision models and their generalization.