  In the theory of open quantum systems, divisibility of the system dynamical
maps is related to memory effects in the dynamics. By decomposing the system
Hilbert space as a direct sum of several Hilbert spaces, we study the
relationship among the corresponding dynamical maps. It is shown that if the
dynamical maps of the open system possess a chain of invariant subspaces, there
exists a divisibility hierarchy for their corresponding dynamics. Two classes
of examples are given for illustrating these hierarchical structures. One is
the pure-dephasing dynamics, and the other is the decay dynamics. Our results
offer a systematic approach to obtaining the divisibility conditions and
non-Markovian witnesses for these dynamics. Moreover, as a new way of
decomposing open quantum systems, it is worthy of further study.
