  The Trotter product formula and the quantum Zeno effect are both
indispensable tools for constructing time-evolutions using experimentally
feasible building blocks. In this work, we discuss assumptions under which
quantitative bounds can be proven in the strong operator topology on Banach
spaces and provide natural bosonic examples. Specially, we assume the existence
of a continuously embedded Banach space, which relatively bounds the involved
generators and creates an invariant subspace of the limiting semigroup with a
stable restriction. The slightly stronger assumption of admissible subspaces is
well-recognized in the realm of hyperbolic evolution systems (time-dependent
semigroups), to which the results are extended. By assuming access to a
hierarchy of continuously embedded Banach spaces, Suzuki-higher-order bounds
can be demonstrated. In bosonic applications, these embedded Banach spaces
naturally arise through the number operator, leading to a diverse set of
examples encompassing notable instances such as the Ornstein-Uhlenbeck
semigroup and multi-photon driven dissipation used in bosonic error correction.
