Operator-Norm Trotter Product Formula on Banach Spaces

Here we collect results concerning the operator-norm convergent Trotter product formula for \(C_0\) -semigroups \(\{\mathrm {e}\,^{- t A}\}_{t\geq 0}\) , \(\{\mathrm {e}\,^{- t B}\}_{t\geq 0}\) on a Banach space. The proof on a Banach space is demonstrated under hypothesis that one of the involved into product formula contraction \(C_0\) -semigroup (e.g. \(\{\mathrm {e}\,^{- t A}\}_{t\geq 0}\) ) is holomorphic and that another one is a \(C_0\) -semigroup generated by operator B, which satisfies conditions of the relative infinitesimally A-smallness.