Stability for Lie-Trotter products for some operator matrix semigroups

_u(t) = (A + B)u(t) with fixed initial value u(0) = u0. The general theory of C0-semigroups yields that the problem is well-posed if and only if A + B is the generator of a C0-semigroup (U(t))t 0, and that in this case the unique solution is given byu(t) = U(t)u0 (we refer to standard monographs on semigroup theory, such as [7]). Now suppose thatA;B generate theC0-semigroups (S(t))t 0, (T (t))t 0, respectively. If ( A B) has dense range and domain D(A +B), then one has the following equivalence, stating the Lie‐Trotter, or sequential splitting, formula: Convergence: U(t)x = lim n!1 h S t n T t n in x for allx2 E.