Sums and commutators of generators of noncommuting strongly continuous groups. (Q1856905)

Let \(A\) and \(B\) be the generators of two strongly continuous groups of Banach space operators. The commutator \([A,B]\) is introduced as a densely defined operator with domain \(D([A, B]):= D(AB)\cap D(BA)\) and it is assumed that \([A,B]\) either commutes with both groups \(e^{At}\), \(e^{Bs}\) for all \(t, s\in\mathbb{R}\), or \([A,B]\) is a multiple of \(A\). The author proves that under some further natural restrictions, the sum \(A+B\) and the commutator \([A,B]\) are closable and their closures generate strongly continuous groups, which are given by explicit representations. In order to achieve this, the author extends a variant of the classical Campbell-Hausdorff formula (for bounded operators) to the case of unbounded operators.