Semigroups of sums of two operators with small commutators

Abstract

Let A be the generator of a $C_0$-semigroup $(e^{At})_{t\ge 0}$ on a Banach space $\mathcal {X}$ and B be a bounded operator in $\mathcal {X}$. Assuming that $\int _{0}^{\infty } \Vert e^{At}\Vert \Vert e^{Bt}\Vert dt<\infty$ and the commutator $AB-BA$ is bounded and has a sufficiently small norm, we show that $\int _{0}^{\infty } \Vert e^{(A+B)t}\Vert dt<\infty$, where $(e^{(A+B)t})_{t\ge 0}$ is the semigroup generated by $A+B$. In addition, estimates for the supremum- and $L^1$-norms of the difference $e^{(A+B)t}-e^{At}e^{Bt}$ are derived.