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.