Norm Estimates for a Semigroup Generated by the Sum of Two Operators with an Unbounded Commutator

Abstract

Let A be the generator of an analytic semigroup $(e^{At})_{t\ge 0}$ on a Banach space ${\mathcal {X}}$, B be a bounded operator in ${\mathcal {X}}$ and $K=AB-BA$ be the commutator. Assuming that there is a linear operator S having a bounded left-inverse operator $S_l^{-1}$, such that $\int _{0}^{\infty } \Vert Se^{At}\Vert \Vert e^{Bt}\Vert dt<\infty$, and the operator $KS_l^{-1}$ 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.