Norm estimates for a semigroup generated by the sum of two operators with an unbounded commutator (Q2302197)

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.