Perturbations of eventually differentiable and eventually norm-continuous semigroups on Banach spaces (Q2506399)

Let \(X\) be a Banach space and let \(\{T(t)\}_{t\geq 0}\) be a semigroup on \(X\) with the infinitesimal generator \(A\). The author obtains perturbation results for eventually differentiable and eventually norm-continuous semigroups. One of the main results of the paper may be stated as follows. Theorem. If \(\{T(t)\}_{t\geq 0}\) is a \(C_0\)-semigroup which is differentiable for \(t \geq t_0 > 0\), \(B\) is relatively bounded with respect to \(A\), i.e., \(D(A)\subset D(B)\) and \(\| Bx\| \leq a\| Ax\| + b\| x\| \) for all \(x \in D(A)\), where \(a, b\geq 0\), and has the properties \(B(D(A)) \subset D(A)\) and \(T(t)B \subset B T(t)\), then the \(C_0\)-semigroup generated by \(A+B\) is differentiable for \(t\geq t_0\).