Relatively bounded extensions of generator perturbations (Q1024979)

Let \(A\) be the generator of a \(C_0\)-semigroup \(T(t)\) on a Banach space \(X\), \(D\subset {\mathcal D}(A)\) be a dense subspace of \(X\) such that \(T(t)D\subset D\) for all \(t\geq 0\), and \(B_0:D\to Y\) be a linear operator from \(D\) to a Banach space \(Y\) such that \(B_0T(\cdot)\) is continuous for each \(x\in D\). The authors characterize the extendability of \(B_0\) to an \(A\)-bounded operator \(B:{\mathcal D}(A)\to Y\) by conditions similar to but weaker than the Miyadera condition (in the well-known Miyadera perturbation theorem). They also treat the relatively bounded extension problem for the case when \(A\) is a Hille-Yosida operator in \(X\).