On the relationship between perturbed semigroups and their generators (Q1583911)

The author investigates the relationship between the perturbation of a semigroup and the corresponding perturbation of its generator. A typical result is the following: Let \(T\) and \(S\) be semigroups generated in a Banach space \(X\) by \(A\) and \(B\), respectively, let \(Y\) be a Banach space such that \(X_A\hookrightarrow Y\hookrightarrow X^A\), and suppose that \(T\) admits a continuous extension to \(Y\). Then \(\|(T(t)- S(t))x\|_Y= O(t)\) for all \(T\) if and only if there exists a continuous operator \(C: X\to (Y^A)_F\) such that \(B= (A^A+ C)|_{D(B)}\), or alternatively, such that \(B|_{D(A)}= A+C\). Here \(D(\cdot)\) denotes a domain, \(X_A\) is \(D(A)\) with graph norm, \(X^A\) is \(X\times X/G(A)\), \(G(\cdot)\) denoting a graph, and \((Y^A)_F\) is the Favard class in \(Y^A\).