Non-uniform stability for bounded semi-groups on Banach spaces (Q1021388)

For a bounded strongly continuous semigroup \(S(t)\) on a Banach space with the generator \(-A\), the authors show that, if \(\|S(t)(A+I)^{-1}\|\) tends to zero, then \(iR\) is contained in \(\rho(A)\). While the converse has been known, they further estimate the rate of convergence in terms of the resolvent of \(A\) on \(iR\), and also give a Laplace transform version of the result. These results in a sense extend some previous theorems.