Decay Rate of $\exp(A^{-1}t)A^{-1}$ on a Hilbert Space and the Crank-Nicolson Scheme with Smooth Initial Data

Abstract

This paper is concerned with the decay rate of e A −1 t A −1 for the generator A of an exponentially stable C0-semigroup (e At ) t≥0 on a Hilbert space. To estimate the decay rate of e A −1 t A −1 , we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. Analogous results are obtained for polynomially stable C0-semigroups whose generator is normal.