Decay Rate of exp(A-1t)A-1 on a Hilbert Space and the Crank–Nicolson Scheme with Smooth Initial Data

This paper is concerned with the decay rate of e A - 1 t A - 1 for the generator A of an exponentially stable C 0 -semigroup 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. A similar argument is applied to a polynomially stable C 0 -semigroup whose generator is normal.