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

Abstract This paper is concerned with the decay rate of $$e^{A^{-1}t}A^{-1}$$ e A - 1 t A - 1 for the generator A of an exponentially stable $$C_0$$ C 0 -semigroup on a Hilbert space. To estimate the decay rate of $$e^{A^{-1}t}A^{-1}$$ 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 $$C_0$$ C 0 -semigroup whose generator is normal.