  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.
