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.

翻译：本文研究Hilbert空间上指数稳定$C_0$-半群生成元$A$的$e^{A^{-1}t}A^{-1}$衰减率。为估计$e^{A^{-1}t}A^{-1}$的衰减率，我们应用有界函数演算。借助该估计与Lyapunov方程，我们进一步研究光滑初值下Crank-Nicolson格式的量化渐近行为。类似论证亦适用于生成元为正规算子的多项式稳定$C_0$-半群。