Let $-A$ be the generator of a bounded $C_0$-semigroup $(e^{-tA})_{t \geq 0}$ on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform $V_{\omega}(A) := (A-\omega I) (A+\omega I)^{-1}$ with $\omega >0$. We give a decay estimate for $\|V_{\omega}(A)^nA^{-1}\|$ when $(e^{-tA})_{t \geq 0}$ is polynomially stable. Considering the case where the parameter $\omega$ varies, we estimate $\|(\prod_{k=1}^n V_{\omega_k}(A))A^{-1}\|$ for exponentially stable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$. Next we show that if the generator $-A$ of the bounded $C_0$-semigroup has a bounded inverse, then $\sup_{t \geq 0} \|e^{-tA^{-1}} A^{-\alpha} \| < \infty$ for all $\alpha >0$. We also present an estimate for the rate of decay of $\|e^{-tA^{-1}} A^{-1} \|$, assuming that $(e^{-tA})_{t \geq 0}$ is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the $\mathcal{B}$-calculus.

翻译：设$-A$为Hilbert空间上有界$C_0$-半群$(e^{-tA})_{t \geq 0}$的生成元。首先研究参数$\omega >0$下Cayley变换$V_{\omega}(A) := (A-\omega I) (A+\omega I)^{-1}$的长期渐近行为。当$(e^{-tA})_{t \geq 0}$具有多项式稳定性时，给出$\|V_{\omega}(A)^nA^{-1}\|$的衰减估计。考虑参数$\omega$变化的情形，对指数稳定的$C_0$-半群$(e^{-tA})_{t \geq 0}$，估计$\|(\prod_{k=1}^n V_{\omega_k}(A))A^{-1}\|$。进一步证明：若有界$C_0$-半群的生成元$-A$具有有界逆，则对所有$\alpha >0$有$\sup_{t \geq 0} \|e^{-tA^{-1}} A^{-\alpha} \| < \infty$。同时，在假设$(e^{-tA})_{t \geq 0}$多项式稳定的条件下，给出$\|e^{-tA^{-1}} A^{-1} \|$的衰减速率估计。上述结果通过一种称为$\mathcal{B}$-演算的函数演算方法提供的算子范数估计而获得。