Decay estimates for Cayley transforms and inverses of semigroup generators via the $\mathcal{B}$-calculus

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.