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

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