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

Abstract

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

23 pages. To appear in Journal of Evolution Equations