Decay Rate of $\exp (A^{-1}t)A^{-1}$ on a Hilbert Space and the Crank-Nicolson Scheme with Smooth Initial Data

Abstract

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.

arXiv comment17 pages. To appear in Integral Equations and Operator Theory

References in corpus (1)

• A Short Counterexample to the Inverse Generator Problem on non-Hilbertian Reflexive $L^p$-spaces