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

TL;DR

This paper investigates the decay rate of a specific operator related to exponentially stable semigroups on Hilbert spaces and analyzes the asymptotic behavior of the Crank-Nicolson scheme with smooth initial data using functional calculus and Lyapunov equations.

Contribution

It provides new decay rate estimates for the operator $e^{A^{-1}t}A^{-1}$ and applies these results to study the Crank-Nicolson scheme's asymptotic behavior for smooth initial data.

Findings

Decay rate estimates for $e^{A^{-1}t}A^{-1}$ using bounded functional calculus.

Quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data.

Extension of analysis to polynomially stable semigroups with normal generators.

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.