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

Abstract

This paper is concerned with the decay rate of $$e^{A^{-1}t}A^{-1}$$ e A - 1 t A - 1 for the generator A of an exponentially stable $$C_0$$ C 0 -semigroup on a Hilbert space. To estimate the decay rate of $$e^{A^{-1}t}A^{-1}$$ 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 $$C_0$$ C 0 -semigroup whose generator is normal.

The Cayley transform of the generator of a polynomially stable $$C_0$$-semigroup

“…as n → ∞ and the decay rate n −1/(2+β) cannot be replaced by a better one in general. In the case where (e −tA ) t≥0 is exponentially stable and (e −tA −1 ) t≥0 is bounded, the following estimate has been derived in [24]:…”

“…This estimate has been extended to bounded C 0 -semigroups on Hilbert spaces with boundedly invertible generators by means of the B-calculus in [4]. The B-calculus has been also applied in [24] in order to obtain the estimate (7) e…”

“…as n → ∞ for all α > 0; see [24]. In [24], simple examples have been also provided to show that the estimates (7) and ( 8) cannot be in general improved.…”

“…as n → ∞ for all α > 0; see [24]. In [24], simple examples have been also provided to show that the estimates (7) and ( 8) cannot be in general improved. We refer to the survey [15] for further literature on the inverse generator problem.…”

“…By the Lyapunov-based approach, the estimate sup t≥0 e −tA −1 A −1 < ∞ has been derived in [24] for polynomially stable C 0 -semigroups with any parameters on Hilbert spaces. We improve this estimate in two ways by means of the B-calculus.…”