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

04/07/2022

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by   Masashi Wakaiki, et al.

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This paper is concerned with the decay rate of e^A^-1tA^-1 for the generator A of an exponentially stable C_0-semigroup (e^At)_t≥ 0 on a Hilbert space. To estimate the decay rate of e^A^-1tA^-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. Analogous results are obtained for polynomially stable C_0-semigroups whose generator is normal.