Characterizations of the Crandall--Pazy Class of $C_0$-semigroups on Hilbert Spaces and Their Application to Decay Estimates

We investigate immediately differentiable $C_0$-semigroups $(e^{-tA}{)}_{t\ge 0}$ satisfying ${\sup }_{0<t<1}{t}^{1\mathrm{/}\beta }\mathrm{\parallel }A{e}^{-tA}\mathrm{\parallel }<\mathrm{\infty }$ for some $0<\beta \le 1$. Such $C_0$-semigroups are referred to as the Crandall–Pazy class of $C_0$-semigroups. In the Hilbert space setting, we present two characterizations of the Crandall–Pazy class. We then apply these characterizations to estimate decay rates for Crank–Nicolson schemes with smooth initial data when the associated abstract Cauchy problem is governed by an exponentially stable $C_0$-semigroup in the Crandall–Pazy class. The first approach is based on a functional calculus called the $\mathcal{B}$-calculus. The second approach builds upon estimates derived from Lyapunov equations and improves the decay estimate obtained in the first approach, under the additional assumption that $-A^{-1}$ generates a bounded $C_0$-semigroup.