Characterizations of the Crandall–Pazy class of C0-semigroups on Hilbert spaces and their application to decay estimates

We investigate immediately differentiable C 0 -semigroups ( e − t A ) t ≥ 0 satisfying sup 0 < t < 1 ⁡ t 1 / β ‖ A e − t A ‖ < ∞ for some 0 < β ≤ 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 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. • The Crandall–Pazy class of Hilbert-space semigroups are characterized. • Decay rates for Crank–Nicolson schemes are estimated via the characterizations. • The semigroup-generation property of inverse generators yields improved estimates.