Crank–Nicolson Scheme for Abstract Linear Systems

This paper studies the Crank–Nicolson discretization scheme for abstract differential equations on a general Banach space. We show that a time-varying discretization of a bounded analytic C₀-semigroup leads to a bounded discrete-time system. On Hilbert spaces, this result can be extended to all bounded C₀-semigroups for which the inverse generator generates a bounded C₀-semigroup. The presentation is based on C₀-semigroup theory and uses a functional analysis approach.