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 $C_0$-semigroups leads to a bounded discrete-time system. On Hilbert spaces this result can be extended to all bounded $C_0$-semigroups for which the inverse generator generates a bounded $C_0$-semigroup. The presentation is based on $C_0$-semigroup theory and uses a functional analysis approach.