Matrix valued versions of a result of von Neumann with an application to time discretization

Abstract We study the size of the operator C ( A ) where C ( z ) is an n × m -matrix with rational elements such that | C ( z )|⩽1 in some norm for all complex z in the unit disc, and A is a linear operator in a Hilbert space with norm at most 1. An application to variable step integration of initial value problems using one-leg methods is given.