Properties of therealization of inner functions

In this paper we investigate state-space realizations of inner functions. We derive necessary and sufficient, conditions on basis of the inner function to have a exactly controllable and exactly observable realization such that the associated C₀-semigroup is exponentially stable. Furthermore. we give necessary and sufficient conditions on the inner function such that the C₀-semigroup is a group. Combining these results, the C₀-semigroup is an exponentially stable C₀- group if and only if the inner function is the product of a constant of modulus one and a Blaschke product for which the zeros satisfy the Carleson-Newman condition and the zeros lie in a vertical strip bounded away from the imaginary axis.