An H∞ Calculus of Admissible Operators

Given a Hilbert space and the generator A of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any g(−s) ∈ H∞ we show that there exists an inf nite-time admissible output operator g(A). If g is rational, then this operator is bounded, and equals the “normal” def nition of g(A). In particular, when g(s) = 1/(s + α), α ∈ C + 0 , then this admissible output operator equals (αI − A) . Although in general g(A)may be unbounded, we always have that g(A) multiplied by the semigroup is a bounded operator for every (strictly) positive time instant. Furthermore, when there exists an admissible output operator C such that (C,A) is exactly observable, then g(A) is bounded for all g’s with g(−s) ∈ H∞.