Toeplitz operators and H∞ calculus

Let A be the generator of a strongly continuous, exponentially stable, semigroup on a Hilbert space. Furthermore, let the scalar function g be bounded and analytic on the left-half plane, i.e., g(−s)∈H∞. By using the Toeplitz operator associated to g, we construct an infinite-time admissible output operator g(A). If g is rational, then this operator is bounded, and equals the “normal” definition of g(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 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 with g(−s)∈H∞, i.e., there exists a bounded H∞-calculus. Moreover, we rediscover some well-known classes of generators also having a bounded H∞-calculus.