Decay of singular values of the Gramians of infinite-dimensional systems

We show that the Gramians of a control system with an analytic semigroup, control and observation operators that are not too unbounded and which have finite-dimensional input and output spaces have singular values which decay exponentially in the square root. As a corollary it is shown that the Hankel singular values of such control systems also decay exponentially in the square root. Another corollary shows that solutions of algebraic Riccati equations for such systems also have singular values which decay exponentially in the square root.