On the eigenvalue decay of solutions to operator Lyapunov equations

This paper is concerned with the eigenvalue decay of the solution to operator Lyapunov equations with right-hand sides of finite rank. We show that the kth (generalized) eigenvalue decays exponentially in k, provided that the involved operator A generates an exponentially stable analytic semigroup, and A is either self-adjoint or diagonalizable with its eigenvalues contained in a strip around the real axis. Numerical experiments with discretizations of 1D and 2D PDE control problems confirm this decay.