Bounds for exponentially stable semigroups Original Research Article

We give bounds for the decay as well as perturbation bounds for an exponentially stable semigroup eAt in a Hilbert space. The bounds are given in terms of only three quantities: the solution X of the corresponding Lyapunov equation A*X+XA=−I and the upper and lower bound of the spectrum of A+A*. Our estimate is a consequence of a stronger, local estimate on eAtψ which nicely depends on the quantity (Xψ,ψ) showing that the spectral geometry of the Lyapunov solution X – which is always Hermitian and positive definite – replaces, at least partly the possibly poor spectral geometry of the generator A. The local estimate remains meaningful also in some cases in which the semigroup is non-exponentially stable and the operator X is unbounded. The Lyapunov solution also gives new perturbation bounds which do not contain any exponentially growing factor. A set of examples illustrates the power of our decay estimate for finite matrices; our bound appears to be never drastically worse and is not seldom drastically better than the existing estimates. We also illustrate our bound on an infinite dimensional example, that of the transport equation.