Local exponential stability and local robustness for limit cycle solutions of ordinary differential equations can be verified using the characteristic multipliers. These well-known results are here generalized to a class of infinite-dimensional systems. Stability and robustness are now verified using certain invertibility conditions on the linear equations that are obtained when the system is linearized along the limit cycle. The new criterion reduces to the classical condition on the characteristic multipliers when we consider a finite-dimensional system which is perturbed by a bounded but possibly infinite-dimensional operator. The computation of a robustness margin, i.e., a bound on the maximally allowed perturbation, is also considered.