Abstract: The extension of the notion of quantum fidelity from the state-space to the operator level can be used to study environment-induced decoherence. The state-dependent operator fidelity susceptibility (OFS), the leading-order term for slightly different operator parameters, is shown to have a nontrivial behavior when the environment is at critical points. Two different contributions to the OFS are identified which have distinct physical origins and temporal dependence. Exact results are obtained for the finite-temperature decoherence caused by a bath described by the Ising model in a transverse field.