An operator-based and sparsity-based approach is proposed to adaptively separate a signal into additive subcomponents. The proposed approach can be formulated as an optimization problem. Since the design of the operator can be adaptively customized to the target signal, we can propose different types of operators for different types of signals. The subcomponents are a kind of local narrow band signals in the null space of an adaptive operator and a residual signal which is a sparse signal in some sense. Our experiments, including simulated signals and a real-life signal, demonstrate the efficacy and accuracy of the proposed approach.