The risks of frequency instability and power imbalance in new power systems have been raised due to the features of high-penetration power electronics and renewable energy sources (RES). To reduce these risks, it is necessary to consider the constraints of frequency security and adequacy in system operation dispatch. However, the two type constraints will significantly increase the computational complexity of the model, making it difficult to solve high accuracy during operation scale. A frequency and adequacy constrained unit commitment (FACUC) is proposed. Firstly, a system frequency response model and an adequacy evaluation model considering operational constraints are established. And the constraints of frequency security and adequacy are constructed in FACUC. Then a decomposition optimization algorithm is proposed to solve the proposed optimization model, which decomposes the original problem into a main problem of mixed integer linear programming (MILP), a simulation-based subproblem for frequency evaluation and an optimization-based subproblem for adequacy evaluation. Finally, the FACUC is solved by iterating among the main problem and subproblems. The example cases based on the modified IEEE 39-bus system are analyzed. The results show that the proposed FACUC model can effectively improve the frequency stability and adequacy of the system while ensuring economic efficiency.