Abstract: The compactness property of operator solutions to certain operator inequalities arising from the Likhtarnikov–Yakubovich frequency theorem for C₀-semigroups is studied. It is shown that the operator solution can be described through solutions to an adjoint problem, as was previously known under a certain regularity condition. Thus, some regularity properties of the semigroup are connected with the compactness of the operator in the general case. Several results are proved that are useful for checking the noncompactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases that arise in applications. As an example, these theorems are applied for a scalar delay equation posed in a proper Hilbert space and it is shown that the operator solution cannot be compact. These results are related to the author’s recent work on the nonlocal reduction principle of cocycles (nonautonomous dynamical systems) in Hilbert spaces.