  We obtain a version of the Frequency Theorem (a theorem on solvability of
certain operator inequalities), which allows to construct quadratic Lyapunov
functionals for semilinear parabolic equations. We show that the well-known
Spectral Gap Condition, which was used in the theory of inertial manifolds by
C. Foias, R. Temam and G. R. Sell, is a particular case of some frequency
inequality, which arises within the Frequency Theorem. In particular, this
allows to construct inertial manifolds for semilinear parabolic equations
(including also some non-autonomous problems) in the context of a more general
geometric theory developed in our adjacent works. This theory is based on
quadratic Lyapunov functionals and generalizes the frequency-domain approach
used by R. A. Smith. We also discuss the optimality of frequency inequalities
and its relationship with known old and recent results in the field.
