  We use methods of harmonic analysis and group representation theory to study
the spectral properties of the abstract parabolic operator $\mathscr L =
-d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for
invertibility of such operators in terms of the spectral properties of the
operator $A$ and the semigroup generated by $A$. We introduce a homogeneous
space of functions with absolutely summable spectrum and prove a generalization
of the Gearhart-Pr\"uss Theorem for such spaces. We use the results to prove
existence and uniqueness of solutions of a certain class of non-linear
equations.
