Harmonic and Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

Abstract: 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.