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

TL;DR

This paper investigates the spectral characteristics of abstract parabolic operators using harmonic analysis and group theory, establishing conditions for invertibility and extending classical theorems to homogeneous function spaces.

Contribution

It introduces a new homogeneous function space with absolutely summable spectrum and generalizes the Gearhart-Prüss Theorem within this framework.

Findings

Provided invertibility conditions based on spectral properties.

Established a generalized Gearhart-Prüss Theorem.

Proved existence and uniqueness of solutions for certain nonlinear equations.

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.