Mediterranean Journal of Mathematics

, Volume 13, Issue 5, pp 2443–2462 | Cite as

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

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Abstract

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator \({\mathcal{L} = -{\rm d}/{\rm d}t + A}\) 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–Prüss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-linear equations.

Keywords

Abstract parabolic operators homogeneous function spaces Green functions Beurling spectrum 

Mathematics Subject Classification

47A10 46H25 
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© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Applied Mathematics and MechanicsVoronezh State UniversityVoronezhRussia
  2. 2.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

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