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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.

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Authors and Affiliations

  1. Department of Applied Mathematics and Mechanics, Voronezh State University, Voronezh, 394693, Russia

    Anatoly G. Baskakov

  2. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115, USA

    Ilya A. Krishtal

Authors
  1. Anatoly G. Baskakov
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  2. Ilya A. Krishtal
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Correspondence to Ilya A. Krishtal.

Additional information

A. G. Baskakov is supported in part by RFBR grant 16-01-00197, RSF grant 14-21-00066, and the Ministry of Education and Science of Russia: Project 1110 of the 2014-16 state program for institutions of higher learning in science. I. A. Krishtal is supported in part by NSF Grant DMS-1322127.

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Baskakov, A.G., Krishtal, I.A. Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces. Mediterr. J. Math. 13, 2443–2462 (2016). https://doi.org/10.1007/s00009-015-0633-0

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  • DOI: https://doi.org/10.1007/s00009-015-0633-0

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