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Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces, II

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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 \(\mathscr {L}= -\mathrm{d}/\mathrm{d}t+A\) in homogeneous function spaces. We focus on the dependency between various invertibility states of such an operator. In particular, we prove that often, a generally weaker state of invertibility implies a stronger state for \(\mathscr {L}\) under mild additional conditions. For example, we show that if the operator \(\mathscr {L}\) is surjective and the imaginary axis is not contained in the interior of the spectrum of A, then \(\mathscr {L}\) is invertible.

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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 the Ministry of Education and Science of the Russian Federation in the frameworks of the project part of the state work quota (Project no. 1.3464.2017/4.6). 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, II. Mediterr. J. Math. 14, 181 (2017). https://doi.org/10.1007/s00009-017-0982-y

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  • DOI: https://doi.org/10.1007/s00009-017-0982-y

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