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Non‐harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary

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  • Michael Ruzhansky
  • J. P. Velasquez‐Rodriguez

Abstract

We use Operator Ideals Theory and Gershgorin theory to obtain explicit information in terms of the symbol concerning the spectrum of pseudo‐differential operators, on a smooth manifold Ω with boundary ∂Ω, in the context of the non‐harmonic analysis of boundary value problems introduced in [29] in terms of a model operator L. For symbols in the Hörmander class S1,00(Ω¯×I), we provide a non‐harmonic version of Gohberg's lemma, and a sufficient and necessary condition to ensure that the corresponding pseudo‐differential operator is a compact operator in L2(Ω), or a Riesz operator in Lp(Ω) in the case of Riemannian manifolds with smooth boundary. We extend to the context of the non‐harmonic analysis of boundary value problems the well known theorems about the exact domain of elliptic operators, and discuss some applications of the obtained results to evolution equations.

Suggested Citation

  • Michael Ruzhansky & J. P. Velasquez‐Rodriguez, 2021. "Non‐harmonic Gohberg's lemma, Gershgorin theory and heat equation on manifolds with boundary," Mathematische Nachrichten, Wiley Blackwell, vol. 294(9), pages 1783-1820, September.
    • Handle: RePEc:bla:mathna:v:294:y:2021:i:9:p:1783-1820
    DOI: 10.1002/mana.201900072
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