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<p>We thoroughly analyse the double-layer potential’s role in approaches to spectral sets in the spirit of Delyon–Delyon, Crouzeix and Crouzeix–Palencia. While the potential is well-studied, we aim to clarify on several of its aspects in light of these references. In particular, we illustrate how the associated integral operators can be used to characterize the convexity of the domain and the inclusion of the numerical range in its closure. We furthermore give a direct proof of a result by Putinar–Sandberg—a generalization of Berger–Stampfli’s mapping theorem—circumventing dilation theory. Finally, we show for matrices that any smooth domain whose closure contains the numerical range admits a spectral constant only depending on the extremal function and vector. This constant is consistent with the so far best known absolute bound <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="20_2025_2800_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">\(1+\sqrt{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math></EquationSource> </InlineEquation>.</p>

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The Double-Layer Potential for Spectral Constants Revisited

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月14日

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  • Felix L. Schwenninger,
  • Jens de Vries

摘要

We thoroughly analyse the double-layer potential’s role in approaches to spectral sets in the spirit of Delyon–Delyon, Crouzeix and Crouzeix–Palencia. While the potential is well-studied, we aim to clarify on several of its aspects in light of these references. In particular, we illustrate how the associated integral operators can be used to characterize the convexity of the domain and the inclusion of the numerical range in its closure. We furthermore give a direct proof of a result by Putinar–Sandberg—a generalization of Berger–Stampfli’s mapping theorem—circumventing dilation theory. Finally, we show for matrices that any smooth domain whose closure contains the numerical range admits a spectral constant only depending on the extremal function and vector. This constant is consistent with the so far best known absolute bound \(1+\sqrt{2}\) 1 + 2 .