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Area of Science:

  • Spectral theory
  • Complex analysis
  • Matrix analysis

Background:

  • The double-layer potential is crucial in spectral theory.
  • Existing literature provides a foundation, but specific aspects require clarification.

Purpose of the Study:

  • To elucidate the role of the double-layer potential in spectral set approaches.
  • To connect integral operators to domain convexity and numerical range inclusion.
  • To provide a direct proof of a generalized mapping theorem and establish a spectral constant for matrices.

Main Methods:

  • Analysis of integral operators associated with the double-layer potential.
  • Direct proof techniques avoiding dilation theory.
  • Investigation of spectral constants for matrices.

Main Results:

  • Integral operators characterize domain convexity and numerical range inclusion.
  • A direct proof of the Putinar-Sandberg result (generalizing Berger-Stampfli's theorem) is presented.
  • A matrix-based spectral constant is derived, consistent with known bounds.

Conclusions:

  • The double-layer potential offers valuable insights into spectral properties.
  • The findings provide new tools for analyzing spectral sets and matrix behavior.
  • The derived spectral constant advances understanding of matrix stability and dynamics.