The double-layer potential for spectral constants revisited

TL;DR

This paper revisits the double-layer potential's role in spectral set analysis, clarifying its properties, characterizing convexity and numerical range inclusion, providing a new proof of a key theorem, and establishing bounds for matrix spectral constants.

Contribution

It offers new insights into the double-layer potential, provides a direct proof of a spectral set theorem, and establishes bounds for spectral constants in matrices.

Findings

Integral operators characterize convexity and numerical range inclusion.

A direct proof of a generalization of Berger--Stampfli's theorem is provided.

Spectral constants for matrices depend only on extremal functions and vectors, matching known bounds.

Abstract

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 $1+\sqrt{2}$.