Double-layer potentials, configuration constants and applications to numerical ranges

Abstract

Given a compact convex planar domain $\hat{I}\text{c◯}$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(\hat{I}\text{c◯})$ is the norm of the Neumann-Poincaré operator $K_{\hat{I}}\text{c◯}$ acting on the space of continuous real-valued functions on the boundary $\partial \hat{I}\text{c◯}$, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(\hat{I}\text{c◯})$ of $K_{\hat{I}}\text{c◯}$ on the corresponding space of complex-valued functions, and the norm $a(\hat{I}\text{c◯})$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(\hat{I}\text{c◯})=c_{\mathbb{C}}(\hat{I}\text{c◯})$, the analytic Neumann-type inequality $a(\hat{I}\text{c◯})<1$, and provide various estimates for these quantities expressed in terms of the geometry of $\hat{I}\text{c◯}$. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\Vert p(T)\Vert \le K{\sup }_{z\in \hat{I}\text{c◯}}|p(z)|$, where $p$ is a polynomial and $\hat{I}\text{c◯}$ is a domain containing the numerical range of the operator $T$. Among other results, we show that the well-known Crouzeix-Palencia bound $K\le 1+\sqrt{2}$ can be improved to $K\le 1+\sqrt{1+a(\hat{I}\text{c◯})}$. In the case that $\hat{I}\text{c◯}$ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse.

Revised version