Double-Layer Potentials, Configuration Constants, and Applications to Numerical Ranges

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