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

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-Poincar\'e 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.