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

TL;DR

This paper explores the properties of certain operator norms related to convex domains, establishing new equalities and inequalities, and applying these results to improve bounds in the holomorphic functional calculus of operators.

Contribution

It introduces new techniques for analyzing operator norms on complex-valued functions, proves the equality of real and complex configuration constants, and improves bounds on the Crouzeix-Palencia constant.

Findings

Proved $c_{ ext{R}}(\Omega) = c_{ ext{C}}(\Omega)$

Established $a(\Omega) < 1$ for analytic functions

Improved the bound on the Crouzeix-Palencia constant to $K \

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-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.