The Kerzman-Stein operator for piecewise continuously differentiable regions

TL;DR

This paper analyzes the spectral properties of the Kerzman-Stein operator on piecewise continuously differentiable regions, revealing how corners affect compactness and spectrum, and providing explicit constructions and norm variation analysis.

Contribution

It provides a complete spectral description for the Kerzman-Stein operator on symmetric wedges and explores the impact of corners and smooth perturbations on the operator's norm.

Findings

Spectral description for symmetric wedges

Corners cause the operator to be noncompact

Explicit example of a smooth curve with large operator norm

Abstract

The Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman-Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman-Stein operator has large norm, and we demonstrate the variation in norm with respect to a continuously differentiable perturbation.