The Kerzman–Stein operator for piecewise continuously differentiable regions

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.

Keywords:

• Cauchy integral
• Kerzman–Stein operator
• eigenvalue
• spectrum
• piecewise C¹

AMS Subject Classifications:

• 45C05
• 45E05
• 30C40

Acknowledgements

Michael Bolt is partially supported by the National Science Foundation [grant number DMS-1002453] and by Calvin College through a Calvin Research Fellowship. Andrew Raich is partially supported by the National Science Foundation [grant number DMS-0855822]; the Simons Foundation [grant number 280164].