An exponential kernel associated with operators that have one-dimensional self-commutators

TL;DR

This paper investigates the properties of an exponential kernel linked to operators with one-dimensional self-commutators, exploring its continuity, integral representations, and applications in operator theory and quadrature domains.

Contribution

It introduces new continuity and integral representation results for the exponential kernel and applies these findings to the study of operators with one-dimensional self-commutators.

Findings

Established continuity properties of the exponential kernel.

Derived integral representation formulas for the kernel.

Applied the kernel to analyze operators with one-dimensional self-commutators.

Abstract

The exponential kernel \[E{g}(\lambda,w) = \exp -\frac{1}{\pi}\int_{\mathbb{C} } \frac{g(u)}{\overline{u-w} (u-\lambda) } da(u ),\] where the compactly supported bounded measurable function $g$ satisfies $0 \leq g\leq 1,$ and suitably defined for all complex $\lambda, w,$ plays a role in the theory of Hilbert space operators with one-dimensional self-commutators and in the theory of quadrature domains. This article studies continuity and integral representation properties of $E_{g}$ with further applications of this exponential kernel to operators with one-dimensional self-commutator.