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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 rolein the theory of Hilbert space operators with one-dimensional self-commutatorsand in the theory of quadrature domains. This article studies continuity andintegral representation properties of $E_{g}$ with further applications of thisexponential kernel to operators with one-dimensional self-commutator.

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