  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.
