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

Abstract

The exponential kernel \[E{g}(λ,w) = \exp -\frac{1}π\int_{\mathbb{C} } \frac{g(u)}{\overline{u-w} (u-λ) } da(u ),\] where the compactly supported bounded measurable function $g$ satisfies $0\le g\le 1,$ 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.