Abstract. The Cauchy transform $C\colon f(z)\to\int _Kf(w)/(z-w) dw$ over $L^1(K)$ (where $K$ is a compact set in ${\msbm C}$) and the multiplication operators $M_h\colon f(z)\to h(z)f(z)$ have compact comutators $[M_h,C]$ for all $h\in\overline {{\cal R}(K)}$, the closure in $C(K)$ of the rational functions with poles off $K$. In the special case of a polynomial $h$, the eigenvalues and eigenfunctions of $[M_h,C]$ are obtained from the eigenvalues and eigenvectors of a related symmetric matrix.

AMS Subject Classification (1991): 47B47, 47A75

Keyword(s): Cauchy transform, commutators, multiplication operator, compact operator, Hardy space, complex moments

Received January 28, 2009. (Registered under 11/2009.)