A structure theorem for the commutant of a class of cyclic subnormal operators

Abstract:

An $m$-measure is defined to be a measure $\mu$ such that the analytic bounded point evaluations of ${P^2}(\mu )$ is the open unit disk ${\mathbf {D}}$ in the complex plane, and the weak* closure of the analytic polynomials in ${L^\infty }(\mu )$ is the set of bounded analytic functions on ${\mathbf {D}}$. A complete characterization of ${P^2}(\mu ) \cap {L^\infty }(\mu )$, the commutant of the cyclic subnormal operator of multiplication by $z$ on ${P^2}(\mu )$, is then obtained.