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

Abstract

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