


Abstract: This paper relates function theory, several complex variables, operator theory and Banach algebras. In the first section a distance estimate is given which shows that if at each point of the boundary of the unit disc D, the cluster set of a bounded analytic function has area zero, then the radial limit function has vanishing mean oscillation. The proof is based on Putnam s theorem on hyponormal operators; it becomes easier for the special class of subnormal operators In the third section we present a proof for this case based on a quantitative version of the HartogsRosenthal theorem from function algebras that is due to H. Alexander. We use Alexander's spectral area estimate to obtain estimates for the BMOAnorms for an analytic function in terms of the area of its image. 
