Sub-Bergman Hilbert spaces in the unit disk, II

For a finite Blaschke product B let TB denote the analytic multiplication operator (also called a Toeplitz operator) on the Bergman space of the unit disk. We show that the defect operators (I−TBTB∗)1/2 and (I−TB∗TB)1/2 both map the Bergman space to the Hardy space and the Hardy space to the Dirichlet space.