Sub-Bergman Hilbert spaces in the unit disk. II. (Q1403851)

Denote by \(T_B\) the multiplication operator defined by a finite Blaschke product \(B\) on the Bergman space (of the unit disk) \(A^2\). Denote by \(D=\sqrt{I-T^*_B T_B}\) and \(D_*=\sqrt{I-T_B T^*_B}\) the defect operators of \(T_B\) and \(T^*_B\), respectively. The main result of this deep and interesting paper states that the corresponding defect spaces are \(D(A^2)=D_*(A^2)=H^2\) and, moreover, that when restricted to \(H^2\), the corresponding defect spaces \(D(H^2)\) and \(D_*(H^2)\) both coincide with the Dirichlet space. The main tool used in the proofs is the theory of sub-Bergman spaces as developed in the first part of this paper [Indiana Univ. Math. J. 45, 165--176 (1996; Zbl 0863.30051)].