Sub-Bergman Hilbert spaces on the unit disk III

For a bounded analytic function f on the unit disk D with ?f ?8= 1, we consider the defect operators D-f and D-f of the Toeplitz operators T-f and T-f, respectively, on the weighted Bergman space A(a)(2). The ranges of D-f and D-f, written as H(f) and H(f) and equipped with appropriate inner products, are called sub-Bergman spaces. We prove the following three results in the paper: for -1 < a = 0, the space H(f) has a complete Nevanlinna-Pick kernel if and only if f isa M & ouml;bius map; for a > -1, we have H(f) = H(f) = A(a-1)(2) if and only if the defect operators D-f and D-f are compact; and for a > -1, we have D-f(2) (A(a)(2)) = D-f(2)(A(a)(2)) = A(a-2)(2) if and only if f is a finite Blaschke product. In some sense, our restrictions on a here are best possible.