Sub-Bergman Hilbert spaces on the unit disk III

For a bounded analytic function $\phi$ on the unit disk $\text{D}$ with $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ we consider the defect operators $D_{\phi }$ and $D_{\bar{\phi }}$ of the Toeplitz operators $T_{\phi }$ and $T_{\bar{\phi }}$, respectively, on the weighted Bergman space $A_{\alpha }^2$. The ranges of $D_{\phi }$ and $D_{\bar{\phi }}$, written as $H(\phi )$ and $H(\bar{\phi })$ and equipped with appropriate inner products, are called sub-Bergman spaces. We prove the following three results in the paper: for $-1<\alpha \le 0$ the space $H(\phi )$ has a complete Nevanlinna-Pick kernel if and only if $\phi$ is a Möbius map; for $\alpha >-1$ we have $H(\phi )=H(\bar{\phi })=A_{\alpha -1}^2$ if and only if the defect operators $D_{\phi }$ and $D_{\bar{\phi }}$ are compact; and for $\alpha >-1$ we have $D_{\phi }^2(A_{\alpha }^2)=D_{\bar{\phi }}^2(A_{\alpha }^2)=A_{\alpha -2}^2$ if and only if $\phi$ is a finite Blaschke product. In some sense our restrictions on $\alpha$ here are best possible.