  For a bounded analytic function $\varphi$ on the unit disk $\D$ with
$\|\varphi\|_\infty\le1$ we consider the defect operators $D_\varphi$ and
$D_{\overline\varphi}$ of the Toeplitz operators $T_\varphi$ and
$T_{\overline\varphi}$, respectively, on the weighted Bergman space
$A^2_\alpha$. The ranges of $D_\varphi$ and $D_{\overline\varphi}$, written as
$H(\varphi)$ and $H(\overline\varphi)$ and equipped with appropriate inner
products, are called sub-Bergman spaces.
  We prove the following three results in the paper: for $-1<\alpha\le0$ the
space $H(\varphi)$ has a complete Nevanlinna-Pick kernel if and only if
$\varphi$ is a M\"{o}bius map; for $\alpha>-1$ we have
$H(\varphi)=H(\overline\varphi)=A^2_{\alpha-1}$ if and only if the defect
operators $D_\varphi$ and $D_{\overline\varphi}$ are compact; and for
$\alpha>-1$ we have $D^2_\varphi(A^2_\alpha)=
D^2_{\overline\varphi}(A^2_\alpha)=A^2_{\alpha-2}$ if and only if $\varphi$ is
a finite Blaschke product. In some sense our restrictions on $\alpha$ here are
best possible.
