Sub-Bergman Hilbert spaces on the unit disk III

Abstract

Abstract For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$ , we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\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 $-1<\alpha \le 0$ , the space $H(\varphi )$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius 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.