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Sub-Bergman Hilbert spaces on the unit disk III
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 5 / October 2024
- Published online by Cambridge University Press:
- 22 August 2023, pp. 1520-1537
- Print publication:
- October 2024
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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.
