Sub-Bergman Hilbert spaces on the unit disk III

TL;DR

This paper investigates sub-Bergman spaces defined via defect operators of Toeplitz operators on weighted Bergman spaces, characterizing when these spaces have complete Nevanlinna-Pick kernels, are equal to certain weighted spaces, or relate to finite Blaschke products.

Contribution

It provides new characterizations of sub-Bergman spaces related to the properties of the symbol function and the defect operators, extending understanding of their structure and kernel properties.

Findings

H() has a complete Nevanlinna-Pick kernel iff  is a Möbius map for -1<.

H()=H(ar{})=A^2_{\u00b1} iff defect operators are compact for .

D^2_(A^2_)=A^2_{} iff  is a finite Blaschke product.

Abstract

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.