The Root Operator on Finite Zero Based Invariant Subspaces of the Weighted Bergman Space

For a nonnegative integer α, we study and compute the root functions ${R_{\alpha}^{I}(z, w) = (1-\overline{w}z)^{2+\alpha}K_{\alpha}^{I}(z, w)}$ of finite zero based invariant subspaces I of the weighted Bergman space ${A_{\alpha}^{2}}$ , where ${K_{\alpha}^{I}}$ is the reproducing kernel of I. Furthermore, we estimate ranks of the corresponding root operators.