Some properties of solutions to the Riccati equations in connection with Bergman spaces

Abstract

We investigate the properties of solutions to operator Riccati equations in connection with Bergman spaces. We characterize some necessary conditions for the solvability of the Riccati equation

$$XAX+XB-CX-D=0$$

on the set $\tau$ of all Toeplitz operators on the Bergman spaces $A_{\alpha }^2\left(\mathbb{𝔻}\right)$. This extends the results of Karaev on Hardy space.