Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball

The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX + XB - CX = D related to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.