Toeplitz–Berezin-type symbols and solutions of some operator equations

Abstract

We use Berezin-type symbols on the Dirichlet space $\mathcal{D}({\mathbb{B}}_n)$ of the unit ball to investigate solutions of some Riccati operator equations of the form $XAX+XB-CX-D=0$. In particular, we discuss the solvability of the operator equation $XAX=AX$ in the set of all bounded Toeplitz operators acting on the Dirichlet space $\mathcal{D}({\mathbb{B}}_n)$. Some other related questions, such as invariant subspaces, are also discussed.