H-Toeplitz operators on the function spaces

In this paper, we study several classes of H-Toeplitz operators (defined below) on the Hardy space \(H^2\) $H^2$ . In particular, we prove that, for \(\varphi \in L^{\infty }\) $\phi \in L^{\infty }$ , the adjoint of H-Toeplitz operators is hyponormal. Next, we investigate several properties of H-Toeplitz operators on the weighted Bergman spaces. Finally, we give necessary and sufficient conditions for H-Toeplitz operators to be contractive and expansive on the weighted Bergman spaces.