A generalization of Toeplitz operators on the Bergman space

If $\mu$ is a finite measure on the unit disc and $k\ge 0$ is an integer, we study a generalization derived from Englis's work, $T_\mu^{(k)}$, of the traditional Toeplitz operators on the Bergman space $A^2$, which are the case $k=0$. Among other things, we prove that when $\mu\ge 0$, these operators are bounded if and only if $\mu$ is a Carleson measure, and we obtain some estimates for their norms.