Toeplitz operators in Bergman space induced by radial measures

We study radial Carleson--Bergman measures on the unit disk and the corresponding Toeplitz operators acting in the Bergman space. First, we show that such Toeplitz operators are diagonal in the canonical basis, and we compute their eigenvalue sequences and Berezin transforms in terms of the radial component of the measure. Next, considering the average values of radial measures near the boundary, we give a simple characterization of radial Carleson--Bergman measures. Finally, we prove that the eigenvalue sequences of such Toeplitz operators are Lipschitz continuous with respect to the logarithmic distance on natural numbers. As a consequence, we describe the commutative C*-algebra generated by Toeplitz operators induced by radial Carleson--Bergman measures.