  We study the boundedness and compactness of the generalized Volterra integral
operator on weighted Bergman spaces with doubling weights on the unit disk. A
generalized Toeplitz operator is defined and the boundedness, compactness and
Schatten class of this operator are investigated on the Hilbert weighted
Bergman space. As an application, Schatten class membership of generalized
Volterra integral operators are also characterized. Finally, we also get the
characterizations of Schatten class membership of generalized Toeplitz operator
and generalized Volterra integral operators on the Hardy space $H^2$.
