On generalized Toeplitz and little Hankel operators on Bergman spaces

We find a concrete integral formula for the class of generalized Toeplitz operators in Bergman spaces , , studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an -symbol a such that fails to be bounded in , although is seen to be bounded by using the generalized definition. We also confirm that the generalized definition coincides with the classical one whenever the latter makes sense.