On generalized Toeplitz and little Hankel operators on Bergman spaces

Abstract

We find a concrete integral formula for the class of generalized Toeplitz operators $$T_a$$ in Bergman spaces $$A^p$$ , $$1<p<\infty $$ , studied in an earlier work by the authors. The result is extended to little Hankel operators. We give an example of an $$L^2$$ -symbol a such that $$T_{|a|} $$ fails to be bounded in $$A^2$$ , although $$T_a : A^2 \rightarrow A^2$$ 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.