# On generalized Toeplitz and little Hankel operators on Bergman spaces

## TL;DR

This paper derives an integral formula for generalized Toeplitz and little Hankel operators on Bergman spaces, extending previous work and clarifying their boundedness properties with concrete examples.

## Contribution

It provides a new integral representation for these operators and extends the theory to include little Hankel operators, connecting generalized and classical definitions.

## Key findings

- Derived a concrete integral formula for generalized Toeplitz operators
- Extended the formula to little Hankel operators
- Provided an example where boundedness differs between classical and generalized definitions

## 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 $T_a : A^2 \to A^2$ 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.