Some integral transformations with reproducing properties

By elementary considerations, families of integral transformations in certain spaces (e.g., in $L_2(\mathbb K)$, where $\mathbb K$ is the unit disk) are constructed that map the elements of certain subspaces to themselves or to their derivatives, respectively. As a special case, a family of integral transformations is obtained, each of which generates a decomposition of $L_2(\mathbb K)$ into a direct sum. By introducing appropriate new scalar products, these direct sums become orthogonal, and then the corresponding integral transformations become operators of $L_2(\mathbb K)$ into itself that are self-adjoint and positive with respect to the new scalar products. In further special cases, these integral transformations possess bounded and injective extensions that map $L_2(\mathbb K)$ onto certain subspaces of $L_2(\mathbb C)$ defined explicitly. The latter is a consequence of the relationship of the above mappings with the complex Hilbert transformation.