Hyperholomorphic bergman spaces and bergman operators associated with domains in ℂ²

The work deals with a version of quaternionic analysis adapted in such a way that the set of arising hyperholomorphic functions includes, as a proper subset, all holomorphic mappings. There are considered, for the theory of the corresponding Bergman spaces and Bergman operators, both classic, commonly treated aspects and more specific ones, in particular, conformally invariant or covariant character of certain objects; a description of an algebra generated by the Bergman projection together with a criterion ensuring the Fredholmness of elements of the algebra; relations with the usual holomorphic mappings theory in two complex variables are demonstrated as well.