Operators with Dense Images Everywhere1

In this paper, the authors introduce the dense-image operators T as those with a wild behavior near the boundary of a domain G, via certain subsets. The relationship with other kinds of operators with wild behavior is studied, proving that the new concept generalizes the earlier idea of omnipresence, but there is no good relationship with the strongly omnipresent operators. We obtain, among other results, that the following kinds of operators are dense-image: onto linear operators, operators with local dense range satisfying soft conditions, Volterra complex integral operators plus infinite order differential operators, and multiplication operators. In addition, holomorphic self-mappings and entire functions generating dense-image right or left composition operators are completely characterized