On presentation of linear operators commuting with differentiation in simply-connected domain

Let be a space of analytic functions of one variable in simply-connected domain G of the complex plane. It is known that a linear complex convolution operator is generated by a one-variable analytic function, a multivalued one in general. A known problem when all such functions are single-valued is solved. As it turned out, the solution to the problem is connected with the geometry of G domain. Set with property is termed residue of G domain. A class of simply connected regions whose residue is a connected set is described. Let the linear operator be continuous in function space, analytical in simply-connected domain G, and let it commute with differentiation. Then it can be reduced to a complex convolution operator. It is proved that the function generating such an operator will always be single-valued for regions with a connected residue. When the residue of region G is not connected, there is always a complex convolution operator with a multivalued function generating a kernel.