ON PRESENTATION OF LINEAR OPERATORS COMMUTING WITH DIFFERENTIATION IN SIMPLY-CONNECTED DOMAIN

Abstract

Let   H G 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   s G with property   s G G G   is termed re sidue 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 diffe rentiation. Then it can be reduced to a complex convoluti on 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 func tion generating a kernel.