CYCLICAL ELEMENTS OF OPERATORS WHICH ARE LEFT-INVERSES TO MULTIPLICATION BY AN INDEPENDENT VARIABLE

Abstract. We study properties of operators which are left-inverses to the operator of multiplication by an independent variable in the space H(G) of functions that are analytic in an arbitrary domain G. This space is endowed with topology of compact convergence. A description of cyclic elements for such operators is obtained. The obtained statements generalize known results in this direction. In the process of studying the structure of invariant subspaces of a fixed linear con-tinuous operator A acting on a space of analytic functions, it is important to have a de-scription of cyclical elements for the operator A, that is, to have necessary and sufficient conditions for completeness, in the given space, of a system of functions (Anr(z))∞n=0, where r(z) is a fixed function from this space. These problems were investigated for a differential operator acting on various spaces of analytic functions by A. F. Leontev and his students. For spaces of analytic functions, the Pomme operator plays an important role and it is close, in a certain sense, to a differential operator. The Pomme problems mentioned above were solved in works of Yu. A. Kazmin, N. I. Nagnibida and other mathematicians. In this paper, we study a description of cyclic elements in spaces of an