Invariant subspaces of the generalized backward shift operator and rational functions

Abstract

The paper is devoted to a complete characterization of proper closed invariant subspaces of the generalized backward shift operator (Pommiez operator) in the Fréchet space of all holomorphic functions in a simply connected domain Ω ∋ 0 \Omega \ni 0 in the complex plane. In the case when the function that generates this operator does not have zeros in Ω \Omega , all such subspaces are finite-dimensional. If in addition Ω \Omega coincides with the entire complex plane, then the generalized backward shift operator under consideration is unicellular. If this function has zeros in Ω \Omega , then the above family of invariant subspaces splits into two classes: the first consists of finite-dimensional subspaces, and the second of infinite-dimensional ones.