Cyclic Vectors and Invariant Subspaces of the Backward Shift Operator in Schwartz Modules

Abstract

Cyclic vectors and proper closed invariant subspaces of the backward shift operator in the Schwartz modules of entire functions of exponential type are described. The results are applied to describe ideals of the algebra of infinitely differentiable functions on a closed or open interval containing \(0\) with Duhamel product as multiplication.