Dense subspaces of entire functions
The paper under review deals with the problem under what conditions the linear span of the set \(\{T^ kg\}_{k\geq 0}\) is dense in a given Fréchet space of entire functions, where g is a special entire function and T a suitable linear operator. For example, let \(E_ 0\) be the Fréchet space of all entire functions of order less than one and in case of order one of minimal type. Then it is shown that the linear span of \(\{T^ kg\}_{k\geq 0}\) is dense in \(E_ 0\) if \(g\in E_ 0\) is not a polynomial and T is the linear operator defined by \(T[g(z)]=f(t)g(z)\), where f(t) is a power series with positive radius of convergence and f'(0)\(\neq 0.\) The proofs are given by considering the annihilator of the linear span in the dual space and representing the dual space as an algebra of power series. The result of Lemma 3.1 that H(\({\bar {\mathbb{D}}})\) is a principal ideal ring may also be found in the paper of \textit{C. U. Jensen} [J. Pure Appl. Algebra 38, 277-283 (1985; Zbl 0584.30046)] In fact, Jensen shows that H(\({\bar {\mathbb{D}}})\) is even Euclidean.
scientific article; zbMATH DE number 4004587