Jointly cyclic polynomials and maximal domains

TL;DR

This paper investigates the conditions under which shift invariant subspaces generated by polynomials are dense in certain topological spaces of holomorphic functions, and characterizes the maximal domain of evaluation functionals.

Contribution

It provides a characterization of when polynomial-generated subspaces are dense and describes the structure of the maximal domain in various function spaces.

Findings

Subspace generated by polynomials is the whole space iff the common zero set has no points with continuous evaluation.

Cyclicity in two variables reduces to a single polynomial's cyclicity.

Maximal domain is an $F__$ set in metrizable spaces, with constructions on the disk.

Abstract

For a (not necessarily locally convex) topological vector space $\mathcal{X}$ of holomorphic functions in one complex variable, we show that the shift invariant subspace generated by a set of polynomials is $\mathcal{X}$ if and only if their common vanishing set contains no point at which the evaluation functional is continuous. For two variables, we show that this problem can be reduced to determining the cyclicity of a single polynomial and obtain partial results for more than two variables. We proceed to examine the maximal domain, i.e., the set of all points for which the evaluation functional is continuous. When $\mathcal{X}$ is metrizable, we show that the maximal domain must be an $F_\sigma$ set, and then construct Hilbert function spaces on the unit disk whose maximal domain is the disk plus an arbitrary subset of the boundary that is both $F_\sigma$ and $G_\delta$.