Jointly cyclic polynomials and maximal domains
Mikhail Mironov, Jeet Sampat
TL;DR
The paper develops a unified framework for understanding cyclicity of polynomials in spaces of holomorphic functions by introducing the maximal domain \Omega_{max} and the enveloping domain \Omega_{env}. It shows that \Omega_{max} equals the joint eigenvalue set of the adjoint shift operators and proves sharp connections to joint cyclicity, yielding explicit criteria in one- and two-variable settings and partial results for higher dimensions. It provides topological characterizations of maximal domains (notably that they are F_σ in metrizable spaces) and constructs Hilbert spaces on the disk with maximal domains equal to the disk plus prescribed boundary subsets that are both F_σ and G_δ, advancing the understanding of the geometric structure of cyclicity problems. The results unify several known instances for Hardy and Dirichlet-type spaces and open several avenues for further exploration in higher dimensions and more general function spaces.
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 $F_σ$ 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 $F_σ$ and $G_δ$.