Shift-cyclicity in analytic function spaces
Jeet Sampat
TL;DR
This survey develops shift-cyclicity for Banach spaces of analytic functions, connecting polynomial-density, bounded point evaluations, and shift operators to the density of polynomial multiples. It synthesizes one- and several-variable Hardy spaces, Dirichlet-type spaces, and complete Pick spaces, highlighting how outerness, factorization, and multiplier algebras govern cyclicity and invariant-subspace structures. A core theme is that shift-cyclicity often reduces to non-vanishing on the domain or its boundary, with nuanced behavior in higher dimensions and on balls or polydisks; complete Pick spaces provide robust Smirnov representations and multiplier-cyclicity tools that unify many cases. The work also links shift-cyclicity to broader topics (e.g., NBBD criteria, capacity theory) and develops a framework for cyclicity-preserving operators, yielding precise, yet largely open, questions about maximal domains and the algebraic structure of cyclic functions in complex spaces.
Abstract
In this survey, we consider Banach spaces of analytic functions in one and several complex variables for which: (i) polynomials are dense, (ii) point-evaluations on the domain are bounded linear functionals, and (iii) the shift operator corresponding to each variable is a bounded linear map. We discuss the problem of determining the shift-cyclic functions in such a space, i.e., functions whose polynomial multiples form a dense subspace. This problem is known to be intimately connected to some deep problems in other areas of mathematics, such as the dilation completeness problem and even the Riemann hypothesis. What makes determining shift-cyclic functions so difficult is that often we need to employ techniques that are specific to the space in consideration. We therefore cover several different function spaces that have frequently appeared in the past such as the Hardy spaces, Dirichlet-type spaces, complete Pick spaces and Bergman spaces. We highlight the similarities and the differences between shift-cyclic functions among these spaces and list some important general properties that shift-cyclic functions in any given analytic function space must share. Throughout this discussion, we also motivate and provide a large list of open problems related to shift-cyclicity.