Shift-cyclicity in analytic function spaces

TL;DR

This survey explores shift-cyclicity in various Banach spaces of analytic functions, highlighting its deep mathematical connections and discussing properties, differences, and open problems across multiple function spaces.

Contribution

It provides a comprehensive overview of shift-cyclicity in analytic function spaces, comparing different spaces and outlining open problems in the field.

Findings

Shift-cyclicity relates to deep mathematical problems like the Riemann hypothesis.

Different function spaces exhibit distinct properties of shift-cyclic functions.

The survey identifies open problems and common properties across 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.