Cyclicity preserving operators on spaces of analytic functions in $\mathbb{C}^n$

TL;DR

This paper characterizes shift-cyclicity preserving operators on certain analytic function spaces in several complex variables, showing they are weighted composition operators, and extends classical functional analysis results to multivariable contexts.

Contribution

It establishes that shift-cyclicity preserving operators are weighted composition operators on various multivariable function spaces and generalizes the Gleason-Kahane-Żelazko theorem.

Findings

Operators preserving shift-cyclic functions are weighted composition operators.

The result applies to Hardy, Drury-Arveson, and Dirichlet-type spaces.

A multivariable version of the Gleason-Kahane-Żelazko theorem is proved.

Abstract

For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space $H^p(\mathbb{D}^n) \, (0 < p < \infty)$, the Drury-Arveson space $\mathcal{H}^2_n$, and the Dirichlet-type space $\mathcal{D}_{\alpha} \, (\alpha \in \mathbb{R})$. We focus on the Hardy spaces and show that when $1 \leq p < \infty$, the converse is also true. The techniques used to prove the main result also enable us to prove a version of the Gleason-Kahane-\.Zelazko theorem for partially multiplicative linear functionals on spaces of analytic functions in more than one variable.