Cyclicity preserving operators on spaces of analytic functions in $\mathbb{C}^n$
Jeet Sampat
TL;DR
The work addresses the problem of characterizing linear operators that preserve shift-cyclicity of analytic functions on several variables, establishing that such operators must be weighted composition operators in natural spaces like $H^p(\mathbb{D}^n)$, the Drury–Arveson space, and Dirichlet-type spaces. It develops a general framework using properties $P1$–$P3$ and their maximal-domain extensions $Q1$–$Q3$, enabling a representation of functionals $\Lambda$ as point-evaluations after exponential testing, and extends these ideas to a GKŻ-type theorem for spaces of analytic functions. In the Hardy setting, the paper proves a full converse: every bounded weighted composition operator from $H^p(\mathbb{D}^n)$ to $H^q(\mathbb{D}^m)$ preserves cyclicity when $1\le q<\infty$, yielding a robust rigidity result for cyclicity under linear maps. An auxiliary GKŻ-type characterization of partially multiplicative functionals and the maximal-domain perspective provide a unifying viewpoint for analyzing cyclicity and outer/cyclic function distinctions in several complex variables.
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 $\mathcal{D}_α \, (α\in \mathbb{R})$. 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-Żelazko theorem for partially multiplicative linear functionals on spaces of analytic functions in more than one variable.