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

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 ${\mathcal{H}}_n^2$, and the Dirichlet-type space ${\mathcal{D}}_{\alpha }(\alpha \in \mathbb{R})$. We focus on the Hardy spaces and show that when $1\le p<\infty$, 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.

Cited by in corpus (2)

• Shift-cyclicity in analytic function spaces
• Jointly cyclic polynomials and maximal domains