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 \(1 \le p < \infty \), 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.

对于在\（{\ mathbb {C}} ^ n \）中的开放集上定义的，满足某些良好属性的解析函数空间，我们证明了保留移位循环函数的算子必然是加权合成算子。该结果成立的空间示例包括Hardy空间\（H ^ p（{\ mathbb {D}} ^ n）\，（0 <p <\ infty）\），Drury–Arveson空间\（ {\ mathcal {H}} ^ 2_n \）和Dirichlet类型空间\（{\ mathcal {D}} _ {\ alpha} \，（\ alpha \ in {\ mathbb {R}}）\）。我们专注于Hardy空间，并证明当\（1 \ le p <\ infty \），反之亦然。用于证明主要结果的技术还使我们能够证明格里森-卡哈内-谢拉兹科定理的一个版本，该定理用于解析函数在多个变量中的空间上的部分乘线性函数。