For a natural number $n$ and $1 \leq p < \infty$, consider the Hardy space $H^p(D^n)$ on the unit polydisk. Beurling’s theorem characterizes all shift cyclic functions in $H^p(D^n)$ when $n = 1$. Such a theorem is not known to exist in most other analytic function spaces, even in the one variable case. Therefore, it becomes natural to ask what properties these functions satisfy to understand them better. The goal of this talk is to showcase some important properties of cyclic functions in two different settings.
Fix $1 \leq p,q < \infty$ and natural numbers $m, n$. Let $T : H^p(D^n) \to H^q(D^m)$ be a bounded linear operator. Then $T$ preserves cyclic functions i.e., $Tf$ is cyclic whenever $f$ is, if and only if $T$ is a weighted composition operator.
Let $H$ be a normalized complete Nevanlinna-Pick (NCNP) space, and let $f, g$ be functions in $H$ such that $fg$ also lies in $H$. Then, $f$ and $g$ are multiplier cyclic if and only if $fg$ is multiplier cyclic.
We also extend (1) to a large class of analytic function spaces. Both properties generalize all previously known results of this type.