# PhD Thesis Defense: "Properties of Cyclic Functions"

*Abstract: 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 in order to understand them better. The goal of this thesis is to showcase some important properties of cyclic functions in two different settings.*

*1. Fix 1 \leq p,q < \infty and natural numbers m, n. Let T : H^p(D^n) --> 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.*
2. Let H be a normalized complete Nevanlinna-Pick space, and let f, g \in H such that fg \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.*

*Host: Greg Knese*