Analysis Seminar: "Multiplicity of cyclic functions"

Abstract: In this talk, we will look at 2 questions about cyclic functions in a space X :-

1) Given f and g cyclic in X, such that their product fg lies in X, is fg also cyclic?

2) Given an f in X such that 1/f lies in X, is f cyclic?

A famous result of Borichev and Hedenmalm shows that the answer to question 2 is negative for the Bergman space. Building on the same ideas, we will discuss these questions in the setting of complete Pick spaces. Examples of complete Pick spaces include the Dirichlet space and the Drury-Arveson space. The answers to 1 and 2 are known to be affirmative in both these spaces, but the proofs do not easily generalize to all complete Pick spaces. Using some recent results about the structure of functions in complete Pick spaces, we show that the answer for 1 and 2 is affirmative if we consider multiplier-cyclic functions, i.e. f such that fM is a dense subspace of X, where M is the multiplier algebra of X.

Host: John McCarthy