Analysis Seminar/Third Year Candidacy Requirement: "Cyclicity preserving operators on spaces of analytic functions"

Speaker: Jeet Sampat, Washington University in Saint Louis

Abstract: Cyclic functions, with respect to the shift operator, have been greatly studied for a variety of spaces. A complete characterization of cyclic functions is only known for a handful of spaces, however. Beurling’s theorem for Hardy spaces on the unit disc is one such example. For spaces of analytic functions in several variables, we have very little information about cyclic functions. The most common way of obtaining cyclic functions is via a linear operator between two spaces of functions. More often than not, these operators are some sort of a composition operator.

In a recent paper of Kou and Liu, it was shown that all linear operators on Hardy space of the unit disc are necessarily weighted composition operators. In this talk, we will discuss how we can generalize this result, using similar techniques, to “reasonable” spaces of functions in a much more general set-up. We shall also discuss about some of the problems with completely characterizing these operators on well-known spaces, which are in turn related to some famous open problems about cyclicity. If time permits, we will also talk about a similar problem for functions cyclic with respect to some different operator(s) instead of the shift operator.

Host: Brett Wick