Analysis Seminar: “Algebraic properties of cyclic functions”

Speaker: Jeet Sampat, Washington University in Saint Louis

Abstract: In reproducing kernel Hilbert spaces with a complete Pick kernel, functions possess a Smirnov representation. That is, given a function f in the space, there exists multipliers h and g such that f = h/g. Moreover, g can be chosen to be multiplier-cyclic. We use this representation to show some interesting algebraic properties that multiplier-cyclic functions have in every complete Pick space. For instance, it can be shown that if f and g are two cyclic functions such that their product fg lies in the space, then fg has to be cyclic as well. This generalizes a lot of known results for shift-cyclicity in spaces like the Drury-Arveson space and the Dirichlet space, both of which are examples of complete Pick spaces. If given time, we will discuss how multiplier-cyclicity in complete Pick spaces compares to shift-cyclicity in spaces of analytic functions.

Host: John McCarthy