Analysis Seminar: "Norm independence of cyclicity and invertible functions"

Abstract: Fix 0 < q < p < ∞ and a natural number n. Let H^p denote the Hardy space over the unit polydisk D^n. It is easy to check that if a function f in H^p is cyclic in H^p, then it is also cyclic in H^q. However, the converse is not known to be true unless n = 1. It is difficult to determine if the converse is true in general, so we present a few other properties of cyclic functions that relate to it. In particular, we show that if the converse holds in general then it implies that if both f and 1/f lie in H^p, then f (and similarly 1/f) is cyclic in H^p. To end the discussion, we will show that if f lies in H^p, and 1/f lies in H^q for some p and q arbitrary, then f is cyclic in H^r for all r < p.

Host: John McCarthy