Analysis and Probability Seminar

Abstract: 

If H is a reproducing kernel Hilbert space and Mult(H) denotes the space of pointwise multipliers of that space, we say that f ∈ H is cyclic if Mult(H) is dense in H. The first important example of cyclic functions are the so called outer functions which emerge from Beurling’s famous theorem about invariant subspaces of the unilateral shift operator on the Hardy space H2. Later, the work of Korenblum, Brown and Shields revealed that in smaller spaces of analytic functions in the disc, cyclicity of an outer function also depends on the size of its zero-set on the boundary. In general, a complete characterization of such functions is lacking, but the area is rich with deep results. Much less is known in the setting of spaces of functions on the unit ball in several complex variables and as is to expect, the situation is quite complicated. For example, there are polynomials without zeros in the ball which are not cyclic in the standard Drury-Arveson space.

The purpose of the talk is to present some recent results in this direction and the material is based on joint work with K.M. Perfekt, S. Richter, C. Sundberg and J. Sunkes.