The common range of co-analytic Toeplitz operators on the Drury-Arveson space

We characterize the common range of the adjoints of cyclic multiplication operators on the Drury-Arveson space. We show that a function belongs to this common range if and only if its Taylor coefficients satisfy a simple decay condition. To achieve this, we introduce the uniform Smirnov class on the ball and determine its dual space. We show that the dual space of the uniform Smirnov class equals the dual space of the strictly smaller Smirnov class of the Drury-Arveson space, and that this in turn equals the common range of the adjoints of cyclic multiplication operators.

Automated summary

We study the common range of adjoints of cyclic multiplication operators on the Drury-Arveson space, and find that a function belongs to this range if and only if its Taylor coefficients satisfy a simple decay condition. To do this, we introduce the uniform Smirnov class on the ball and determine its dual space. We show that the dual space of the uniform Smirnov class equals the dual space of the smaller Smirnov class of the Drury-Arveson space, which in turn equals the common range of adjoints of cyclic multiplication operators.