On the Brown–Shields conjecture for cyclicity in the Dirichlet space

Abstract

Let D be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function f ∈ D to be cyclic, i.e. for { p f : p a polynomial } to be dense in D . This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in D iff it is outer and its zero set (defined appropriately) is of capacity zero.