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

Details On the Brown--Shields conjecture for cyclicity in the Dirichlet space On the Brown--Shields conjecture for cyclicity in the Dirichlet space

2008-09-26

arxiv.org/abs/0809.4557v1

Mathematics

Complex Variables

Scientific paper

Let $\cD$ 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 $f\in\cD$ to be {\em cyclic}, i.e. for $\{pf: p\text{a polynomial}\}$ to be dense in $\cD$. This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in $\cD$ iff it is outer and its zero set (defined appropriately) is of capacity zero.

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