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arXiv · 1411.4977

Cyclicity and invariant subspaces in the Dirichlet spaces

Omar El-Fallah·Youssef Elmadani·Karim Kellay
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Abstract

Let $\mu$ be a positive finite measure on the unit circle and $\mathcal{D} (\mu)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f \in \mathcal{D} (\mu )$ is cyclic if and only if $c\_\mu (Z (f))= 0$, where $c\_\mu$ is the capacity associated with $\mathcal{D} (\mu)$ and $Z(f)$ is the zero set of $f$. In this paper we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces.

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Omar El-Fallah, Youssef Elmadani, Karim Kellay. 2014-11-18. Cyclicity and invariant subspaces in the Dirichlet spaces. https://arxiv.org/abs/1411.4977

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