Cyclicity and invariant subspaces in the Dirichlet spaces

Omar El-Fallah (1) , Youssef Elmadani (1) , Karim Kellay (2)

1 Département de Mathématiques
2 IMB - Institut de Mathématiques de Bordeaux

Let $\mu$ be a positive finite measure on the unit circle and $\cD (\mu)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f \in \cD (\mu )$ is cyclic if and only if $c_\mu (Z (f))= 0$, where $c_\mu$ is the capacity associated with $\cD (\mu)$ 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.

Domaines: Variables complexes [math.CV]; Analyse fonctionnelle [math.FA]; Analyse classique [math.CA]

Omar El-Fallah, Youssef Elmadani, Karim Kellay. Cyclicity and invariant subspaces in the Dirichlet spaces. 2014. ⟨hal-01084268v2⟩