CYCLICITY AND INVARIANT SUBSPACES IN DIRICHLET SPACES

O El-Fallah (1) , Y Elmadani (1) , K Kellay (2)

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

Let µ be a positive finite measure on the unit circle and D(µ) the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function f ∈ D(µ) is cyclic if and only if c_µ (Z(f)) = 0, where c µ is the capacity associated with D(µ) 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]

O El-Fallah, Y Elmadani, K Kellay. CYCLICITY AND INVARIANT SUBSPACES IN DIRICHLET SPACES. 2014. ⟨hal-01084268v1⟩