CYCLICITY AND INVARIANT SUBSPACES IN DIRICHLET SPACES
Résumé
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.
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