{"paper":{"title":"Cyclicity and invariant subspaces in the Dirichlet spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Karim Kellay (IMB), Omar El-Fallah, Youssef Elmadani","submitted_at":"2014-11-18T19:40:50Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.4977","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}