{"paper":{"title":"Positive semigroups and abstract Lyapunov equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Sergiy Koshkin","submitted_at":"2014-01-28T01:30:55Z","abstract_excerpt":"We consider abstract equations of the form Ax=-z on a locally convex space, where A generates a positive semigroup and z is a positive element. This is an abstract version of the operator Lyapunov equation A*P+PA=-Q from control theory. It is proved that under suitable assumptions existence of a positive solution implies that -A has a positive inverse, and the generated semigroup is asymptotically stable. We do not require that z is an order unit, or that the space contains any order units. As an application, we generalize Wonham's theorem on the operator Lyapunov equations with detectable rig"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7059","kind":"arxiv","version":1},"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"}