{"paper":{"title":"On Enflo and narrow operators acting on $L_p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"B. Randrianantoanina, M. Popov, V. Mykhaylyuk","submitted_at":"2012-01-19T12:32:54Z","abstract_excerpt":"The first part of the paper is inspired by a theorem of H. Rosenthal, that if an operator on $L_1[0,1]$ satisfies the assumption that for each measurable set $A \\subseteq [0,1]$ the restriction $T \\bigl|_{L_1(A)}$ is not an isomorphic embedding, then the operator is narrow.\n  (Here $L_1(A) = \\bigl\\{x \\in L_1: \\,\\, {\\rm supp} \\, x \\subseteq A \\bigr\\}$.) This leads to a natural question of finding mildest possible assumptions for operators on a given space $X$, which will imply that the operator is narrow. We find a partial answer to this question for operators on $L_p(0,1)$ with $1<p<2$. Namely"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.4041","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"}