{"paper":{"title":"Rigidity of composition operators on the Hardy space $H^p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Eero Saksman, Hans-Olav Tylli, Jussi Laitila, Pekka J. Nieminen","submitted_at":"2016-07-01T05:42:58Z","abstract_excerpt":"Let $\\phi$ be an analytic map taking the unit disk $\\mathbb{D}$ into itself. We establish that the class of composition operators $f \\mapsto C_\\phi(f) = f \\circ \\phi$ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space $H^p$, for $1\\le p < \\infty$ and $p \\neq 2$. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) $C_\\phi$ is a compact operator $H^p \\to H^p$, (ii) $C_\\phi$ fixes a (linearly isomorphic) copy of $\\ell^p$ in $H^p$, but $C_\\phi$ does not fix any copies of $\\ell^2$ in $H^p$, (iii) $C_\\phi$ fix"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00113","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"}