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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1607.00113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-01T05:42:58Z","cross_cats_sorted":[],"title_canon_sha256":"75ceebd0f693714e6224b8c2e6fad76e6baae8c41a2f671c3a4c34e9d7149c6a","abstract_canon_sha256":"8ef142d38db598c67e70dfdb958d6422bc59eabaf6d62cdbd3b6b0182e8ea243"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:44.127037Z","signature_b64":"HixP6597ou/3Eb726UmKwgoBxYQb6RXu9D5RqRvtaPJrtqFuE7phswALIsud1fOaLZKQQr9j6n6jGjWaRNAaDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"121b3f150ab11c65e4bc748ae97396dea5ea3c18d6e9fccb818e02cbbcbb6729","last_reissued_at":"2026-05-18T00:33:44.126285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:44.126285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1607.00113","created_at":"2026-05-18T00:33:44.126420+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.00113v1","created_at":"2026-05-18T00:33:44.126420+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.00113","created_at":"2026-05-18T00:33:44.126420+00:00"},{"alias_kind":"pith_short_12","alias_value":"CINT6FIKWEOG","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"CINT6FIKWEOGLZF4","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"CINT6FIK","created_at":"2026-05-18T12:30:09.641336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32","json":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32.json","graph_json":"https://pith.science/api/pith-number/CINT6FIKWEOGLZF4OSFOS44W32/graph.json","events_json":"https://pith.science/api/pith-number/CINT6FIKWEOGLZF4OSFOS44W32/events.json","paper":"https://pith.science/paper/CINT6FIK"},"agent_actions":{"view_html":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32","download_json":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32.json","view_paper":"https://pith.science/paper/CINT6FIK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.00113&json=true","fetch_graph":"https://pith.science/api/pith-number/CINT6FIKWEOGLZF4OSFOS44W32/graph.json","fetch_events":"https://pith.science/api/pith-number/CINT6FIKWEOGLZF4OSFOS44W32/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32/action/storage_attestation","attest_author":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32/action/author_attestation","sign_citation":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32/action/citation_signature","submit_replication":"https://pith.science/pith/CINT6FIKWEOGLZF4OSFOS44W32/action/replication_record"}},"created_at":"2026-05-18T00:33:44.126420+00:00","updated_at":"2026-05-18T00:33:44.126420+00:00"}