{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:ZISAHUNX7W7257N3Q67LGLNFIB","short_pith_number":"pith:ZISAHUNX","schema_version":"1.0","canonical_sha256":"ca2403d1b7fdbfaefdbb87beb32da540492eec9b87c40848a8a1f184623b89a0","source":{"kind":"arxiv","id":"1706.03570","version":2},"attestation_state":"computed","paper":{"title":"Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Daniel Li (LML), Herv\\'e Queff\\'elec (LPP), Luis Rodr\\'iguez-Piazza","submitted_at":"2017-06-12T11:26:32Z","abstract_excerpt":"We give examples of composition operators $C\\_\\Phi$ on $H^2 (\\D^2)$ showing that the condition $\\|\\Phi \\|\\_\\infty = 1$ is not sufficient for their approximation numbers $a\\_n (C\\_\\Phi)$ to satisfy $\\lim\\_{n \\to \\infty} [a\\_n (C\\_\\Phi) ]^{1/\\sqrt{n}} = 1$, contrary to the $1$-dimensional case. We also give a situation where this implication holds. We make a link with the Monge-Amp\\`ere capacity of the image of $\\Phi$."},"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":"1706.03570","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-06-12T11:26:32Z","cross_cats_sorted":[],"title_canon_sha256":"59211998af1a9e689b3b4404111d84f5445ae4cd07ad311908f24f48827fe23f","abstract_canon_sha256":"03f2e2ea628447b41b0c9f794179d77f819d58dad4533eaa2593e2f58ce20c87"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:10.929568Z","signature_b64":"58qomQgA20+VnxRUANyXJUI8By+lVWNdfNskCFxCl5gWecp5eKUAcmSLovtdSW+/+Luem82MfYeA4GYqMAh9Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca2403d1b7fdbfaefdbb87beb32da540492eec9b87c40848a8a1f184623b89a0","last_reissued_at":"2026-05-18T00:22:10.929016Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:10.929016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Daniel Li (LML), Herv\\'e Queff\\'elec (LPP), Luis Rodr\\'iguez-Piazza","submitted_at":"2017-06-12T11:26:32Z","abstract_excerpt":"We give examples of composition operators $C\\_\\Phi$ on $H^2 (\\D^2)$ showing that the condition $\\|\\Phi \\|\\_\\infty = 1$ is not sufficient for their approximation numbers $a\\_n (C\\_\\Phi)$ to satisfy $\\lim\\_{n \\to \\infty} [a\\_n (C\\_\\Phi) ]^{1/\\sqrt{n}} = 1$, contrary to the $1$-dimensional case. We also give a situation where this implication holds. We make a link with the Monge-Amp\\`ere capacity of the image of $\\Phi$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03570","kind":"arxiv","version":2},"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":"1706.03570","created_at":"2026-05-18T00:22:10.929103+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.03570v2","created_at":"2026-05-18T00:22:10.929103+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03570","created_at":"2026-05-18T00:22:10.929103+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZISAHUNX7W72","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZISAHUNX7W7257N3","created_at":"2026-05-18T12:31:59.375834+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZISAHUNX","created_at":"2026-05-18T12:31:59.375834+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/ZISAHUNX7W7257N3Q67LGLNFIB","json":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB.json","graph_json":"https://pith.science/api/pith-number/ZISAHUNX7W7257N3Q67LGLNFIB/graph.json","events_json":"https://pith.science/api/pith-number/ZISAHUNX7W7257N3Q67LGLNFIB/events.json","paper":"https://pith.science/paper/ZISAHUNX"},"agent_actions":{"view_html":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB","download_json":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB.json","view_paper":"https://pith.science/paper/ZISAHUNX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.03570&json=true","fetch_graph":"https://pith.science/api/pith-number/ZISAHUNX7W7257N3Q67LGLNFIB/graph.json","fetch_events":"https://pith.science/api/pith-number/ZISAHUNX7W7257N3Q67LGLNFIB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB/action/storage_attestation","attest_author":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB/action/author_attestation","sign_citation":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB/action/citation_signature","submit_replication":"https://pith.science/pith/ZISAHUNX7W7257N3Q67LGLNFIB/action/replication_record"}},"created_at":"2026-05-18T00:22:10.929103+00:00","updated_at":"2026-05-18T00:22:10.929103+00:00"}